# Cumulants asymptotics for the zeros counting measure of real Gaussian
processes

Louis Gass

###### Abstract

We compute the exact asymptotics for the cumulants of linear statistics
associated with the zeros counting measure of a large class of real Gaussian
processes. Precisely, we show that if the underlying covariance function is
regular and square integrable, the cumulants of order higher than two of these
statistics asymptotically vanish. This result implies in particular that the
number of zeros of such processes satisfies a central limit theorem. Our
methods refines the recent approach by T. Letendre and M. Ancona and allows us
to prove a stronger quantitative asymptotics, under weaker hypotheses on the
underlying process. The proof exploits in particular the elegant interplay
between the combinatorial structures of cumulants and factorial moments in
order to simplify the determination of the asymptotics of nodal observables.
The class of processes addressed by our main theorem englobes as motivating
examples random Gaussian trigonometric polynomials, random orthogonal
polynomials and the universal Gaussian process with sinc kernel on the real
line, for which the asymptotics of higher moments of the number of zeros were
so far only conjectured.

00footnotetext: Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes,
France.00footnotetext: This work was supported by the ANR grant UNIRANDOM,
ANR-17-CE40-0008.00footnotetext: Email: louis.gass(at)ens-rennes.fr

###### Contents

  1. 1 Introduction
    1. 1.1 Statement of the main results
    2. 1.2 Outline of the proof
  2. 2 Basics and notations
    1. 2.1 Partition and cumulants
    2. 2.2 Diagonal set and factorial power measure
    3. 2.3 Matrix notations
    4. 2.4 Divided differences
  3. 3 Kac–Rice formula for Gaussian processes
    1. 3.1 Kac density and cumulants of the zeros counting measure
    2. 3.2 Matrix representation of the Kac density and factorization property
    3. 3.3 Decay of the cumulant Kac density
  4. 4 Asymptotics of the cumulants of the zeros counting measure
    1. 4.1 Uniform non degeneracy of the covariance matrix
    2. 4.2 Asymptotics of the cumulants

##  1 Introduction

The study of the number of zeros of smooth Gaussian processes has a long
history and is in particular motivated by the pioneering works of Kac and
Rice, see e.g. [8] for a general introduction to this topic. The asymptotics
for the expectation and the variance of the number of zeros of a stationary
Gaussian process on an interval growing interval $[0,R]$ as $R$ grows to
infinity has been known since [10], where a central limit theorem (CLT) for
the number of zeros is also proved. The variance asymptotics is there
established using the celebrated Kac–Rice method and the CLT is proved using
approximation by an $m$-dependent process.

With similar methods, the variance of the number of zeros of random Gaussian
trigonometric polynomials with large degree has been studied in [14], as well
as the associated CLT. Later on, the machinery of Wiener chaos expansion was
then successfully used in order to compute the variance asymptotics as well as
establishing CLTs for the number of zeros of various models of stochastic
processes, see for instance [7, 5, 12]. Central limit theorems for the number
of real roots of random algebraic polynomials have also been investigated, see
for example [18] and the references therein.

In the recent paper [13], focusing on the asymptotics of the Kac density
rather than on the full integral Kac–Rice formula, the author managed to avoid
some of the technical computations inherent to the use of Kac–Rice method.
This allowed him to get a unifying point of view, make explicit the needed
decorrelation estimates and then deduce the variance asymptotics for the
number of zeros of many models of Gaussian processes. It has been then
conjectured that the same heuristics could be applied to treat the asymptotics
of the higher central moments of the number of zeros of a Gaussian process,
which is the goal of the present paper.

Up to now, very few results about the asymptotics of the higher central
moments are known. The best result so far is the one by T. Letendre and M.
Ancona [1], where it has been proved that the $p$-th central moment, when
properly rescaled, converges towards the $p$-th moment of a Gaussian random
variable, under the very restrictive condition that the covariance function is
in the Schwartz class of fast decreasing functions. This last result then
yields another proof of the CLT for the number of zeros of this particular
class of processes by the method of moments. Note that the stationary sine
process, i.e. the Gaussian process with $\operatorname{sinc}$ covariance
function, which plays a central role in probability theory and mathematical
physics, is ruled out from their framework, due to the slow decay of the
$\operatorname{sinc}$ kernel.

In this paper, we prove the exact asymptotics of the higher central moments of
the number of zeros of large class of Gaussian processes, under the only
hypothesis, apart from regularity, that the covariance function as well as its
derivatives are square integrable. Our results apply in particular for
Gaussian trigonometric and orthogonal polynomials, as well as the stationary
process with $\operatorname{sinc}$ kernel and other Gaussian stationary
processes on the real line with possibly slow decaying kernels. We prove in
fact a more general theorem by computing the exact asymptotics of the
cumulants of linear statistics associated with the zeros counting measure of
the underlying processes. The use of cumulants instead of central moments
simplifies the rather intricate combinatorics involved when estimating higher
order moments via the Kac–Rice method.

Our result in turn implies the convergence of associated moments of any order
and thus a CLT, with an exact rate of convergence. As a corollary, we deduce a
polynomial concentration of any order of the number of zeros, and by a
Borel–Cantelli argument, the almost sure convergence of the number of zeros.
Note that these last facts cannot be deduced from chaos expansion methods.
More generally, in the context of linear statistics, we prove the almost sure
equidistribution of the zeros set at the limit for a large class of smooth
Gaussian processes.

###  1.1 Statement of the main results

####  1.1.1 Cumulants asymptotics and central limit theorems

In the following, all the random variables considered are defined on a common
abstract probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathbb{E}$
will denote the associated expectation. In the sequel, $W$ stands for a
standard Gaussian random variable, i.e. centered with unit variance. We denote
by $\kappa_{p}(Z)$ the $p$-th cumulant of a random variable $Z$, given by the
expression

$\kappa_{p}(Z)=\sum_{\mathcal{I}\in\mathcal{P}_{p}}(-1)^{|I|-1}(|I|-1)!\prod_{I\in\mathcal{I}}\mathbb{E}\left[Z^{|I|}\right],$

where the sum is indexed by the set $\mathcal{P}_{p}$ of all the partitions of
the finite set $\\{1,\ldots,p\\}$. We refer to [19] or Section 2.1.3 below for
more details on the cumulants of a random variable. The following theorem
describes the asymptotics of all the cumulants of the number of zeros of a
Gaussian trigonometric polynomial with independent coefficients.

######  Theorem 1.1.

Let $(a_{k})_{k\geq 0}$ and $(b_{k})_{k\geq 0}$ be two iid sequences of
standard Gaussian variables. Let $Z_{n}$ be the number of zeros on $[0,2\pi]$
of the process
$h_{n}(x):=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}a_{k}\cos(kx)+b_{k}\sin(kx).$ For
$p$ a positive integer, there is an explicit constant $\gamma_{p}$ such that
$\lim_{n\rightarrow+\infty}\frac{\kappa_{p}(Z_{n})}{n}=\gamma_{p}.$ The
constants $\gamma_{1}$ and $\gamma_{2}$ are positive. The above theorem
implies in particular that

$\lim_{n\rightarrow+\infty}\frac{\operatorname{Var}(Z_{n})}{n}=\gamma_{2}\quad\text{and}\quad\forall
p\geq
3,\quad\frac{\kappa_{p}(Z_{n})}{n^{p/2}}=O\left(\frac{1}{n^{\frac{p}{2}-1}}\right).$

Given the expression of the central moments in terms of cumulants and the fact
that the cumulants of a Gaussian random variable are zero for $p\geq 3$, the
asymptotics (1.1.1) imply in fact that for every positive integer $p$,

$\mathbb{E}\left[\left(\frac{Z_{n}-\mathbb{E}[Z_{n}]}{\sqrt{\operatorname{Var}(Z_{n})}}\right)^{p}\right]=\mathbb{E}[W^{p}]+O\left(\frac{1}{\sqrt{n}}\right).$

Note that the exact asymptotics of the cumulants given by Theorem 1.1 is in
nature stronger than the cruder bound given by (1.1.1) and thus the central
moment asymptotics (1.1.1).

As a consequence, it therefore implies a central limit theorem for the number
of zeros, as well as a polynomial concentration to any order of the number of
zeros around its mean.

######  Corollary 1.2.

As $n$ goes to infinity, we have the convergence in distribution
$\frac{Z_{n}-\mathbb{E}[Z_{n}]}{\sqrt{\operatorname{Var}(Z_{n})}}\underset{n\rightarrow+\infty}{\overset{d}{\longrightarrow}}\mathcal{N}(0,1).$
For all $p\geq 2$, there is a constant $C_{p}$ such that
$\mathbb{P}\left(\left|Z_{n}-\mathbb{E}[Z_{n}]\right|\geq
n\varepsilon\right)\leq\frac{C_{p}}{(\sqrt{n}\varepsilon)^{p}}.$

Note that the variance estimate in Equation (1.1.1) and the associated CLT
were first established in [14] by the Kac–Rice method and in [7] by the Wiener
chaos expansion. So far the exact asymptotics of the $p$-th central moment or
cumulants of $Z_{n}$ has never been computed for $p>2$. Theorem 1.1 shows that
it asymptotically behaves like the $p$-th moment of a Gaussian random
variable, which is expected from the already existing central limit theorem
for the random variable $Z_{n}$. The polynomial concentration of the number of
zeros and a Borel–Cantelli argument implies the almost sure convergence

$\lim_{n\rightarrow+\infty}\frac{Z_{n}}{n}=\gamma_{0}\quad\text{a.s},$

a result already known from [4], using a derandomization method. Exponential
concentration has been established in [17] for this particular model but the
proof is of very different nature and strongly use the trigonometric nature of
the random process $h_{n}$. Our proof only uses the fact that the process is
of class $\mathcal{C}^{\infty}$ and is adaptable to many other models.

The error term in (1.1.1) is new and cannot be deduced from the Wiener chaos
expansion. It implies in particular a rate of convergence towards the Gaussian
random variable of order $1/\sqrt{n}$ for the moment metric. It is reminiscent
of the Berry–Essen bound for more classical CLT.

The independence hypothesis above on the Gaussian random coefficients can be
relaxed. Namely, we can extend the previous Theorem 1.1 to the case where the
Gaussian sequences $(a_{k})_{k\geq 0}$ and $(b_{k})_{k\geq 0}$ are independent
and stationary.

######  Theorem 1.3.

Let $(a_{k})_{k\geq 0}$ and $(b_{k})_{k\geq 0}$ be two independent sequences
of standard Gaussian variables, such that
$\mathbb{E}[a_{k}a_{l}]=\mathbb{E}[b_{k}b_{l}]=\rho(k-l).$ We assume that the
spectral measure associated with the correlation function $\rho$ has a
continuous positive density on the torus $\mathbb{T}$. Let $Z_{n}$ be the
number of zeros on $[0,2\pi]$ of the process
$h_{n}(x):=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}a_{k}\cos(kx)+b_{k}\sin(kx).$
Then the conclusion of Theorem 1.1 and its Corollary 1.2 holds.

The expectation of the number of zeros in this model has been studied in [2,
3] and the variance in [13]. The above Theorem 1.3 gives the asymptotics of
every cumulants and therefore, as discussed above in the independent case, it
proves a central limit theorem for the number of zeros, which is a new result
in this dependent framework, as well as concentration around the mean and a
quantification of rate of convergence.

In another direction, one can replace the functions $\cos$ and $\sin$ by more
general functions. A standard framework is then the following model of random
orthogonal polynomials, for which we can give a similar statement.

######  Theorem 1.4.

Let $(a_{k})_{k\geq 0}$ be an iid sequence of standard Gaussian variables. Let
$(P_{k})_{k\geq 0}$ be a sequence of orthogonal polynomials associated with a
measure $\mu$ on the line, and let $[a^{\prime},b^{\prime}]$ be an interval.
We assume that the measure $\mu$ and the interval $[a^{\prime},b^{\prime}]$
satisfies the hypotheses of [12, Thm. 1.1]. Let $Z_{n}$ be the number of zeros
on $[a^{\prime},b^{\prime}]$ of the process
$h_{n}(x):=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}a_{k}P_{k}(x).$ Then the
conclusion of Theorem 1.1 and its Corollary 1.2 holds.

The expectation, the variance and a central limit theorem for this model have
been very recently studied with the method of chaos in [12]. Here again, we
extend this result by determining the asymptotics of higher cumulants and thus
higher moments. As already discussed after the previous statements above, from
the cumulants asymptotics established in Theorem 1.4, we can also deduce
concentration around the expected number of zeros and as well as a rate of
convergence in the associated CLT, these two last results cannot be obtained
with the Wiener chaos method.

At last, we extend known results about the number of zeros of a stationary
Gaussian process on a growing interval, establishing in particular a CLT under
the sole square integrability of the associated correlation function and its
derivatives.

######  Theorem 1.5.

Let $f$ be a stationary Gaussian process with $\mathcal{C}^{\infty}$ paths and
covariance function $r$. For $R>0$ we define $Z_{R}$ to be the number of zeros
on $[0,R]$ of the process $f$. • If the covariance function $r$ and its
derivatives are in $L^{2}(\mathbb{R})$, then for all $p\geq 2$
$\lim_{R\rightarrow+\infty}\mathbb{E}\left[\left(\frac{Z_{R}-\mathbb{E}[Z_{R}]}{\sqrt{\operatorname{Var}(Z_{R})}}\right)^{p}\right]=\mathbb{E}[W^{p}].$
• If the covariance function $r$ and its derivatives are in
$L^{q}(\mathbb{R})$ for all $q\geq 1$, then for $p$ a positive integer, there
is an explicit constant $\gamma_{p}$ such that
$\lim_{R\rightarrow+\infty}\frac{\kappa_{p}(Z_{R})}{n}=\gamma_{p}.$

As mentioned above, the CLT which is obtained from the above moments
asymptotics by the method of moments is already known in the particular case
of stationary Gaussian processes with covariance function belonging to the
Schwartz class, see [1]. Here the assumption on the decay of the correlation
function is greatly relaxed and we only need to assume the square
integrability of the covariance kernel as well as its derivatives.

As a particular and representative case, Theorem 1.5 englobes the example of
the stationary Gaussian process $f$ with $\operatorname{sinc}$ kernel, which
is a completely new result. This process plays a central role in the study of
determinantal point processes, and appears as the limit of the local
statistics of various random models, from eigenvalues of random matrices to
random integer partitions. For this particular process, the asymptotic of the
expectation and the variance of $Z_{R}$, as well as a CLT were known since the
pioneering works of [10] and the references therein. But so far, the exact
asymptotics of higher central moments or cumulants of $Z_{R}$ remained
unknown.

Under the stronger hypothesis that the covariance function and its derivatives
are in $L^{p}$ for all $p>1$, we deduce a polynomial concentration around the
mean to any order for the number of zeros, which appears to be a new result.
We also deduce the exact asymptotic of the cumulants of any order. This
integrability hypothesis in particular holds true for processes whose
covariance functions $r$ and their derivatives satisfy the bound

$\forall x\in\mathbb{R},\quad r^{(u)}(x)\leq\frac{C_{u}}{1+|x|},$

which is the case for a stationary Gaussian process with $\operatorname{sinc}$
covariance function.

####  1.1.2 A more general and unifying statement

In fact Theorems 1.1, 1.3, 1.4 and 1.5 are all corollaries of a single, more
general statement given below. In order to state it, we need to introduce
first a few notations that will be used for the rest of the paper.

Let $U$ be a non-empty open interval of the real line $\mathbb{R}$ or of the
one-dimensional torus $\mathbb{T}$, endowed with their canonical distance
$|\,.\,|$. Let $n\in\mathbb{R}_{+}^{*}\cup\\{+\infty\\}$. If $n$ is finite
then $nU$ is a non-empty open subset of $\mathbb{R}$ or of the one-dimensional
torus $n\mathbb{T}$ of length $n$. For $n=+\infty$ we use the convention
$(+\infty)U=\mathbb{R}$. This setting allows us to give a unified exposition
for processes defined on the torus (e.g. random trigonometric polynomials) and
on the real line (e.g. the sinc process).

Let $\mathfrak{N}$ be an unbounded subset of $\mathbb{R}_{+}^{*}$ and
$\overline{\mathfrak{N}}=\mathfrak{N}\sqcup\\{+\infty\\}$. For each
$n\in\overline{\mathfrak{N}}$, we consider a centered Gaussian process $f_{n}$
defined on $nU$, and we assume that the process $f_{\infty}$ is a non
degenerate stationary process on $\mathbb{R}$. Note that for
$n\in\mathfrak{N}$ the process $h_{n}=f_{n}(n\,.\,)$ is a Gaussian process on
$U$. For $n\in\overline{\mathfrak{N}}$ and $s,t\in nU$ we define the
covariance function

$r_{n}(s,t)=\mathbb{E}[f_{n}(s)f_{n}(t)]\quad\text{and}\quad
r_{\infty}(s-t)=r_{\infty}(s,t).$

If the process $f_{n}$ is of class $\mathcal{C}^{k}(U)$ for $k\geq 0$ then the
covariance function $r_{n}$ is also of class $\mathcal{C}^{k}$ in each
coordinates, and one has for $u,v\leq k$ and $x,y\in nU$

$r_{n}^{(u,v)}(x,y)=\mathbb{E}[f_{n}^{(u)}(x)f_{n}^{(v)}(y)].$

For $n\in\mathfrak{N}$ we define

$Z_{n}=\left\\{x\in U\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
f_{n}(nx)=0\right\\}\quad\text{and}\quad\nu_{n}:=\sum_{x\in Z_{n}}\delta_{x},$

the random counting measure on $Z_{n}$. Note that
$(\nu_{n})_{n\in\mathfrak{N}}$ is a family of measures on $U$. Assume for now
(it will be a consequence of Bulinskaya Lemma) that for each
$n\in\mathfrak{N}$ the set $Z_{n}$ is almost surely locally finite. For a
bounded function $\phi:U\rightarrow\mathbb{R}$, with compact support in $U$,
we define the bracket

$\langle\nu_{n},\phi\rangle=\sum_{x\in Z_{n}}\phi(x).$

For instance, if $U=\mathbb{R}$ and $\phi=\mathds{1}_{[0,1]}$ then

$\langle\nu_{n},\phi\rangle=\mathrm{Card}\left\\{x\in[0,n]\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ f_{n}(x)=0\right\\}.$

Note the Kac–Rice formula implies that in expectation, the counting measure
has a density with respect to the Lebesgue measure. It allows us to extend the
domain of definition of tests functions $\phi$ to the space

$L^{(\infty)}(U)=\bigcup_{1\leq p<+\infty}\\!\\!\\!L^{p}(U).$

For $q\geq 1$ we define the two following hypotheses.

  * $\bullet$

$H_{1}(q)\,:$ The sequence of processes
$(f_{n})_{n\in\overline{\mathfrak{N}}}$ is of class $\mathcal{C}^{q}(U)$, and
there is a uniformly continuous function $\psi$ on $U$, bounded below and
above by positive constants, such that for $u,v\leq q$, the following
convergence holds uniformly for $x\in U$ and locally uniformly for
$s,t\in\mathbb{R}$,

$\lim_{n\rightarrow+\infty}r_{n}^{(u,v)}(nx+s,nx+t)=\psi(x)r_{\infty}^{(u,v)}(s,t).$

  * $\bullet$

$H_{2}(q)\,:$ For $u,v\leq q$, there is a function $g$ bounded and going to
zero near infinity, such that for $n\in\overline{\mathfrak{N}}$ and $s,t\in
nU$,

$|r_{n}^{(u,v)}(s,t)|\leq g(|s-t|),$

and for some positive constant $\varepsilon$ the function $g_{\varepsilon}$ is
in $L^{2}(\mathbb{R})$, where

$g_{\varepsilon}:x\;\mapsto\;\sup_{|u|\leq\varepsilon}g(x+u).$

######  Theorem 1.6.

Let $p\geq 2$ and $q=2p-1$. We assume that the sequence of processes
$(f_{n})_{n\in\overline{\mathfrak{N}}}$ satisfies hypotheses $H_{1}(q)$ and
$H_{2}(q)$ defined above. Then for every function $\phi\in L^{(\infty)}(U)$,
$\lim_{n\rightarrow+\infty}\mathbb{E}\left[\left(\frac{\langle\nu_{n},\phi\rangle-\mathbb{E}\left[\langle\nu_{n},\phi\rangle\right]}{\sqrt{\operatorname{Var}\left(\langle\nu_{n},\phi\rangle\right)}}\right)^{p}\right]=\mathbb{E}[W^{p}].$
Assume moreover that $g_{\varepsilon}\in L^{\frac{p}{p-1}}(\mathbb{R})$. Then
there is an explicit constant $\gamma_{p}$ depending only on the process
$f_{\infty}$, such that
$\lim_{n\rightarrow+\infty}\frac{\kappa_{p}(\langle\nu,\phi\rangle)}{n}=\gamma_{p}\,.\,\left(\int_{U}\phi(x)^{p}\mathrm{d}x\right).$

The assumption $H_{1}(q)$ characterizes the convergence of the family of
processes $(f_{n})_{n\in\mathfrak{N}}$ towards a limit stationary process in
$\mathcal{C}^{q}$ norm. This hypothesis is natural and arises in many models.
For instance the covariance function of random trigonometric polynomials
converges towards the $\operatorname{sinc}$ function. The regularity of the
process $f_{n}$ ensures the well-definiteness of the $p$-th moment. The
convergence towards a non-degenerate stationary process ensures the uniform
non-degeneracy on the process $f_{n}$, as well as the explicit asymptotics for
the cumulants.

The assumption $H_{2}(q)$ is much weaker than the one present in [1], where
the authors require a function $g$ that decrease like $x^{-8p}$. Here we show
that the asymptotics of higher moments is independent of the rate of decay of
the covariance function, and must only satisfy some uniform square
integrability condition.

Let us briefly now show that the unifying Theorem 1.6 indeed implies the
collection of theorems of the previous subsection. First, Theorem 1.1 and 1.3
are a consequence of Theorem 1.6, by setting $U=\mathbb{T}$,
$\mathfrak{N}=\mathbb{N}^{*}$, $\phi=\mathds{1}_{\mathbb{T}}$ and

$f_{n}(x)=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}a_{k}\cos\left(\frac{kx}{n}\right)+b_{k}\sin\left(\frac{kx}{n}\right).$

Let $\psi$ be the spectral density of the correlation function $\rho$ of the
stationary Gaussian sequences $(a_{k})_{k\geq 0}$ and $(b_{k})_{k\geq 0}$,
which is assumed to be continuous and positive on $\mathbb{T}$. Assumptions
$H_{1}(q)$ and $H_{2}(q)$ are proved for all $q>0$ for this model in the paper
[13] with limit process having $\operatorname{sinc}$ covariance function, and

$g=\frac{C_{\alpha}}{1+|x|^{\alpha}},$

where the exponent $\alpha$ can be taken in $]1/2,1[$. Note that Theorem 1.1
is a particular case of Theorem 1.3 with $\psi=1$ (in that case, one can take
$\alpha=1$ above).

Similarly, Theorem 1.4 is an consequence of Theorem 1.6. Let $\mu$ be a
measure with compact support on the real line. We set $U$ a subinterval of
$\mathbb{R}$ such that $\mu$ has a positive continuous density on
$\overline{U}$. It is proved in [12] under mild assumption on the measure
$\mu$ that for the model of random orthogonal polynomials with respect to the
measure $\mu$, the assumption $H_{2}(q)$ holds true for all $q>0$ and

$g(x)=\frac{C}{1+|x|}.$

Also, a slight modification of assumption $H_{1}(q)$ holds true for all $q>0$
with limit process having $\operatorname{sinc}$ covariance function, and
$\psi$ the inverse of the density of the measure $\mu$. This minor difference
involves the equilibrium measure of the support of the measure $\mu$, through
some rescaling of the ambient space. The proof of Theorem 1.6 adapts verbatim
to this setting. Note that if $\operatorname{supp}\mu=[0,1]$, then after a
change of variable, the equilibrium measure is simply the Lebesgue measure on
the torus $\mathbb{T}$ and hypothesis $H_{1}(q)$ then exactly holds true.

At last, Theorem 1.5 is again a consequence of Theorem 1.6 with
$U=\mathbb{R}$, $\mathfrak{N}=\mathbb{R}_{+}^{*}$, $f_{n}=f$, and test
function $\phi=\mathds{1}_{[0,1]}$.

####  1.1.3 Asymptotics for the linear statistics

Let $\nu_{\infty}$ denotes the Lebesgue measure on the interval $U$. Theorem
1.6 implies a strong law of large number and a central limit theorem for the
sequence of random measure $(\nu_{n})_{n\in\mathfrak{N}}$. The two following
Corollaries 1.7 and 1.8 extend the results of [1, Sec. 1.4] to our framework,
and we refer to this paper for a more thorough discussion.

######  Corollary 1.7 (Law of large numbers).

Assume that the hypotheses $H_{1}(q)$ and $H_{2}(q)$ are satisfied for all
$q\geq 1$, and either $\mathfrak{N}=\mathbb{N}^{*}$, or
$\mathfrak{N}=\mathbb{R}_{+}^{*}\;$ and for $n\in\mathbb{R}_{+}^{*}$,
$f_{n}=f_{\infty}$. Then we have the following almost-sure convergence for the
vague topology
$\lim_{n\rightarrow+\infty}\frac{1}{n}\nu_{n}=\gamma_{1}\,\nu_{\infty}\quad\mathrm{a.s.}\quad.$

Corollary 1.7 shows that zeros of the of the process $f_{n}(n\,.\,)$ tend to
be equidistributed on the set $U$ as $n$ goes to $+\infty$. When
$\mathfrak{N}=\mathbb{N}^{*}$, the proof follows from an application of the
Borel–Cantelli Lemma. When $\mathfrak{N}=\mathbb{R}_{+}^{*}\;$ and
$\;\;\forall n\in\mathbb{R}_{+}^{*},\;\;f_{n}=f_{\infty}$, we can apply the
Borel–Cantelli Lemma to prove the almost sure convergence on a polynomial
subsequence. It is then a standard fact that the monotonicity of $Z_{n}$
ensures the almost sure convergence of the whole sequence.

######  Corollary 1.8 (Central limit theorem).

Assume that the hypotheses $H_{1}(q)$ and $H_{2}(q)$ are satisfied for all
$q\geq 1$. Then we have the following convergence in distribution
$\forall\phi\in
C_{c}(U),\quad\frac{\sqrt{n}}{\gamma_{2}}\left\langle\left(\frac{1}{n}\nu_{n}-\gamma_{1}\nu_{\infty}\right),\phi\right\rangle\underset{n\rightarrow+\infty}{\sim}\mathcal{N}\left(0,\|\phi\|_{2}^{2}\right).$

Corollary 1.8 implies that the fluctuations around the mean of the counting
measure $\nu_{n}$ is comparable to a Gaussian white noise.

###  1.2 Outline of the proof

Before giving a complete and detailed proof of Theorem 1.6, let us briefly
expose its main ingredients and arguments. It mainly relies on a careful
analysis of the Kac–Rice formula, which asserts that for a test function
$\phi$,

$\mathbb{E}[\langle\nu_{n},\phi\rangle^{p}]=\int_{(nU)^{p}}\left(\prod_{i=1}^{p}\phi\left(\frac{x_{i}}{n}\right)\right)\rho_{p,n}(x_{1},\ldots,x_{p})\mathrm{d}x_{1}\ldots\mathrm{d}x_{p}\;+\;\text{extra
terms},$

where

$\rho_{p,n}(x_{1},\ldots,x_{p}):=T_{n}(x_{1},\ldots,x_{p})\mathbb{E}\left[\prod_{i=1}^{p}|f_{n}^{\prime}(x_{i})|\,\middle|\,f_{n}(x_{1})=\ldots=f_{n}(x_{p})=0\right],$

and $T_{n}(x_{1},\ldots x_{p})$ is the density at zero of the Gaussian vector
$(f_{n}(x_{1}),\ldots,f_{n}(x_{p}))$. The extra terms appearing in Equation
(1.2) are of combinatorial nature and can be treated the exact same way as the
first term, so we will omit them in the following heuristics. The function
$\rho_{p,n}$ is called the Kac density of order $p$ associated with the
process $f_{n}$. Observe that the function $\rho_{p,n}$ is ill defined when
two of its arguments collapse. This issue is solved by using the technique of
divided differences, that appeared in [11] and was developed in [1]. Let us
give an example with $p=2$. The idea is to replace in (1.2) the quantity

$f_{n}(x_{1})=f_{n}(x_{2})=0\quad\quad\text{by}\quad\quad
f_{n}(x_{1})=\frac{f_{n}(x_{2})-f_{n}(x_{1})}{x_{2}-x_{1}}=0.$

If the variables $x_{1}$ and $x_{2}$ collapse, the second expression becomes
$f_{n}(x_{1})=f_{n}^{\prime}(x_{1})=0$. The regularity of the process $f_{n}$
given by the assumption $H_{1}(q)$ and the non-degeneracy of the limit process
$f_{\infty}$ ensures that the Gaussian vector $(f_{n}(x),f_{n}^{\prime}(x))$
is non-degenerate and gives an alternative non-singular expression of the
function $\rho_{2,n}$ near the diagonal. For higher integers $p$, the
reasoning is the same. For each partition $\mathcal{I}$ of the set
$\\{1,\ldots,p\\}$, we will thus give an alternative and non-singular
expression of the density $\rho_{p,n}$, that extends by continuity on points
$(x_{1},\ldots,x_{p})$ such that $x_{i}$ and $x_{j}$ are equals if $i$ and $j$
belong to the same cell of the partition $\mathcal{I}$. This procedure is
explained in Section 2.4.

Now we develop the expression of the cumulant of order $p$ as a function of
the moments in order to get

$\displaystyle\kappa_{p}(\langle\nu_{n},\phi\rangle)$
$\displaystyle=\sum_{\mathcal{J}}(|\mathcal{J}|-1)!(-1)^{|\mathcal{J}|-1}\prod_{J\in\mathcal{J}}\mathbb{E}\left[\langle\nu_{n},\phi\rangle^{|J|}\right]$
$\displaystyle=\int_{(nU)^{p}}\prod_{i=1}^{p}\phi\left(\frac{x_{i}}{n}\right)F_{p,n}(x_{1}\ldots,x_{p})\mathrm{d}x_{1}\ldots\mathrm{d}x_{p},+\;\text{extra
terms},$ (9)

where the sum indexed by $\mathcal{J}$ runs over all the partitions of the set
$\\{1,\ldots,p\\}$, and with

$F_{p,n}(x_{1}\ldots,x_{p})=\sum_{\mathcal{J}}(|\mathcal{J}|-1)!(-1)^{|\mathcal{J}|-1}\prod_{J\in\mathcal{J}}\rho_{|J|,n}(\underline{x}_{J}).$

Now let $\mathcal{I}$ be a partition of $\\{1,\ldots,p\\}$, and assume that
for $i$ and $j$ belonging to two different cells of the partition
$\mathcal{I}$, the variable $x_{i}$ and $x_{j}$ are far from each other. Then
the decay hypothesis $H_{2}(q)$ implies that the Gaussian random variable
$f_{n}(x_{i})$ and $f_{n}(x_{j})$ are almost independent, and from the
definition of the Kac density $\rho_{p,n}$ we deduce that for
$A\in\subset\\{1,\ldots,p\\}$,

$\rho_{|A|,n}((x_{a})_{a\in A})\simeq\prod_{I\in\mathcal{I}}\rho_{|A\cap
I|,n}((x_{a})_{a\in A\cap I}),$

Note that the function $\rho_{p,n}$ depends on $f_{n}$ only through the
covariance matrix of the vector
$(f(x_{1}),\ldots,f(x_{p}),f^{\prime}(x_{1}),\ldots,f^{\prime}(x_{p}))$. This
matrix representation formula allows us to give a precise error term in (1.2),
proportional to the square of the magnitude of $r_{n}^{(u,v)}(x_{i},x_{j})$,
where $i$ and $j$ belong to different cells of the partition $\mathcal{I}$. We
refer to Section 2.3 for matrix notations and to Section 3.2 for the matrix
representation of the Kac Density.

The combinatoric properties of cumulants and (1.2) imply that

$F_{p,n}(x_{1}\ldots,x_{p})\simeq 0,$

as soon as the variables $(x_{i})_{1\leq i\leq p}$ are clustered with respect
to some partition $\mathcal{I}$ with at least two cells. A refinement of
Taylor expansion using graph theoretic arguments (see Section 3.3), gives a
much more precise error in (1.2) than the approach taken in [1]. We then show
that far from the diagonal $(x,\ldots,x)$ the function $F_{p,n}$ is small and
will have sufficiently nice integrability properties on $(nU)^{p}$ in order to
show in (9) that for $p\geq 3$,

$\lim_{n\rightarrow+\infty}\frac{1}{n^{p/2}}\int_{(nU)^{p}}\prod_{i=1}^{p}\phi\left(\frac{x_{i}}{n}\right)F_{p,n}(x_{1}\ldots,x_{p})\mathrm{d}x_{1}\ldots\mathrm{d}x_{p}=0.$

Given the link between cumulants and central moments, this fact leads to the
convergence of the central moment of order $p$ to the central moment of a
Gaussian random variable. If moreover, the function $g_{\varepsilon}$ is in
$L^{\frac{p}{p-1}}(\mathbb{R})$ then the function
$F_{p,n}(0,x_{2},\ldots,x_{p})$ is integrable on $(nU)$, uniformly for
$n\in\overline{\mathfrak{N}}$. This fact leads to the exact asymptotics of the
$p$-th cumulant of the random variable $\langle\nu_{n},\phi\rangle$.

Despite its apparent simplicity, the detailed proof is quite technical and the
diversity of arguments used justifies the following section, which introduces
several notions and associated notations for the rest of the paper. In
particular, the notion of partition of a finite set plays a central role in
this article. From a combinatoric point of view, it appears in the Kac–Rice
formula when expressing moments of the factorial power counting measure in
terms of moments of the usual power measure, but also from the interpretation
of cumulants in the context of Möebius inversion in the lattice of partition.
The interplay between these last two combinatoric facts leads to an elegant
expression of the cumulants of the zeros counting measure (given by
Proposition 3.5), and simplifies the approach taken by the authors in [1],
where they computed directly the asymptotics of central moments.

A novelty of this paper is also the intensive use of the matrix representation
of the Kac density which allows us to dissociate the probabilistic setting,
and facts concerning pure matrix analysis. We believe that this approach,
already taken by the author in [13] to treat the asymptotic of the variance,
greatly simplifies the exposition of proofs using Kac–Rice formula.

##  2 Basics and notations

We define a few notations that will be of use and simplify the exposition. In
following, $A$ is a finite set. The letter $a,b,\ldots$ denote elements of
$A$. The letters $B,C,\ldots$ denote subset of $A$. The letters
$\mathcal{I},\mathcal{J},\ldots$ denote subsets of the power set of $A$. The
letters $\mathsf{C}$, $\mathsf{D}$, $\ldots$ denote multisets whose elements
lie in $A$.

###  2.1 Partition and cumulants

####  2.1.1 Set theory

We denote $|A|$ the cardinal of the set $A$ and $\mathcal{P}(A)$ the power set
of $A$. For a set $E$, we define $E^{A}$ the product of $|A|$ copies of $E$. A
generic element of $E^{A}$ is denoted

$\underline{x}_{A}=(x_{a})_{a\in A}$

to avoid any confusion when elements of $A$ are also sets. For a function
$f:E\mapsto\mathbb{R}$ and $\underline{x}_{A}\in E^{A}$ we write

$f(\underline{x}_{A})=(f(x_{a}))_{a\in A}.$

Let $\underline{\phi}_{A}=(\phi_{a})_{a\in A}$ be functions from $E$ to
$\mathbb{R}$. We define

$\underline{\phi}_{A}^{\otimes}:\underline{x}_{A}\mapsto\prod_{a\in
A}\phi(x_{a}).$

At last, we denote

$2A=\\{1,2\\}\times A$

The set $2A$ should be seen as the disjoint union of $A$ and a copy of itself.
For an element $\underline{x}_{A}\in E^{A}$ we denote $\underline{x}_{A,A}$
the element $(\underline{x}_{A},\underline{x}_{A})\in E^{2A}$.

####  2.1.2 Multisets and partial derivatives

A multiset is a set with possible repetition of its element. Alternatively, it
is an unordered sequence of elements. We write them with double braces
$\\{\\!\\!\\{\;\\}\\!\\!\\}$ to distinguish them from usual sets. The
collection of multisets with elements in the set $A$ is denoted
$\mathcal{P}^{m}(A)$. For instance, if $A=\\{1,2\\}$ then

$\mathcal{P}^{m}(A)=\\{\text{\O},\\{\\!\\!\\{1\\}\\!\\!\\},\\{\\!\\!\\{2\\}\\!\\!\\},\\{\\!\\!\\{1,1\\}\\!\\!\\},\\{\\!\\!\\{1,2\\}\\!\\!\\},\\{\\!\\!\\{2,2\\}\\!\\!\\},\\{\\!\\!\\{1,1,1\\}\\!\\!\\},\\{\\!\\!\\{1,1,2\\}\\!\\!\\},\ldots\\}.$

In this paper, the motivation for the introduction multiset arises from the
symmetry of partial derivatives. If $(a_{1},\ldots,a_{n})$ is a sequence of
elements of the set $A$ and $f:\mathbb{R}^{A}\rightarrow\mathbb{R}$ is an
infinitely differentiable function, then for any permutation $\sigma$ of
$\\{1,\ldots,n\\}$,

$\partial_{a_{1}},\ldots\partial_{a_{n}}f=\partial_{a_{\sigma(1)}},\ldots\partial_{a_{\sigma(n)}}f,$

an we can define the quantity $\partial_{\mathsf{C}}f$ for a multiset
$\mathsf{C}$ in $\mathcal{P}^{m}(A)$.

The cardinal of a multiset $\mathsf{C}$ is the number of element with
repetition in this multiset and is denoted $|\mathsf{C}|$. For instance,
$|\\{\\!\\!\\{1,1,2\\}\\!\\!\\}|=3$. We define a few operations on the
multisets. Let $\mathsf{C}$ and $\mathsf{D}$ be multiset with elements in $A$.
We define the disjoint union $\mathsf{C}\sqcup\mathsf{D}$ as the reunion of
the collections of elements in $\mathsf{C}$ and $\mathsf{D}$. For instance

$\\{\\!\\!\\{1,1,2\\}\\!\\!\\}\sqcup\\{\\!\\!\\{1,2,2,3\\}\\!\\!\\}=\\{\\!\\!\\{1,1,1,2,2,2,3\\}\\!\\!\\}.$

If $B$ is a subset of $A$, we define

$\mathsf{C}_{B}=\\{\\!\\!\\{c\in\mathsf{C}\;|\;c\in B\\}\\!\\!\\},$

which is an element of $\mathcal{P}^{m}(B)$. For instance, if
$\mathsf{C}=\\{\\!\\!\\{1,1,2,3,3\\}\\!\\!\\}$ and $B=\\{1,2\\}$ then
$\mathsf{C}_{B}=\\{\\!\\!\\{1,1,2\\}\\!\\!\\}$.

####  2.1.3 The lattice of partitions and cumulants

We define $\mathcal{P}_{A}$ the set of partitions of $A$. The partition of $A$
into singletons will be denoted $\overline{A}$. In the following, $B$ is a
subset of $A$ and $\mathcal{I}$ is a partition of $A$. For $a\in A$ we denote
$[a]_{\mathcal{I}}$ the cell of $\mathcal{I}$ in which the element $a$
belongs, and $\mathcal{I}_{B}$ the partition of $B$ induced by the partition
$\mathcal{I}$ of $A$. For instance, if
$\mathcal{I}=\\{\\{1,2\\},\\{3,4\\},\\{5\\}\\}$, $a=1$ and $B=\\{1,2,3\\}$
then

$[a]_{\mathcal{I}}=\\{1,2\\}\quad\text{and}\quad\mathcal{I}_{B}=\\{\\{1,2\\},\\{3\\}\\}.$

The partition $\mathcal{I}$ induce a partition on the set $2A$ via the
relation

$\left\\{2I\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
I\in\mathcal{I}\right\\},$

and we will still denote $\mathcal{I}$ this partition.

The set $\mathcal{P}_{A}$ has a natural structure of poset (partially ordered
set). Given $\mathcal{I}$ and $\mathcal{J}$ two partition of $A$, we say that
$\mathcal{I}$ is finer than $\mathcal{J}$ (or that $\mathcal{J}$ is coarser
than $\mathcal{I}$) and we denote it $\mathcal{I}\preceq\mathcal{J}$ (or
$\mathcal{J}\succeq\mathcal{I}$), if

$\forall I\in\mathcal{I},\;\exists J\in\mathcal{J}\;\text{such that}\;I\subset
J.$

We then have

$\mathcal{I}_{J}=\left\\{I\in\mathcal{I}\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ I\subset J\right\\}\quad\text{and}\quad
J=\bigsqcup_{I\in\mathcal{I}_{J}}I.$

Note that there is a one-to-one correspondence between the set of partitions
of $A$ coarser than a partition $\mathcal{I}$, and the set of partition of
$\mathcal{I}$, given by the application

$\mathcal{J}\mapsto\left\\{\mathcal{I}_{J}\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ J\in\mathcal{J}\right\\}.$

Following this observation, we denote $[I]_{\mathcal{J}}$ the cell of
$\mathcal{J}$ in which the set $\mathcal{I}$ is included. For instance, if
$\mathcal{I}=\\{\\{1\\},\\{2,3\\},\\{4\\},\\{5\\}\\}$ and
$\mathcal{J}=\\{\\{1,2,3\\}\\{4,5\\}\\}$ then $\mathcal{I}\preceq\mathcal{J}$
and

$[\\{2,3\\}]_{\mathcal{J}}=\\{1,2,3\\}\quad\text{and}\quad\mathcal{I}_{\\{4,5\\}}=\\{\\{4\\},\\{5\\}\\}.$

Note that two partitions $\mathcal{I}$ and $\mathcal{J}$ have a greatest lower
bound and a least upper bound for this partial order, which turns
$(\mathcal{P}_{A},\preceq)$ into a finite lattice. Let $(m_{B})_{B\subset A}$
and $(\kappa_{B})_{B\subset A}$ two families of numbers. In our case of
interest, the Möebius inversion on this particular lattice takes the form

$\left(\forall\mathcal{B}\subset A,\quad
m_{B}=\sum_{\mathcal{I}\in\mathcal{P}_{B}}\prod_{I\in\mathcal{I}}\kappa_{I}\right)\quad\text{iff}\quad\left(\forall
B\subset
A,\quad\kappa_{B}=\sum_{\mathcal{I}\in\mathcal{P}_{B}}(-1)^{|I|-1}(|I|-1)!\prod_{I\in\mathcal{I}}m_{I}\right).$

We will make use of the following cancellation property of the cumulants.

######  Lemma 2.1.

Let $(m_{B})_{B\subset A}$ and $(\kappa_{B})_{B\subset A}$ be two families of
numbers related by one of the equivalent formulas in (2.1.3). Assume the
existence of a partition $\mathcal{I}\neq\\{A\\}$ such that $\forall B\subset
A,\quad m_{B}=\prod_{I\in\mathcal{I}}m_{I\cap B}.$ Then $\kappa_{A}=0.$

###### Proof.

See [19]. ∎

If $(X_{a})_{a\in A}$ is a family of random variables, we can define for a
subset $B$ of $A$

$m_{B}((X_{b})_{b\in B})=\mathbb{E}\left[\prod_{b\in
B}X_{B}\right]\quad\text{and}\quad\kappa_{B}((X_{b})_{b\in
B})=\sum_{\mathcal{I}\in\mathcal{P}_{B}}(-1)^{|I|-1}(|I|-1)!\prod_{I\in\mathcal{I}}\mathbb{E}\left[\prod_{i\in
I}X_{i}\right].$

The quantity $m_{b}((X_{b})_{b\in B})$ (resp. $\kappa_{B}((X_{b})_{b\in B})$)
is the joint moment (resp. cumulant) of the family of random variables
$(X_{a})_{a\in A}$. The previous Lemma 2.1 translates in the following
property for the cumulant. If there is a partition $\mathcal{I}$ with at least
two cells such that the random variables $X_{i}$ and $X_{j}$ are independent
whenever $[i]_{\mathcal{I}}\neq[j]_{\mathcal{I}}$, then the joint cumulant of
the family $(X_{a})_{a\in A}$ is zero.

The joint cumulants are a convenient tool in the Gaussian framework, since for
a Gaussian vector $(X_{a})_{a\in A}$, the joint cumulant
$\kappa_{A}((X_{a})_{a\in A})$ cancels as soon as $|A|\geq 3$. Reciprocally, a
random variable $X$ such that $\kappa_{p}(X,\ldots,X)=0$ for all $p\geq 3$ is
Gaussian.

###  2.2 Diagonal set and factorial power measure

We will see in Section 3 that the Kac–Rice formula gives an integral formula
for the $p$-th factorial power measure of the zero set of a Gaussian process.
The expression of the Kac density degenerates near the diagonal and it
motivates the introduction of a few notations for the diagonal of a set and
factorial power measure. In the following, $A$ is a finite set and $(E,d)$ is
a metric space.

####  2.2.1 Diagonal set and diagonal inclusion

We define the (large) diagonal of $E^{A}$ the subset

$\Delta:=\Delta^{A}=\left\\{\underline{x}_{A}\in E^{A}\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ \exists a,b\in A\quad\text{with}\quad a\neq
b\quad\text{and}\quad x_{a}=x_{b}\right\\}.$

Let $\mathcal{I}$ be a partition of the set $A$. We define

$\Delta_{\mathcal{I}}=\left\\{\underline{x}_{A}\in E^{A}\leavevmode\nobreak\
\middle|\leavevmode\nobreak\
x_{a}=x_{b}\;\Longleftrightarrow\;[a]_{\mathcal{I}}=[b]_{\mathcal{I}}\right\\}.$

From this definition, one has the following decomposition of the space $E$

$E^{A}=\bigsqcup_{\mathcal{I}\in\mathcal{P}_{A}}\Delta_{\mathcal{I}},\quad\Delta=\bigsqcup_{\begin{subarray}{c}\mathcal{I}\in\mathcal{P}_{A}\\\
\mathcal{I}\neq\overline{A}\end{subarray}}\Delta_{\mathcal{I}}\quad\text{and}\quad
E^{A}\setminus\Delta=\Delta_{\overline{A}},$

where $\overline{A}$ is the partition of $A$ in singletons. We also define

$\Delta_{\mathcal{I}^{+}}:=\bigsqcup_{\mathcal{J}\preceq\mathcal{I}}\Delta_{\mathcal{J}}=\left\\{\underline{x}_{A}\in
E^{A}\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
x_{a}=x_{b}\;\Longrightarrow\;[a]_{\mathcal{I}}=[b]_{\mathcal{I}}\right\\}.$

####  2.2.2 Enlargement of the diagonal set

We fix a number $\eta\geq 0$ and $\underline{x}_{A}\in E^{A}$. We define the
graph $G_{\eta}(\underline{x}_{A})$ with set vertices $A$, and where two
vertices $a$ and $b$ are connected by an edge if $d(x_{a},x_{b})\leq\eta$.
Denote by $\mathcal{I}_{\eta}(\underline{x}_{A})$ the partition of $A$ induced
by the connected components of $G_{\eta}(\underline{x}_{A})$. It allows us to
define the subset

$\Delta_{\mathcal{I},\eta}=\left\\{\underline{x}_{A}\in
E^{A}\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
\mathcal{I}_{\eta}(\underline{x}_{A})=\mathcal{I}\right\\}.$

If $\eta=0$ then $\Delta_{\mathcal{I},0}=\Delta_{\mathcal{I}}$. In the case
where $\eta>0$ we have
$\overline{\Delta_{\mathcal{I},\eta}}\subset\Delta_{\mathcal{I}^{+}}$. As in
the case $\eta=0$, we also have

$E^{k}=\bigsqcup_{\mathcal{I}\in\mathcal{P}_{A}}\Delta_{\mathcal{I},\eta}.$

The fundamental property of this construction is the following. Let $a,b\in A$
and $\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$. If
$[a]_{\mathcal{I}}=[b]_{\mathcal{I}}$ then

$d(x_{a},x_{b})\leq|A|\eta,$

and if $[a]_{\mathcal{I}}\neq[b]_{\mathcal{I}}$ then

$d(x_{a},x_{b})\geq\eta.$

####  2.2.3 The factorial power measure

We define the diagonal inclusion

$\displaystyle\iota_{\mathcal{I}}:E^{\mathcal{I}}$
$\displaystyle\longrightarrow E^{A}$
$\displaystyle\underline{x}_{\mathcal{I}}$
$\displaystyle\longrightarrow(x_{[a]_{\mathcal{I}}})_{a\in A}.$

For instance, if $\mathcal{I}=\\{\\{1,3\\},\\{2\\}\\}$ then
$\iota_{\mathcal{I}}(x,y)=(x,y,x)$. A direct consequence of this definition is
that the mapping $\iota_{\mathcal{I}}$ is a bijection between
$E^{\mathcal{I}}\setminus\Delta$ and $\Delta_{\mathcal{I}}$.

Let $Z$ be a locally finite subset of the metric space $E$. We set
$\nu:=\sum_{x\in Z}\delta_{x}$ the counting measure on $Z$, and

$\nu^{A}=\sum_{x\in
Z^{A}}\delta_{x}\quad\quad\text{and}\quad\quad\nu^{[A]}=\sum_{x\in
Z^{A}\setminus\Delta}\delta_{x}.$

The measure $\nu^{A}$ (resp. $\nu^{[A]}$) is the power (resp. factorial power)
measure of the measure $\nu$. Both measures are linked by the following lemma.

######  Lemma 2.2.

With the notations as above, one has
$\nu^{A}=\sum_{\mathcal{I}\in\mathcal{P}_{A}}\iota_{\mathcal{I}}\,\\!{}_{*}\nu^{[\mathcal{I}]}.$

###### Proof.

We have

$\sum_{x\in
Z^{A}}\delta_{x}=\sum_{\mathcal{I}\in\mathcal{P}_{A}}\left(\sum_{x\in
Z^{A}\cap\Delta_{\mathcal{I}}}\delta_{x}\right).$

Using the fact that the mapping $\iota_{\mathcal{I}}$ is a bijection between
$E^{\mathcal{I}}\setminus\Delta$ and $\Delta_{\mathcal{I}}$, one gets

$\sum_{x\in Z^{A}\cap\Delta_{\mathcal{I}}}\delta_{x}=\sum_{y\in
Z^{\mathcal{I}}\setminus\Delta}\delta_{\iota_{\mathcal{I}}(y)}=\iota_{\mathcal{I}}\,\\!{}_{*}\nu^{[\mathcal{I}]}.$

∎

###  2.3 Matrix notations

The Kac density (see Section 3 and Lemma 3.6) is expressed in term of the
covariance matrix of the underlying Gaussian process and its derivatives. This
fact allows us to consider the Kac density as a function defined on the set of
positive definite matrices, evaluated in some covariance matrix related to our
underlying Gaussian process. To this end, we introduce a few useful notations

####  2.3.1 Basic matrix notations

In the following, we define $\mathcal{M}_{A}(\mathbb{R})$,
$\mathcal{S}_{A}(\mathbb{R})$ and $\mathcal{S}_{A}^{+}(\mathbb{R})$
respectively the sets of square, symmetric and symmetric positive definite
matrices acting on the space $\mathbb{R}^{A}$ equipped with its canonical
basis. If $B$ is another finite set, we define $\mathcal{M}_{A,B}(\mathbb{R})$
the set of matrices from $\mathbb{R}^{A}$ to $\mathbb{R}^{B}$. The open subset
of matrices in $\mathcal{M}_{A,B}(\mathbb{R})$ with maximal rank is denoted
$\mathcal{M}_{A,B}^{*}(\mathbb{R})$. For a matrix
$\Gamma\in\mathcal{M}_{A,B}(\mathbb{R})$ we define

$\|\Gamma\|=\sup_{i,j}|\Gamma_{i,j}|.$

Given a matrix $\Sigma\in\mathcal{M}_{2A,2B}(\mathbb{R})$, we write

$\Sigma=\begin{pmatrix}[c|c]\Sigma^{11}&\Sigma^{12}\\\
\hline\cr\Sigma^{21}&\vphantom{\int^{\int^{a}}}\Sigma^{22}\quad\end{pmatrix},\quad\Sigma^{11},\Sigma^{12},\Sigma^{21},\Sigma^{22}\in\mathcal{M}_{A,B}(\mathbb{R}).$

Let $\Sigma\in\mathcal{M}_{2A}(\mathbb{R})$. If the matrix $\Sigma^{11}$ is
invertible, we define the matrix $\Sigma^{c}\in\mathcal{M}_{A}(\mathbb{R})$ to
be the Schur complement of $\Sigma^{11}$ in $\Sigma$ :

$\Sigma^{c}=\Sigma^{22}-\Sigma^{21}(\Sigma^{11})^{-1}\Sigma^{12}.$

This matrix arises from the identity

$\begin{pmatrix}[c|c]\mathrm{Id}&0\\\
\hline\cr-\Sigma^{21}(\Sigma^{11})^{-1}&\vphantom{\int^{\int^{a}}}\;\mathrm{Id}\;\quad\end{pmatrix}\begin{pmatrix}[c|c]\Sigma^{11}&\Sigma^{12}\\\
\hline\cr\Sigma^{21}&\vphantom{\int^{\int^{a}}}\Sigma^{22}\quad\end{pmatrix}=\begin{pmatrix}[c|c]\Sigma^{11}&\Sigma^{12}\\\
\hline\cr 0&\vphantom{\int^{\int^{a}}}\Sigma^{c}\quad\end{pmatrix}.$

In particular

$\det(\Sigma)=\det(\Sigma^{11})\det(\Sigma^{c}).$

If $\Sigma\in\mathcal{S}_{2A}^{+}(\mathbb{R})$ then
$\Sigma^{c}\in\mathcal{S}_{A}^{+}(\mathbb{R})$ and

$(\Sigma^{c})^{-1}=(\Sigma^{-1})^{22}.$

####  2.3.2 Covariance matrix and Gaussian conditioning

Let $\underline{X}_{A}=(X_{a})_{a\in A}$ and $\underline{Y}_{A}=(Y_{a})_{a\in
A}$ two sequences of jointly Gaussian vectors. We assume that the Gaussian
vector $(\underline{X}_{A},\underline{Y}_{A})$ is non-degenerate. We define

$\Sigma^{11}=\operatorname{Cov}(\underline{X}_{A}),\quad\Sigma^{22}=\operatorname{Cov}(\underline{Y}_{A}),\quad\Sigma^{12}=\operatorname{Cov}(\underline{X}_{A},\underline{Y}_{A}),$

and

$\Sigma:=\operatorname{Cov}\left[\left(\underline{X}_{A},\underline{Y}_{A}\right)\right]=\begin{pmatrix}[c|c]\Sigma^{11}&\Sigma^{12}\\\
\hline\cr{}^{T}\Sigma^{12}&\vphantom{\int^{\int^{a}}}\Sigma^{22}\quad\end{pmatrix}.$

######  Lemma 2.3.

We have
$\operatorname{\mathrm{Law}}(\underline{Y}_{A}|\underline{X}_{A}=0)\sim\mathcal{N}(0,\Sigma^{c}).$

###### Proof.

We define the Gaussian vector

$\underline{Y}_{A}^{c}=\underline{Y}_{A}-\,\\!^{T}\Sigma^{12}(\Sigma^{11})^{-1}\underline{X}_{A}.$

Then

$\operatorname{Cov}(\underline{X}_{A},\underline{Y}_{A}^{c})=0\quad\text{and}\quad\operatorname{Cov}(\underline{Y}_{A}^{c})=\Sigma^{c}.$

Since decorrelation implies independence for Gaussian vectors, we have the
following equality of conditional distributions

$\operatorname{\mathrm{Law}}(\underline{Y}_{A}|\underline{X}_{A})\sim\mathcal{N}(\,\\!^{T}\Sigma^{12}(\Sigma^{11})^{-1}\underline{X}_{A},\Sigma^{c})\quad\text{and}\quad\operatorname{\mathrm{Law}}(\underline{Y}_{A}|\underline{X}_{A}=0)\sim\mathcal{N}(0,\Sigma^{c}).$

∎

####  2.3.3 Compactness in matrix sets

The following lemmas give equivalent conditions to being compact in several
matrix spaces.

######  Lemma 2.4.

A set $K$ is relatively compact in $\mathcal{S}_{A}^{+}(\mathbb{R})$ if and
only if one can find positive constants $c_{K}$ and $C_{K}$ such that
$\forall\Sigma\in K,\quad\det\Sigma\geq c_{K},\quad\|\Sigma\|\leq C_{K}.$

######  Lemma 2.5.

Let $B$ be a subset of $A$. A set $K$ is relatively compact in
$\mathcal{M}_{B,A}^{*}(\mathbb{R})$ if and only if one can find positive
constants $c_{K}$ and $C_{K}$ such that $\forall Q\in K,\quad\det
Q\,\\!^{T}\\!Q\geq c_{K},\quad\|Q\|\leq C_{K}.$

######  Lemma 2.6.

Let $K$ be a relatively compact subset of
$\mathcal{M}_{B,A}^{*}(\mathbb{R})\times\mathcal{S}_{A}^{+}(\mathbb{R})$. Then
the set $\left\\{Q\Sigma\,\\!^{T}\\!Q\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ (\Sigma,Q)\in K\right\\}$ is relatively compact
in $\mathcal{S}_{B}^{+}(\mathbb{R})$.

###### Proof of the lemmas.

The proof of Lemma 2.4 is a direct consequence of the continuity of the
determinant. For Lemma 2.5, note that a matrix $Q$ of
$\mathcal{M}_{B,A}(\mathbb{R})$ is of maximal rank if and only if $\det
Q\,\\!^{T}\\!Q>0$. The conclusion follows again from the continuity of the
determinant. As for Lemma 2.6, let $\Sigma\in\mathcal{S}_{A}^{+}(\mathbb{R})$
and $Q\in\mathcal{M}_{B,A}^{*}(\mathbb{R})$. A direct computation shows that
the matrix $Q\Sigma\,\\!^{T}\\!Q$ is positive definite. The conclusion then
follows from the continuity of the application $(Q,\Sigma)\mapsto
Q\Sigma\,\\!^{T}\\!Q$. ∎

####  2.3.4 Block diagonal matrix with respect to a partition

Let $B$ and $C$ be subsets of $A$, and
$\Gamma\in\mathcal{M}_{B,C}(\mathbb{R})$. For $I$ and $J$ subset of $A$ we
define

$\Gamma_{I,J}=(\Gamma_{i,j})_{i\in I\cap B,j\in J\cap
C}\quad\text{and}\quad\Gamma_{I}=\Gamma_{I,I}.$

Now let $\Sigma\in\mathcal{M}_{2B,2C}(\mathbb{R})$. We define similarly

$\Sigma_{I,J}=\begin{pmatrix}[c|c]\Sigma^{11}_{I,J}&\Sigma^{12}_{I,J}\\\
\hline\cr\Sigma^{21}_{I,J}&\vphantom{\int^{\int^{a}}}\Sigma^{22}_{I,J}\quad\end{pmatrix}\quad\text{and}\quad\Sigma_{I}=\Sigma_{I,I}.$

For a partition $\mathcal{I}$ of the set $A$ and a matrix
$\Gamma\in\mathcal{M}_{B,C}(\mathbb{R})$ we define $\Gamma_{\mathcal{I}}$ to
be the block diagonal matrix with blocks $(\Gamma_{I})_{I\in\mathcal{I}}$.
Similarly, for a matrix $\Sigma\in\mathcal{M}_{2B,2C}(\mathbb{R})$ we define

$\Sigma_{\mathcal{I}}=\begin{pmatrix}[c|c]\Sigma^{11}_{\mathcal{I}}&\Sigma^{12}_{\mathcal{I}}\\\
\hline\cr\Sigma^{21}_{\mathcal{I}}&\vphantom{\int^{\int^{a}}}\Sigma^{22}_{\mathcal{I}}\quad\end{pmatrix}.$

Note that if $\Sigma\in\mathcal{S}_{2A}^{+}(\mathbb{R})$ such that
$\Sigma=\Sigma_{\mathcal{I}}$, then

$(\Sigma^{c})_{\mathcal{I}}=(\Sigma_{\mathcal{I}})^{c},$

but the equality is not true in all generality.

####  2.3.5 Power product space

We introduce a technical matrix space, that will be central in the alternative
expression of the cumulant Kac density in Section 3.2.3. We define the sets

$\widetilde{\mathcal{M}}_{A}(\mathbb{R}):=\prod_{B\subset
A}\left(\mathcal{M}_{A}(\mathbb{R})\times\mathcal{M}_{2B,2A}(\mathbb{R})\right)\quad\text{and}\quad\widetilde{\mathcal{M}}_{A}^{*}(\mathbb{R}):=\prod_{B\subset
A}\left(\mathcal{M}_{A}(\mathbb{R})\times\mathcal{M}_{2B,2A}^{*}(\mathbb{R})\right).$

The space $\widetilde{\mathcal{M}}_{A}^{*}(\mathbb{R})$ is an open subset of
$\widetilde{\mathcal{M}}_{A}(\mathbb{R})$. An element $(M^{B},Q^{B})_{B\subset
A}$ of $\widetilde{\mathcal{M}}_{A}(\mathbb{R})$ will be denoted
$(\widetilde{M},\widetilde{Q})$, with

$\widetilde{M}=(M^{B})_{B\subset
A}\quad\text{and}\quad\widetilde{Q}=(Q^{B})_{B\subset A}.$

If $C$ is a subset of $A$, we denote $(\widetilde{M}_{C},\widetilde{Q}_{C})$
the element of $\widetilde{\mathcal{M}}_{C}(\mathbb{R})$ defined by

$\widetilde{M}_{C}=\left(M^{B}_{C}\right)_{B\subset
C}\quad\text{and}\quad\widetilde{Q}_{C}=\left(Q^{B}_{C}\right)_{B\subset C}.$

At last, if $\mathcal{I}$ is a partition of $A$, we define

$\widetilde{M}_{\mathcal{I}}=\left(M^{B}_{\mathcal{I}}\right)_{B\subset
A}\quad\text{and}\quad\widetilde{Q}_{\mathcal{I}}=\left(Q^{B}_{\mathcal{I}}\right)_{B\subset
A}.$

###  2.4 Divided differences

We now introduce the notion of divided differences. Classically used in
interpolation theory, we use it in order to give a non degenerate expression
of the Kac density near the diagonal. This approach was first taken in [11] in
order to give a necessary and sufficient condition for the finiteness of the
$p$-th moment of the number of zeros on an interval, and has been extensively
used in [1]. The results of this section is standard material about
interpolation, but we will recall basic properties of divided differences.

####  2.4.1 Definition and basic properties

##### Definition

Let $f$ be a continuous function defined on an open interval $U$ of
$\mathbb{R}$ or $\mathbb{T}$. We use the notations of Section 2.2. In
particular, we consider $\Delta$ the large diagonal of $U^{A}$. Let $A$ be a
finite set and $\underline{x}_{A}\in U^{A}\setminus\Delta$. We define
$\mathbb{R}^{A}[X]$ the space of polynomials of degree $|A|-1$. The polynomial

$L[f;\underline{x}_{A}]:x\mapsto\sum_{a\in A}f(x_{a})\prod_{b\neq
a}\frac{x-x_{b}}{x_{a}-x_{b}}$

interpolates the function $f$ at the point $(x_{a})_{a\in A}$. It is the only
polynomial in $\mathbb{R}^{A}[X]$ with this property, since the difference of
two such polynomials cancels at least $|A|$ times and thus must be zero. The
application

$\pi_{\underline{x}_{A}}:f\mapsto L[f;\underline{x}_{A}]$

is a projector onto the space $\mathbb{R}^{A}[X]$, whose kernel is the space
of functions that cancels on $\underline{x}_{A}$. We then define the divided
difference of $f$ as the coefficient of degree $|A|-1$ in
$L[f;\underline{x}_{A}]$. More explicitly,

$f[\underline{x}_{A}]=\sum_{a\in A}f(x_{a})\prod_{b\neq
a}\frac{1}{x_{a}-x_{b}}.$

For instance,

$f[x]=f(x),\quad f[x,y]=\frac{f(x)-f(y)}{x-y},\quad
f[x,y,z]=\frac{f(x)(z-y)+f(y)(x-z)+f(z)(y-x)}{(x-y)(y-z)(z-x)},$

and so on. The following lemma is an analogous of Taylor expansion theorem, in
the context of divided differences.

######  Lemma 2.7.

For $a\in A$, one has
$L[f;\underline{x}_{A}](x)=L[f;\underline{x}_{A\setminus\\{a\\}}](x)+f[\underline{x}_{A}]\prod_{b\neq
a}(x-x_{b}),$ and
$f(x)=L[f;\underline{x}_{A}](x)+f[\underline{x}_{A},x]\prod_{a\in
A}(x-x_{a}).$

###### Proof.

For $a\in A$, the polynomial

$x\mapsto L[f;\underline{x}_{A}](x)-f[\underline{x}_{A}]\prod_{b\neq
a}(x-x_{b})$

interpolates the points $(x_{b})_{b\neq a}$ and is of degree $|A|-2$. By
uniqueness of the interpolating polynomial, it coincides with the polynomial
$L[f;\underline{x}_{A\setminus\\{a\\}}]$. Hence the first statement. An
application of this formula with interpolating points $A\cup\\{x\\}$ yields
the second statement. ∎

##### Continuity property of the divided differences

We will now assume that the function $f$ is of class $\mathcal{C}^{|A|-1}$ on
$U$. Define $C_{A}$ to be the standard simplex of dimension $|A|-1$ :

$C_{A}=\left\\{\underline{t}_{A}\in\mathbb{R}^{A}\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ \sum_{a\in A}t_{a}=1\right\\}.$

We can equip the simplex $C_{A}$ with the induced Lebesgue measure
$\mathrm{d}m$. We then have the following integral representation for the
divided differences, which is analogous to the integral rest in Taylor
expansion in the context of divided differences.

######  Lemma 2.8 (Hermite–Genocchi formula).

We have $f[\underline{x}_{A}]=\int_{C_{A}}f^{(|A|-1)}\left(\sum_{a\in
A}t_{a}x_{a}\right)\mathrm{d}m(\underline{t}_{A}).$ In particular, the
application $\underline{x}_{A}\rightarrow f[\underline{x}_{A}]$ continuously
extends to the whole space $U^{A}$.

###### Proof.

See [9]. ∎

This proposition allows us to extend by continuity the functions

$\underline{x}_{A}\mapsto
f[\underline{x}_{A}]\quad\text{and}\quad\underline{x}_{A}\mapsto
L[f;\underline{x}_{A}],$

from $U^{A}\setminus\Delta$ to the whole space $U^{A}$, as long as $f$ is of
class $\mathcal{C}^{|A|-1}$. For instance, if $x_{a}=y$ for all $a\in A$, then

$f[\underline{x}_{A}]=\frac{f^{|A|-1}(y)}{(|A|-1)!}\quad\text{and}\quad
L_{f}[\underline{x}_{A}](x)=\sum_{j=0}^{|A|-1}\frac{f^{(i)}(y)}{i!}(x-y)^{i}.$

This expression coincides with the Taylor expansion of order $|A|$ of the
function $f$ at the point $y$. The continuity property of the divided
differences allows us to extend Lemma 2.7 and 2.8 by taking
$\underline{x}_{A}$ in the whole space $U^{A}$ and $x\in\mathbb{R}$, provided
that $f$ is sufficiently regular.

##### Divided difference of a polynomial

In this section, $P$ denotes a polynomial. The definition of the divided
differences implies that the quantity $P[\underline{x}_{A}]$ is a rational
fraction in the variables $\underline{x}_{A}$. From the Hermite–Genocchi
formula, we can extend the definition of the divided difference to the whole
complex space $\mathbb{C}^{A}$ by analyticity. It implies the following lemma.

######  Lemma 2.9.

The quantity $P[\underline{x}_{A}]$ is a polynomial expression of the
coefficients $\underline{x}_{A}$. For instance, if $P(x)=x^{4}$ then

$P[x,y]=x^{3}+x^{2}y+xy^{2}+y^{3}.$

If $\deg P<|A|$ then $L_{P}[\underline{x}_{A},x]=P$, and the coefficient of
degree $|A|$ of this polynomial is zero, which implies that
$P[\underline{x}_{A},x]=0$. In the following we assume that $\deg P\geq|A|$.

######  Lemma 2.10.

The polynomial $x\mapsto P[\underline{x}_{A},x]$ is of degree $\deg P-|A|$,
and the leading coefficient this polynomial is the leading coefficient of the
polynomial $P$. From the definition of the divided differences,

$P[\underline{x}_{A},x]\underset{x\rightarrow+\infty}{\simeq}\frac{P(x)}{\prod_{a\in
A}(x-x_{a})},$

which implies that the polynomial $x\mapsto P[\underline{x}_{A},x]$ is of
degree $\deg P-|A|$, and its leading coefficient is the leading coefficient of
$P$.

##### Iterated divided differences

Let $B$ be a subset of $A$ and $\underline{x}_{B}\in U^{B}\setminus\Delta$. We
define the function on $U\setminus\underline{x}_{B}$

$f^{[\underline{x}_{B}]}:x\mapsto f[\underline{x}_{B},x].$

######  Lemma 2.11.

Let $\underline{x}_{A}\in U^{A}\setminus\Delta$. Then
$L[f;\underline{x}_{A}][\underline{x}_{B},\,.\,]=L[f^{[\underline{x}_{B}]},\underline{x}_{A\setminus
B}],$ and
$f[\underline{x}_{A}]=f^{[\underline{x}_{B}]}[\underline{x}_{A\setminus B}].$
Let $x\in U^{A}\setminus\Delta$. Then
$f[\underline{x}_{B},x]=L[f;\underline{x}_{A}][\underline{x}_{B},x]+f[\underline{x}_{A},x]\prod_{a\in
A\setminus B}(x-x_{a}).$

###### Proof.

Consider the two polynomials

$P_{1}=L[f^{[\underline{x}_{B}]},\underline{x}_{A\setminus
B}]\quad\quad\text{and}\quad\quad
P_{2}=L[f;\underline{x}_{A}]^{[\underline{x}_{B}]}.$

They both interpolate the points $\underline{x}_{A\setminus B}$ at values
$(f[\underline{x}_{B},x_{a}])_{a\in A\setminus B}$. The polynomial $P_{1}$ is
in $\mathbb{R}^{A\setminus B}[X]$, and
$L_{f}[\underline{x}_{A},\underline{y}_{B}]$ is in $\mathbb{R}^{A}[X]$. By
Lemma 2.10, the polynomial $P_{2}$ is also in $\mathbb{R}^{B}[X]$. By
uniqueness of the interpolating polynomial, these two polynomials are equals,
hence the first statement.

The coefficient of degree $|B|-1$ in $P_{1}$ is
$f^{[\underline{x}_{B}]}[\underline{x}_{A\setminus B}]$, and the one in
$P_{2}$ is $f[\underline{x}_{A}]$ according to lemma 2.10. By the previous
equality these two coefficients are equals, which yields the second formula.
The last formula is a direct application of Lemma 2.7 applied to the function
$f^{[\underline{x}_{B}]}$. ∎

####  2.4.2 Matrix viewpoint of the divided differences

In order to describe the divided differences of a Gaussian process and the
induced transformation on the covariance matrix, we rewrite the operation of
taking divided differences from a matrix viewpoint. From now, we equip $A$
with an arbitrary total order $\leq$ and we introduce the notation

$a\mid A=\left\\{b\in A\leavevmode\nobreak\ \middle|\leavevmode\nobreak\ b\leq
a\right\\}.$

Thus, $\underline{x}_{\,a\mid A}=(x_{b})_{b\leq a}$.

##### Basis of polynomials adapted to the divided difference

For $\underline{x}_{A}\in\mathbb{R}^{A}$ we define the polynomial

$P^{a}_{\underline{x}_{A}}:x\mapsto\prod_{b<a}(x-x_{b}).$

For any subset $B$ of $A$, the family $(P^{b}_{\underline{x}_{B}})_{b\in B}$
is a basis of the space $\mathbb{R}^{B}[X]$ and we will always equip the space
$\mathbb{R}^{B}[X]$ with this basis.

######  Remark 2.12.

The family $(P^{a}_{\underline{x}_{A}})_{a\in A}$ is a family of monic
polynomials of increasing degrees. Thus, if we equip $A$ with another total
order, the transformation matrix is of determinant $\pm 1$ and depends
continuously on the quantities $(x_{a}-x_{b})_{a,b\in A}$. It justifies in the
following why the the order can be chosen arbitrarily.

A direct induction based on Lemma 2.7 implies the following lemma.

######  Lemma 2.13.

Let $\underline{x}_{A}\in U^{A}\setminus\Delta$. Then
$L[f;\underline{x}_{A}]=\sum_{a\in A}f[\underline{x}_{\,a\mid
A}]P^{a}_{\underline{x}_{A}}.$

The finite differences of $f$ are thus the coefficients of the Lagrange
interpolation polynomial in the basis $(P^{a}_{\underline{x}_{A}})_{a\in A}$.
We define

$f_{A}[\underline{x}_{A}]=(f[\underline{x}_{a|A}])_{a\in
A}\quad\text{and}\quad
M(\underline{x}_{A})=(P^{b}_{\underline{x}_{A}}(x_{a}))_{a,b\in A}.$

Then Lemma 2.13 rewrites matricially

$f(\underline{x}_{A})=M(\underline{x}_{A})f_{A}[\underline{x}_{A}].$

For instance,

$\begin{pmatrix}f(x)\\\ f(y)\\\ f(z)\end{pmatrix}=\begin{pmatrix}1&0&0\\\
1&y-x&0\\\ 1&z-x&(z-x)(z-y)\end{pmatrix}\begin{pmatrix}f[x]\\\ f[x,y]\\\
f[x,y,z]\end{pmatrix}.$

The matrix $M(\underline{x}_{A})$ is lower triangular, thus

$\det M(\underline{x}_{A})=\prod_{a\in
A}P^{a}_{\underline{x}_{A}}(x_{a})=\prod_{a<b}(x_{b}-x_{a}).$

##### Divided differences with respect to a partition

In the following, $\mathcal{I}$ is a partition of the set $A$. We define

$f_{\mathcal{I}}[\underline{x}_{A}]=(f_{I}[\underline{x}_{I}])_{I\in\mathcal{I}}=(f[\underline{x}_{i|I}])_{i\in
I,I\in\mathcal{I}}$

We can perform the divided difference independently on each element of the
partition. That is, we can write for all $I\in\mathcal{I}$

$f_{I}(\underline{x}_{I})=M(\underline{x}_{I})f_{I}[\underline{x}_{I}]\quad\text{and}\quad
f(\underline{x}_{A})=M^{\mathcal{I}}(\underline{x}_{A})f_{\mathcal{I}}[\underline{x}_{A}];$

where $M^{\mathcal{I}}(\underline{x}_{A})$ is the block diagonal matrix with
blocks $(M(\underline{x}_{I}))_{I\in\mathcal{I}}$. For instance, if
$\mathcal{I}=\\{\\{1,2\\},\\{3,4\\}\\}$ then

$\begin{pmatrix}f(w)\\\ f(x)\\\ f(y)\\\
f(z)\end{pmatrix}=\begin{pmatrix}1&0&0&0\\\ 1&x-w&0&0\\\ 0&0&1&0\\\
0&0&1&z-y\end{pmatrix}\begin{pmatrix}f[w]\\\ f[w,x]\\\ f[y]\\\
f[y,z]\end{pmatrix}.$

From Equation (2.4.2),

$\det M^{\mathcal{I}}(\underline{x}_{A})=\prod_{I\in\mathcal{I}}\det
M_{I}(\underline{x}_{A})=\prod_{I\in\mathcal{I}}\prod_{\begin{subarray}{c}i,j\in
I\\\ i<j\end{subarray}}(x_{j}-x_{i}).$

######  Lemma 2.14.

Let $\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$. There is a constant
$C(\eta)$ such that $\|M^{\mathcal{I}}(\underline{x}_{A})\|\leq C(\eta).$

###### Proof.

Let $a,b\in A$. If $[a]_{\mathcal{I}}\neq[b]_{\mathcal{I}}$, then
$(M^{\mathcal{I}}(\underline{x}_{A}))_{a,b}=0$. Else, there is
$I\in\mathcal{I}$ such that $a,b\in I$. Then

$(M^{\mathcal{I}}(\underline{x}_{A}))_{a,b}=P^{b}_{\underline{x}_{I}}(x_{a})=\prod_{\begin{subarray}{c}c<b\\\
c\in I\end{subarray}}(x_{a}-x_{c}).$

For $a,c\in I$, one has $|x_{a}-x_{c}|\leq|A|\eta$. The conclusion follows. ∎

For $\underline{x}_{A}\in U^{A}$ we consider the mapping

$\displaystyle\pi_{\underline{x}_{A}}^{\mathcal{I}}:\mathbb{R}^{A}[X]$
$\displaystyle\longrightarrow\prod_{I\in\mathcal{I}}\mathbb{R}^{I}[X]$
$\displaystyle P$
$\displaystyle\longrightarrow(L[P;\underline{x}_{I}])_{I\in\mathcal{I}}.$

It is well defined for $\underline{x}_{A}\in U^{A}$, since a polynomial is
infinitely differentiable. For a subset $I$ of $A$, we equip
$\mathbb{R}^{I}[X]$ with the basis of polynomials
$(P^{i}_{\underline{x}_{I}})_{i\in I}$. Let
$Q^{\mathcal{I}}(\underline{x}_{A})$ the matrix of the application
$\pi_{\underline{x}_{A}}^{\mathcal{I}}$ in that basis. For instance, when
$\mathcal{I}=\\{\\{1,2\\},\\{3\\}\\}$ then

$\begin{pmatrix}f[x]\\\ f[x,y]\\\ f[z]\end{pmatrix}=\begin{pmatrix}1&0&0\\\
0&1&0\\\ 1&z-x&(z-x)(z-y)\end{pmatrix}\begin{pmatrix}f[x]\\\ f[x,y]\\\
f[x,y,z]\end{pmatrix}.$

Let $\underline{x}_{A}\in U^{A}\setminus\Delta$. Then
$M^{\mathcal{I}}(\underline{x}_{A})$ is invertible and from Equation (2.4.2)
and (2.4.2), one has

$f_{\mathcal{I}}[\underline{x}_{A}]=[M^{\mathcal{I}}(\underline{x}_{A})]^{-1}M(\underline{x}_{A})f_{A}[\underline{x}_{A}],$

and thus

$Q^{\mathcal{I}}(\underline{x}_{A})=[M^{\mathcal{I}}(\underline{x}_{A})]^{-1}M(\underline{x}_{A}).$

It implies that

$\left|\det
Q^{\mathcal{I}}(\underline{x}_{A})\right|=\prod_{\begin{subarray}{c}\\{I,J\\}\subset\mathcal{I}\\\
I\neq J\end{subarray}}\prod_{i\in I}\prod_{j\in J}|x_{i}-x_{j}|.$

By continuity with respect to $\underline{x}_{A}$ of the application
$\pi_{\underline{x}_{A}}^{\mathcal{I}}$ this formula remain true for
$\underline{x}_{A}\in\mathbb{R}^{A}$. We deduce that the application
$\pi_{\underline{x}_{A}}^{\mathcal{I}}$ is invertible for
$\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$.

Now let $\mathcal{J}$ be another partition of $A$, finer than the partition
$\mathcal{I}$. For $\underline{x}_{A}\in U^{A}$ we consider the mapping

$\displaystyle\pi_{\underline{x}_{A}}^{\mathcal{I},\mathcal{J}}:\prod_{I\in\mathcal{I}}\mathbb{R}^{I}[X]$
$\displaystyle\longrightarrow\prod_{J\in\mathcal{J}}\mathbb{R}^{J}[X]$
$\displaystyle(P_{I})_{I\in\mathcal{I}}$
$\displaystyle\longrightarrow(L[P_{[J]_{I}};\underline{x}_{J}])_{J\in\mathcal{J}}.$

Restricted to $\mathbb{R}^{I}[X]$, the application
$\pi_{\underline{x}_{A}}^{\mathcal{I},\mathcal{J}}$ coincides with
$\pi_{\underline{x}_{I}}^{\mathcal{J}_{I}}$. Let
$Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})$ be the matrix of the
application $\pi_{\underline{x}_{A}}^{\mathcal{I},\mathcal{J}}$, so that

$f_{\mathcal{J}}[\underline{x}_{A}]=Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})f_{\mathcal{I}}[\underline{x}_{A}].$

######  Lemma 2.15.

$\left|\det
Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A}))\right|=\prod_{I\in\mathcal{I}}\prod_{\begin{subarray}{c}\\{J_{1},J_{2}\\}\subset\mathcal{J}_{I}\\\
J_{1}\neq J_{2}\end{subarray}}\prod_{j_{1}\in J_{1}}\prod_{j_{2}\in
J_{2}}|x_{j_{1}}-x_{j_{2}}|.$ In particular, the application
$\pi_{\underline{x}_{A}}^{\mathcal{I},\mathcal{J}}$ is invertible in the case
where $\underline{x}_{A}\in\Delta_{J^{+}}$.

###### Proof.

We have from Equation (2.4.2)

$\left|\det
Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A}))\right|=\prod_{I\in\mathcal{I}}\left|\det
Q^{\mathcal{J}_{I}}(\underline{x}_{I}))\right|=\prod_{I\in\mathcal{I}}\prod_{\begin{subarray}{c}\\{J_{1},J_{2}\\}\subset\mathcal{J}_{I}\\\
J_{1}\neq J_{2}\end{subarray}}\prod_{j_{1}\in J_{1}}\prod_{j_{2}\in
J_{2}}|x_{j_{1}}-x_{j_{2}}|.$

And this expression does not cancel when $\underline{x}_{A}\in\Delta_{J^{+}}$.
∎

##### Divided difference on a subset

In this section, $B$ is a subset of $A$ and $\mathcal{I}$ is a partition of
$A$. From Equation (2.4.2) and (2.4.2), one has

$f(\underline{x}_{B})=M_{B,A}(\underline{x}_{A})f_{A}[\underline{x}_{A}]\quad\text{and}\quad
f(\underline{x}_{B})=M_{B,A}^{\mathcal{I}}(\underline{x}_{A})f_{\mathcal{I}}[\underline{x}_{A}],$

where for a matrix $M$, the matrix $M_{B,A}$ is defined in (2.3.4). For
$\underline{x}_{A}\in U^{A}$ we consider the mapping

$\displaystyle\pi_{\underline{x}_{A}}^{\mathcal{I},B}:\prod_{I\in\mathcal{I}}\mathbb{R}^{I}[X]$
$\displaystyle\longrightarrow\prod_{I\in\mathcal{I}}\mathbb{R}^{I\cap B}[X]$
$\displaystyle(P_{I})_{I\in\mathcal{I}}$
$\displaystyle\longrightarrow(L[P_{I};\underline{x}_{I\cap
B}])_{I\in\mathcal{I}},$ (32)

and let $Q^{\mathcal{I},B}(\underline{x}_{A})$ be the matrix of this
application.

######  Lemma 2.16.

The set $\left\\{Q^{\mathcal{I},B}(\underline{x}_{A})\leavevmode\nobreak\
\middle|\leavevmode\nobreak\
\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}\right\\}$ is relatively compact
in $\mathcal{M}_{B,A}^{*}(\mathbb{R})$.

###### Proof.

Let $\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$. The matrix
$Q^{\mathcal{I},B}(\underline{x}_{A})$ is block diagonal with respect to the
partition $\mathcal{I}$, with blocks $(Q^{\\{I\\},B\cap
I}(\underline{x}_{I}))_{I\in\mathcal{I}}$. For $I\in\mathcal{I}$, the
quantities $(x_{i}-x_{j})_{i,j\in I}$ are bounded by $|A|\eta$. According to
remark 2.12, we can change the order on $I$ and assume that $B\cap I=b|I$ for
some $b\in B\cap I$. In that case, one has

$Q^{\\{I\\},B\cap I}(\underline{x}_{I})=\begin{pmatrix}[c|c]\mathrm{Id}_{B\cap
I}\vphantom{\int^{\int}}&0\vphantom{\int^{\int}}\end{pmatrix},$

and the conclusion follows. ∎

##### Doubling divided differences

In the paragraph, we consider the divided difference on the set $2A$ defined
in (2.1.1). We equip the set $2A$ with the lexicographic order inherited from
the order on $\\{1,2\\}$ and the arbitrary order on $A$. Note that

$(f[x],f[x,x])=(f(x),f^{\prime}(x)).$

The interest of doubling divided differences is to consider simultaneous
interpolation of the function $f$ and $f^{\prime}$ on prescribed points
$(\underline{x}_{a})_{a\in A}$. This part coincides with the classical Hermite
interpolation.

Let $\mathcal{I}$ be a partition of $A$ (and thus of $2A$, following the
notation (2.1.3)). As a consequence of Definition (32), one has

$f_{\mathcal{I}_{B}}[\underline{x}_{B,B}]=Q^{\mathcal{I},2B}(\underline{x}_{A,A})f_{\mathcal{I}}[\underline{x}_{A,A}].$

We define

$\widetilde{M}^{\mathcal{I}}(\underline{x}_{A})=(M^{\mathcal{I}_{B}}(\underline{x}_{B}))_{B\subset
A}\quad\text{and}\quad\widetilde{Q}^{\mathcal{I}}(\underline{x}_{A})=(Q^{\mathcal{I},2B}(\underline{x}_{A,A}))_{B\subset
A}.$

Recall the definition of $\widetilde{\mathcal{M}}_{A}(\mathbb{R})$ and
$\widetilde{\mathcal{M}}_{A}^{*}(\mathbb{R})$ in Paragraph 2.3.5. Then

$(\widetilde{M}^{\mathcal{I}}(\underline{x}_{A}),\widetilde{Q}^{\mathcal{I}}(\underline{x}_{A}))\in\widetilde{\mathcal{M}}_{A}(\mathbb{R}),$
$\widetilde{M}^{\mathcal{I}}(\underline{x}_{A})=\widetilde{M}_{\mathcal{I}}^{\mathcal{I}}(\underline{x}_{A})\quad\text{and}\quad\widetilde{Q}^{\mathcal{I}}(\underline{x}_{A})=\widetilde{Q}^{\mathcal{I}}_{\mathcal{I}}(\underline{x}_{A}).$

Moreover one has the following key proposition.

######  Proposition 2.17.

The set
$\left\\{(\widetilde{M}^{\mathcal{I}}(\underline{x}_{A}),\widetilde{Q}^{\mathcal{I}}(\underline{x}_{A}))\leavevmode\nobreak\
\middle|\leavevmode\nobreak\
\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}\right\\}$ is relatively compact
in $\widetilde{\mathcal{M}}^{*}_{A}(\mathbb{R})$.

###### Proof.

Let $\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$. According to Lemma 2.14,
the coefficients of the matrix $M^{\mathcal{I}_{B}}(\underline{x}_{B})$ are
bounded by a constant $C(\eta)$. We also have
$\underline{x}_{A,A}\in\Delta_{\mathcal{I},\eta}$. Lemma 2.16 applies, and the
set of matrices
$(Q^{\mathcal{I},2B}(\underline{x}_{A,A}))_{\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}}$
is relatively compact in $\mathcal{M}_{2B,2A}^{*}(\mathbb{R})$. The conclusion
follows. ∎

##### Divided differences of a Gaussian process

At last we describe the covariance matrix of the divided difference vector of
a Gaussian process. The integral representation given by the Hermite–Genocchi
formula gives a convenient expression for the coefficients of the covariance
matrix.

Let $f$ be a Gaussian process of class $\mathcal{C}^{|A|-1}$ on the interval
$U$. We denote $r$ the covariance function of $f$, which is differentiable
$|A|-1$ times in each variable. We define

$\Sigma^{\mathcal{I}}(\underline{x}_{A})=\operatorname{Cov}(f_{\mathcal{I}}[\underline{x}_{A}]).$

######  Lemma 2.18.

Let $I,J\in\mathcal{I}$, $a\in I$ and $b\in J$. Then
$\Sigma^{\mathcal{I}}(\underline{x}_{A})_{ab}=\int_{C_{a|I}\times
C_{b|J}}r^{(|\,a|I\,|-1\,,\,|\,b|J\,|-1)}\left(\sum_{i\in
a|I}s_{i}x_{i},\sum_{j\in
b|J}t_{j}x_{j}\right)\mathrm{d}m(\underline{s}_{i|I})\mathrm{d}m(\underline{t}_{i|J}).$

###### Proof.

The Hermite Genocchi formula 2.8 asserts that

$\displaystyle\Sigma^{\mathcal{I}}[\underline{x}_{A}]_{ab}$
$\displaystyle=\mathbb{E}[f[\underline{x}_{a|I}]f[\underline{x}_{b|J}]]$
$\displaystyle=\int_{C_{a|I}\times
C_{b|J}}\mathbb{E}\left[f^{(|\,a|I\,|-1)}\left(\sum_{i\in
a|I}s_{i}x_{i}\right)f^{(|\,b|J\,|-1)}\left(\sum_{j\in
b|J}t_{j}x_{j}\right)\right]\mathrm{d}m(\underline{s}_{a|I})\mathrm{d}m(\underline{t}_{b|J})$
$\displaystyle=\int_{C_{a|I}\times
C_{b|J}}r^{(|\,a|I\,|-1,|\,b|J\,|-1)}\left(\sum_{i\in
a|I}s_{i}x_{i},\sum_{j\in
b|J}t_{j}x_{j}\right)\mathrm{d}m(\underline{s}_{a|I})\mathrm{d}m(\underline{t}_{b|J}).$

∎

##  3 Kac–Rice formula for Gaussian processes

For now on, $A$ is a finite set and $f$ is a centered Gaussian process defined
on an interval $U$ of $\mathbb{R}$ or $\mathbb{T}$, with covariance function
$r$. We assume for this section that the process $f$ is of class
$\mathcal{C}^{2|A|-1}$, and for every partition $\mathcal{I}$ of $A$ and
$\underline{y}_{\mathcal{I}}\in U^{\mathcal{I}}\setminus\Delta$, the following
non-degeneracy condition holds

$\det\left[\operatorname{Cov}\left(\left(f^{(k)}(y_{I})\right)_{\begin{subarray}{c}0\leq
k\leq 2|I|-1\\\ I\in\mathcal{I}\end{subarray}}\right)\right]>0.$

It has been shown for instance in [8] that this condition ensures the
finiteness of the $|A|$-th moment of the number of zeros of a Gaussian process
on a bounded interval.

###  3.1 Kac density and cumulants of the zeros counting measure

In this section we give the expression of the $p$-th factorial moment and
cumulant of the zeros counting measure. The first step is to lift the
degeneracy of the Kac–Rice formula near the diagonal.

####  3.1.1 Non-degeneracy of the Kac density near the diagonal

We apply the method of divided difference to lift the degeneracy of the Kac
density and give an alternative formula near the diagonal. The material is
this section is quite standard, see for instance [8] and [1]. Only Lemma 3.4
is new and allows us to express the Kac density as a function of a non
degenerate Gaussian vector.

We fix for the rest of this paragraph a partition $\mathcal{I}$ of $A$.

######  Lemma 3.1.

The Gaussian vectors $f_{\mathcal{I}}[\underline{x}_{A}]$ and
$f_{\mathcal{I}}[\underline{x}_{A,A}]$ are non degenerate for
$\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$.

###### Proof.

We prove first the non degeneracy of $f_{\mathcal{I}}[\underline{x}_{A}]$. The
process $f$ is of class $\mathcal{C}^{|A|-1}$, and the Gaussian vector
$f_{\mathcal{I}}[\underline{x}_{A}]$ is well-defined. Let
$\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$. By definition of the set
$\Delta_{\mathcal{I}^{+}}$, there is a partition $\mathcal{J}$ finer than the
partition $\mathcal{I}$ and such that
$\underline{x}_{A}\in\Delta_{\mathcal{J}}$. We write
$\underline{x}_{A}=\iota_{\mathcal{J}}(\underline{y}_{\mathcal{J}})$ for some
$\underline{y}_{\mathcal{J}}\in\mathbb{R}^{\mathcal{J}}\setminus\Delta$. For
$J\in\mathcal{J}$, we have from equation (2.4.1)

$f_{\mathcal{J}}[\underline{x}_{A}]=\left(\frac{f^{(k)}(y_{J})}{k!}\right)_{\begin{subarray}{c}0\leq
k\leq|J|-1\\\ J\in\mathcal{J}\end{subarray}},$

which is non-degenerate by the hypothesis (3). Now from Equation (2.4.2),

$f_{\mathcal{J}}[\underline{x}_{A}]=Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})f_{\mathcal{I}}[\underline{x}_{A}].$

According to Lemma 2.15, the matrix
$Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})$ is invertible when
$\underline{x}_{A}\in\Delta_{\mathcal{J}}$, which implies that the Gaussian
vector $f_{\mathcal{I}}[\underline{x}_{A}]$ is non degenerate for
$\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$.

Now for the Gaussian vector $f_{\mathcal{I}}[\underline{x}_{A,A}]$, the
process $f$ is of class $\mathcal{C}^{2|A|-1}$ and the Gaussian vector
$f_{\mathcal{I}}[\underline{x}_{A,A}]$ is well-defined. Moreover, if
$\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$ then
$\underline{x}_{A,A}\in\Delta_{\mathcal{I}^{+}}$ and we can apply the previous
case to the set $2A$ to deduce the non degeneracy of the Gaussian vector
$f_{\mathcal{I}}[\underline{x}_{A,A}]$. ∎

We define the random set

$Z=\left\\{x\in U\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
f(x)=0\right\\}.$

By Bulinskaya lemma (see [8]) and the assumption on $f$, the subset $Z$ is
almost surely a closed discrete subset of $U$ and we can define the random
measure $\nu$ to be the counting measure of $Z$. Kac–Rice formula (see [8,
Thm. 3.2] and [1, Prop. 3.6]) asserts that for a positive function
$\Phi:U^{A}\rightarrow\mathbb{R}$, one has, following the notations of Section
2.2,

$\mathbb{E}[\langle\nu^{[A]},\Phi\rangle]=\int_{U^{A}}\rho(\underline{x}_{A})\Phi(\underline{x}_{A})\mathrm{d}\underline{x}_{A},$

with

$\rho(\underline{x}_{A}):=\rho_{|A|}(\underline{x}_{A})=\frac{N(\underline{x}_{A})}{D(\underline{x}_{A})},$

where

$N(\underline{x}_{A})=\mathbb{E}\left[\prod_{a\in
A}|f^{\prime}(x_{a})|\;\middle|\;\forall a\in
A,\;f(x_{a})=0\right]\quad\text{and}\quad
D(\underline{x}_{A})=\sqrt{\det\left[2\pi\operatorname{Cov}(f(x_{A}))\right]}.$

The function $\rho$ is only defined for $\underline{x}_{A}\in
U^{A}\setminus\Delta$. Along the diagonal $\Delta$, the function $N$ is ill-
defined and the function $D$ cancels. The first step consists in giving an
alternative non singular formula for $\rho$ in a neighborhood of the diagonal
$\Delta$. Let $\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$. We define

$D^{\mathcal{I}}(\underline{x}_{A})=\sqrt{\det[2\pi\operatorname{Cov}(f_{\mathcal{I}}[\underline{x}_{A}])]},\quad
N^{\mathcal{I}}(\underline{x}_{A})=\mathbb{E}\left[\prod_{I\in\mathcal{I}}\prod_{i\in
I}\left|f[\underline{x}_{I},x_{i}]\right|\;\middle|\;f_{\mathcal{I}}[\underline{x}_{A}]=0\right],$

and

$\rho^{\mathcal{I}}(\underline{x}_{A})=\frac{N(\underline{x}_{A})}{D(\underline{x}_{A})}.$

Lemma 3.1 implies that the three quantities above are well defined.

######  Remark 3.2.

If $\mathcal{I}=\overline{A}$, then
$f_{\overline{A}}(\underline{x}_{A})=f(\underline{x}_{A})$ and
$f[x_{a},x_{a}]=f^{\prime}(x_{a})$, which implies that

$\rho(\underline{x}_{A})=\rho^{\overline{A}}(\underline{x}_{A}).$

One has the following relations.

######  Lemma 3.3.

Let $\mathcal{J}$ be a finer partition than $\mathcal{I}$. Then for
$\underline{x}_{A}\in\Delta_{\mathcal{J}^{+}}$ one has $\displaystyle
D^{\mathcal{J}}(\underline{x}_{A})$ $\displaystyle=\left|\det
Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})\right|D^{\mathcal{I}}(\underline{x}_{A}),$
$\displaystyle N^{\mathcal{J}}(\underline{x}_{A})$ $\displaystyle=\left(\det
Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})\right)^{2}N^{\mathcal{I}}(\underline{x}_{A}),$
$\displaystyle\rho^{\mathcal{J}}(\underline{x}_{A})$ $\displaystyle=\left|\det
Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})\right|\rho^{\mathcal{I}}(\underline{x}_{A}).$

It implies that the function $\rho^{\mathcal{J}}$ can be extended by
continuity from $\Delta_{\mathcal{J}^{+}}$ to $\Delta_{\mathcal{I}^{+}}$ via
these relations. By taking $\mathcal{I}=\\{A\\}$ and
$\mathcal{J}=\overline{A}$, it implies that the function the function $\rho$
can be extended by continuity to the whole space $U^{A}$. Moreover, one has

$\det Q^{\\{A\\},\overline{A}}(\underline{x}_{A})=\det
M(\underline{x}_{A})=\prod_{a<b}(x_{b}-x_{a}),$

thus the function $\rho$ vanishes on the diagonal $\Delta$.

###### Proof.

Let $\underline{x}_{A}\in\Delta_{\mathcal{J}^{+}}$. From Equation (2.4.2),

$f_{\mathcal{J}}[\underline{x}_{A}]=Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})f_{\mathcal{I}}[\underline{x}_{A}].$

We deduce that

$\displaystyle D^{\mathcal{J}}(\underline{x}_{A})$
$\displaystyle=\sqrt{\det\left[2\pi\operatorname{Cov}(f_{\mathcal{J}}[\underline{x}_{A}])\right]}$
$\displaystyle=\left|\det
Q^{\mathcal{I},\mathcal{J}}(\underline{x}_{A})\right|D^{\mathcal{I}}(\underline{x}_{A}).$
(38)

The quantities $N_{A}^{\mathcal{I}}(\underline{x}_{A})$ and
$N_{A}^{\mathcal{J}}(\underline{x}_{A})$ are well defined. The Gaussian
vectors $f_{\mathcal{I}}[\underline{x}_{A}]$ and
$f_{\mathcal{J}}[\underline{x}_{A}]$ are equals up to a linear invertible
change of variable, and they cancels simultaneously. In other words, one has

$\displaystyle N^{\mathcal{J}}(\underline{x}_{A})$
$\displaystyle=\mathbb{E}\left[\prod_{J\in\mathcal{J}}\prod_{j\in
J}\left|f[\underline{x}_{J},x_{j}]\right|\;\middle|\;f_{\mathcal{J}}[\underline{x}_{A}]=0\right]$
$\displaystyle=\mathbb{E}\left[\prod_{J\in\mathcal{J}}\prod_{j\in
J}\left|f[\underline{x}_{J},x_{j}]\right|\;\middle|\;f_{\mathcal{I}}[\underline{x}_{A}]=0\right]$
$\displaystyle=\mathbb{E}\left[\prod_{I\in\mathcal{I}}\prod_{J\in\mathcal{J}_{I}}\prod_{j\in
J}\left|f[\underline{x}_{J},x_{j}]\right|\;\middle|\;f_{\mathcal{I}}[\underline{x}_{A}]=0\right].$

Let $I\in\mathcal{I}$, $J\in\mathcal{J}_{I}$ and $j\in J$. From Lemma 2.11,
conditionally to $f_{I}[\underline{x}_{I}]=0$ one has

$f[\underline{x}_{J},x_{j}]=\left(\prod_{\begin{subarray}{c}i\in I\setminus
J\end{subarray}}(x_{i}-x_{j})\right)f[\underline{x}_{I},x_{j}],$

from which we deduce

$\displaystyle N^{\mathcal{J}}(\underline{x}_{A})$
$\displaystyle=\left(\prod_{I\in\mathcal{I}}\prod_{J\in\mathcal{J}_{I}}\prod_{j\in
J}\prod_{i\in I\setminus
J}|x_{i}-x_{j}|\right)\mathbb{E}\left[\prod_{I\in\mathcal{I}}\prod_{i\in
I}|f[\underline{x}_{I},x_{i}]|\;\middle|\;f_{\mathcal{I}}[\underline{x}_{A}]=0\right]$
$\displaystyle=\left(\prod_{I\in\mathcal{I}}\prod_{\begin{subarray}{c}J_{1},J_{2}\subset\mathcal{J}_{I}\\\
J_{1}\neq J_{2}\end{subarray}}\prod_{j_{1}\in J_{1}}\prod_{j_{2}\in
J_{2}}|x_{j_{1}}-x_{j_{2}}|\right)N^{\mathcal{I}}(\underline{x}_{A}).$ (39)

We deduce the alternative expression for $\rho^{\mathcal{J}}$ from (38) and
(39). ∎

When the points $(\underline{x}_{a})_{a\in A}$ collapse on the diagonal
$\Delta_{\mathcal{I}}$ the vector
$(f[\underline{x}_{I},\underline{x}_{i}])_{I\in\mathcal{I},i\in I}$ becomes
degenerate, it makes unpractical the analysis of regularity of the function
$\rho^{\mathcal{I}}$ in a neighborhood of the diagonal $\Delta_{\mathcal{I}}$.
The following lemma gives another expression of the quantity
$N^{\mathcal{I}}(\underline{x}_{A})$ that depends fully on a non degenerate
Gaussian vector. Recall the definition of the function
$f^{[\underline{x}_{B}]}$ for a subset $B$ of $A$ in (2.4.1).

######  Lemma 3.4.

One has
$N^{\mathcal{I}}(\underline{x}_{A})=\mathbb{E}\left[\prod_{I\in\mathcal{I}}\prod_{i\in
I}\left|\left(M(\underline{x}_{I})f^{[\underline{x}_{I}]}_{I}[\underline{x}_{I}]\right)_{i}\vphantom{\int}\right|\;\middle|\;f_{\mathcal{I}}[\underline{x}_{A}]=0\right].$

###### Proof.

Let $I\in\mathcal{I}$. According to formula (2.4.2),

$f^{[\underline{x}_{I}]}(\underline{x}_{I})=M(\underline{x}_{I})f^{[\underline{x}_{I}]}_{I}[\underline{x}_{I}].$

The conclusion follows from the definition of
$N^{\mathcal{I}}(\underline{x}_{A})$. ∎

####  3.1.2 Expression of the cumulants of the zeros counting measure

We are now ready to give the expression of the cumulant of order $|A|$ of the
linear statistics associated to zeros counting measure. Let $(\phi_{a})_{a\in
A}$ be a collection of bounded functions with compact support. We define

$\kappa_{A}(\nu)(\underline{\phi}_{A})=\kappa\left(\left(\langle\nu,\phi_{a}\rangle\right)_{a\in
A}\right),$

the joint cumulant of the family of random variables
$(\langle\nu,\phi_{a}\rangle)_{a\in A}$. We define the cumulant Kac density
associated with the set $A$ to be the function

$\displaystyle F_{A}:\mathbb{R}^{A}$ $\displaystyle\longrightarrow\mathbb{R}$
$\displaystyle\underline{x}_{A}$
$\displaystyle\longrightarrow\sum_{\mathcal{J}\in\mathcal{P}_{A}}(|\mathcal{J}|-1)!(-1)^{|\mathcal{J}|-1}\prod_{J\in\mathcal{J}}\rho(\underline{x}_{J}).$
(40)

The following Proposition 3.5 express the cumulant of order $|A|$ of the
linear statistics associated to zeros counting measure. It is key step in
towards proof of Theorem 1.6, and reveals the elegant interplay between the
factorial power counting measure and the combinatorics of cumulants.

######  Proposition 3.5.

We have
$\kappa_{A}(\nu)(\underline{\phi}_{A})=\sum_{\mathcal{I}\in\mathcal{P}_{A}}\int_{U^{\mathcal{I}}}\left(\prod_{I\in\mathcal{I}}\prod_{i\in
I}\phi_{i}(x_{I})\right)F_{\mathcal{I}}(\underline{x}_{\mathcal{I}})\mathrm{d}\underline{x}_{\mathcal{I}}.$

###### Proof.

We have, using the expression of cumulants in terms of moments given by
(2.1.3) and the notation (2.1.1)

$\kappa_{A}(\nu)(\underline{\phi}_{A})=\sum_{\mathcal{I}\in\mathcal{P}_{A}}(|\mathcal{I}|-1)!(-1)^{|\mathcal{I}|-1}\prod_{I\in\mathcal{I}}\mathbb{E}\left[\left\langle\nu^{I},\underline{\phi}_{I}^{\otimes}\right\rangle\right].$

The link between the power measure and factorial power measure given by Lemma
2.2 implies that

$\mathbb{E}\left[\left\langle\nu^{I},\underline{\phi}_{I}^{\otimes}\right\rangle\right]=\sum_{\mathcal{J}\in\mathcal{P}_{I}}\mathbb{E}\left[\left\langle\nu^{[\mathcal{J}]},\underline{\phi}_{I}^{\otimes}\circ\iota_{\mathcal{J}}\right\rangle\right].$

The bijection given by (2.1.3) then implies

$\displaystyle\kappa_{A}(\nu)(\underline{\phi}_{A})$
$\displaystyle=\sum_{\mathcal{I}\in\mathcal{P}_{A}}(|\mathcal{I}|-1)!(-1)^{|\mathcal{I}|-1}\sum_{\mathcal{J}\preceq\mathcal{I}}\prod_{I\in\mathcal{I}}\mathbb{E}\left[\left\langle\nu^{[\mathcal{J}_{I}]},\underline{\phi}_{I}^{\otimes}\circ\iota_{\mathcal{J}_{I}}\right\rangle\right]$
$\displaystyle=\sum_{\mathcal{J}\in\mathcal{P}_{A}}\sum_{\mathcal{I}\succeq\mathcal{J}}(|\mathcal{I}|-1)!(-1)^{|\mathcal{I}|-1}\prod_{I\in\mathcal{I}}\mathbb{E}\left[\left\langle\nu^{[\mathcal{J}_{I}]},\underline{\phi}_{I}^{\otimes}\circ\iota_{\mathcal{J}_{I}}\right\rangle\right]$

Kac–Rice formula then asserts that

$\mathbb{E}\left[\left\langle\nu^{[\mathcal{J}_{I}]},\underline{\phi}_{I}^{\otimes}\circ\iota_{\mathcal{J}_{I}}\right\rangle\right]=\int_{U^{\mathcal{J}_{I}}}\left(\prod_{\begin{subarray}{c}J\in\mathcal{J}\\\
J\subset I\end{subarray}}\prod_{j\in
J}\phi_{j}(x_{J})\right)\rho(\underline{x}_{\mathcal{J}_{I}})\mathrm{d}\underline{x}_{\mathcal{J}_{I}},$

from which we deduce that

$\displaystyle\kappa_{A}(\nu)(\underline{\phi}_{A})$
$\displaystyle=\sum_{\mathcal{J}\in\mathcal{P}_{A}}\int_{U^{\mathcal{J}}}\left(\prod_{I\in\mathcal{I}}\prod_{i\in
I}\phi_{i}(x_{I})\right)\sum_{\mathcal{I}\succeq\mathcal{J}}(|\mathcal{I}|-1)!(-1)^{|\mathcal{I}|-1}\prod_{I\in\mathcal{I}}\rho(\underline{x}_{\mathcal{J}_{\mathcal{I}}})\mathrm{d}\underline{x}_{\mathcal{J}_{\mathcal{I}}}$
$\displaystyle=\sum_{\mathcal{J}\in\mathcal{P}_{A}}\int_{U^{\mathcal{J}}}\left(\prod_{I\in\mathcal{I}}\prod_{i\in
I}\phi_{i}(x_{I})\right)F_{\mathcal{J}}(\underline{x}_{\mathcal{J}})\mathrm{d}\underline{x}_{\mathcal{J}},$

where the last equality follows from the bijection given by (2.1.3). ∎

For instance if $|A|=2$ then the second order cumulant coincides with the
variance and

$\kappa_{A}(\nu)(\underline{\phi}_{A})=\left(\int_{U^{2}}\phi_{1}(x)\phi_{2}(y)\left[\rho_{2}(x,y)-\rho_{1}(x)\rho_{1}(y)\right]\mathrm{d}x\mathrm{d}y\right)+\left(\int_{U}\phi_{1}(x)\phi_{2}(x)\rho_{1}(x)\mathrm{d}x\right).$

###  3.2 Matrix representation of the Kac density and factorization property

In this section we prove a matrix representation for the Kac density and the
cumulant Kac density. It allows us to dissociate the analysis of the
covariance matrix of divided differences associated with the Gaussian process
$f$, and of the Kac density seen as a functional of the covariance matrix.

####  3.2.1 Matrix representation of the Kac density

We define the mapping

$\displaystyle\widetilde{\rho}:\mathcal{M}_{A}(\mathbb{R})\times\mathcal{S}_{2A}^{+}(\mathbb{R})\,$
$\displaystyle\longrightarrow\mathbb{R}$ $\displaystyle M\times\Sigma\quad$
$\displaystyle\longrightarrow\frac{1}{\sqrt{\det\left[(2\pi)^{2}\Sigma\right]}}\int_{\mathbb{R}^{A}}\prod_{a\in
A}\left|(M\underline{x}_{A})_{a}\right|\exp\left(-\frac{{}^{T}\\!\underline{x}_{A}(\Sigma^{c})^{-1}\underline{x}_{A}}{2}\right)\mathrm{d}\underline{x}_{A}.$

Recall from definition (2.4.2) that

$\Sigma^{\mathcal{I}}(\underline{x}_{A})=\operatorname{Cov}(f_{\mathcal{I}}[\underline{x}_{A}])\quad\text{and}\quad\Sigma^{\mathcal{I}}(\underline{x}_{A,A})=\operatorname{Cov}(f_{\mathcal{I}}[\underline{x}_{A,A}]).$

The following lemma gives an alternative expression of $\rho$ as a function of
the covariance matrix $\Sigma^{\mathcal{I}}(\underline{x}_{A,A})$, and the
matrix of divided differences $M^{\mathcal{I}}(\underline{x}_{A})$ defined in
(2.4.2).

######  Lemma 3.6.

For $\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$,
$\rho(\underline{x}_{A})=\left|\det
M^{\mathcal{I}}(\underline{x}_{A})\right|\widetilde{\rho}\left(M^{\mathcal{I}}(\underline{x}_{A}),\Sigma^{\mathcal{I}}(\underline{x}_{A,A})\right).$

###### Proof.

Note first that Remark 3.2 and Lemma 3.3 implies that

$\rho(\underline{x}_{A})=\left|\det
Q^{\mathcal{I},\overline{A}}(\underline{x}_{A})\right|\rho^{\mathcal{I}}(\underline{x}_{A})=\left|\det
M^{\mathcal{I}}(\underline{x}_{A})\right|\rho^{\mathcal{I}}(\underline{x}_{A}).$

Let $I\in\mathcal{I}$. In virtue of Lemma 2.11, one has

$f^{[\underline{x}_{I}]}_{I}[\underline{x}_{I}]=(f[\underline{x}_{I},x_{i|I}])_{i\in
I}\quad\text{and
thus}\quad\left(f_{I}[\underline{x}_{I}],f^{[\underline{x}_{I}]}_{I}[\underline{x}_{I}]\right)=f_{2I}[\underline{x}_{I,I}].$

Following the notations of Section 2.3 it implies that

$\Sigma^{\mathcal{I}}(\underline{x}_{A,A})^{11}=\operatorname{Cov}(f_{\mathcal{I}}[\underline{x}_{A}]).$

From Equation (2.3.1), one has

$\det\left[(2\pi)^{2}\Sigma^{\mathcal{I}}(\underline{x}_{A,A})\right]=\det\left[2\pi\Sigma^{\mathcal{I}}(\underline{x}_{A,A})^{11}\right]\det\left[2\pi\Sigma^{\mathcal{I}}(\underline{x}_{A,A})^{c}\right].$

Using the alternative expression of $N^{\mathcal{I}}$ given by Lemma 3.4 and
the conditional formula of Lemma 2.3, we deduce

$\displaystyle\rho^{\mathcal{I}}(\underline{x}_{A})$
$\displaystyle=\frac{1}{\sqrt{\det\Sigma^{\mathcal{I}}(\underline{x}_{A,A})^{11}}}\mathbb{E}\left[\prod_{I\in\mathcal{I}}\prod_{i\in
I}\left|\left(M(\underline{x}_{I})f^{[\underline{x}_{I}]}_{I}[\underline{x}_{I}]\right)_{i}\vphantom{\int}\right|\;\middle|\;f_{\mathcal{I}}[\underline{x}_{A}]=0\right]$
$\displaystyle=\frac{1}{\sqrt{\left[(2\pi)^{2}\Sigma^{\mathcal{I}}(\underline{x}_{A,A})\right]}}\int_{\mathbb{R}^{A}}\prod_{a\in
A}\left|(M^{\mathcal{I}}(\underline{x}_{A})\underline{x}_{A})_{a}\right|\exp\left(-\frac{{}^{T}\\!\underline{x}_{A}(\Sigma^{\mathcal{I}}(\underline{x}_{A,A})^{c})^{-1}\underline{x}_{A}}{2}\right)\mathrm{d}\underline{x}_{A}$
$\displaystyle=\widetilde{\rho}\left(M(\underline{x}_{I}),\Sigma^{\mathcal{I}}(\underline{x}_{A,A})\right).$

The conclusion follows. ∎

######  Lemma 3.7.

Let $\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$. For a subset $B$ of $A$,
$\rho(\underline{x}_{B})=\left|\det
M^{\mathcal{I}_{B}}(\underline{x}_{B})\right|\widetilde{\rho}\left(M^{\mathcal{I}_{B}}(\underline{x}_{B}),Q^{\mathcal{I},2B}(\underline{x}_{A,A})\Sigma^{\mathcal{I}}(\underline{x}_{A,A})\,\\!^{T}\\!Q^{\mathcal{I},2B}(\underline{x}_{A,A})\right).$

###### Proof.

Recall that the partition $\mathcal{I}_{B}$ of $B$ is the partition induced by
the partition $\mathcal{I}$ on the subset $B$. If
$\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$ then
$\underline{x}_{B}\in\Delta_{\mathcal{I}_{B}^{+}}$. We can thus apply the
previous Lemma 3.6 to get

$\rho(\underline{x}_{B})=\left|\det
M^{\mathcal{I}_{B}}(\underline{x}_{B})\right|\widetilde{\rho}\left(M^{\mathcal{I}_{B}}(\underline{x}_{B}),\Sigma^{\mathcal{I}_{B}}[\underline{x}_{B}]\right).$

From Equation (2.4.2),

$f_{\mathcal{I}_{B}}[\underline{x}_{B,B}]=Q^{\mathcal{I},2B}(\underline{x}_{A,A})f_{\mathcal{I}}[\underline{x}_{A,A}],$

thus

$\Sigma^{\mathcal{I}_{B}}[\underline{x}_{B}]=Q^{\mathcal{I},2B}(\underline{x}_{A,A})\Sigma^{\mathcal{I}}(\underline{x}_{A,A})\,\\!^{T}\\!Q^{\mathcal{I},2B}(\underline{x}_{A,A}).$

∎

Given two open subsets $\Omega_{1}$ and $\Omega_{2}$ of finite dimensional
vector spaces, we define the function space
$\mathcal{C}^{0,\infty}(\Omega_{1},\Omega_{2})$ to be the set of functions
from $\Omega_{1}\times\Omega_{2}$ to $\mathbb{R}$, that are infinitely
differentiable with respect to the second argument and such that the partial
derivatives (with respect to the second argument) are continuous with respect
to the first argument. Note that this space can be turned into a Frechet space
with the semi-norms

$\forall
K\subset\Omega_{1}\times\Omega_{2}\;\text{compact},\;\forall\alpha\in\mathbb{N}^{\dim\Omega_{2}},\quad\quad\|f\|_{\alpha,K}=\sup_{(x,y)\in
K}|\partial_{2}^{\alpha}f(x,y)|.$

######  Lemma 3.8.

The application $\widetilde{\rho}$ belongs to
$\mathcal{C}^{0,\infty}(\mathcal{M}_{A}(\mathbb{R}),\mathcal{S}_{2A}^{+}(\mathbb{R}))$.

###### Proof.

Let

$h(\Sigma,\underline{x}_{A})=\frac{1}{\sqrt{\det\left[(2\pi)^{2}\Sigma\right]}}\exp\left(-\frac{{}^{T}\\!\underline{x}_{A}(\Sigma^{c})^{-1}\underline{x}_{A}}{2}\right).$

The function $\Sigma\mapsto h(\Sigma,\underline{x}_{A})$ is infinitely
differentiable on $\mathcal{S}_{2A}^{+}(\mathbb{R})$ and its partial
derivatives are also exponentially decreasing with respect to the variable
$\underline{x}_{A}$. By differentiability under the integral, it implies that
$\widetilde{\rho}$ belongs to
$\mathcal{C}^{0,\infty}(\mathcal{M}_{A}(\mathbb{R}),\mathcal{S}_{2A}^{+}(\mathbb{R}))$.
∎

####  3.2.2 Factorization of the Kac density and error term

In this section, we show that the function $\widetilde{\rho}$ satisfies a nice
factorization property. This is a rigorous statement of the approximation
(1.2) stated in introduction. For the rest of this section, $\mathcal{I}$ is a
partition of the set $A$, $M$ is a matrix of $\mathcal{M}_{A}(\mathbb{R})$ and
$\Sigma$ is a matrix of $\mathcal{S}_{2A}^{+}(\mathbb{R})$ such that

$M=M_{\mathcal{I}}\quad\text{and}\quad\Sigma=\Sigma_{\mathcal{I}}.$

######  Lemma 3.9.

$\widetilde{\rho}(M,\Sigma)=\prod_{I\in\mathcal{I}}\widetilde{\rho}(M_{I},\Sigma_{I}).$

###### Proof.

Since the matrix $\Sigma$ is block diagonal with respect to the partition
$\mathcal{I}$,

$\sqrt{\det\left[(2\pi)^{2}\Sigma\right]}=\prod_{I\in\mathcal{I}}\sqrt{\det\left[(2\pi)^{2}\Sigma_{I}\right]}.$

Similarly, for $\underline{x}_{A}\in\mathbb{R}^{A}$,

$\exp\left(-\frac{{}^{T}\\!\underline{x}_{A}(\Sigma^{c})^{-1}\underline{x}_{A}}{2}\right)=\prod_{I\in\mathcal{I}}\exp\left(-\frac{{}^{T}\\!\underline{x}_{I}((\Sigma_{I})^{c})^{-1}\underline{x}_{I}}{2}\right).$

The matrix $M_{\mathcal{I}}$ is also block diagonal with respect to the
partition $\mathcal{I}$ and

$\prod_{a\in A}|(M\underline{x}_{A})_{a}|=\prod_{I\in\mathcal{I}}\prod_{a\in
A}|M_{I}\underline{x}_{I}|.$

The conclusion follows from the definition of $\widetilde{\rho}$. ∎

We want to describe the error term in Lemma 3.9 when the matrix $\Sigma$ is
not block diagonal with respect to the partition $\mathcal{I}$. We start with
the following lemma.

######  Lemma 3.10.

Let $H$ be a matrix of $\mathcal{S}_{2A}(\mathbb{R})$, such that
$H_{\mathcal{I}}=0$. Then
$\widetilde{\rho}(M,\Sigma+H)=\widetilde{\rho}(M,\Sigma)+O(\|H\|^{2}).$

###### Proof.

For such matrix $H$ small enough, the matrix $\Sigma+H$ belongs to
$\mathcal{S}_{2A}^{+}(\mathbb{R})$. The matrix $\Sigma$ is block diagonal,
whereas the matrix $H$ satisfies $H_{\mathcal{I}}=0$. It implies that

$(\Sigma^{-1}H)_{\mathcal{I}}=(\Sigma^{-1}H\Sigma^{-1})_{\mathcal{I}}=0\quad\text{and
thus}\quad\operatorname{Tr}(\Sigma^{-1}H)=0.$

One has from identity (2.3.1)

$\displaystyle[(\Sigma+H)^{c}]^{-1}$ $\displaystyle=[(\Sigma+H)^{-1}]^{22}$
$\displaystyle=(\Sigma^{c})^{-1}+H_{\Sigma}+O(\|H\|^{2}),$

where $H_{\Sigma}=[\Sigma^{-1}H\Sigma^{-1}]^{22}$. By (3.2.2), one has
$(H_{\Sigma})_{\mathcal{I}}=0$. Differentiation under the integral sign gives

$\widetilde{\rho}(M,\Sigma+H)=\widetilde{\rho}(M,\Sigma)+\mathrm{d}\widetilde{\rho}_{(M,\Sigma)}.H+O(\|H\|^{2}),$

where

$\displaystyle\mathrm{d}\widetilde{\rho}_{(M,\Sigma)}.H$
$\displaystyle=-\frac{1}{2\sqrt{\det\left[(2\pi)^{2}\Sigma\right]}}\int_{\mathbb{R}^{A}}\prod_{a\in
A}\left|(M\underline{x}_{A})_{a}\right|\left({}^{T}\\!\underline{x}_{A}H_{\Sigma}\underline{x}_{A}\right)\exp\left(-\frac{{}^{T}\\!\underline{x}_{A}(\Sigma^{c})^{-1}\underline{x}_{A}}{2}\right)\mathrm{d}\underline{x}_{A}$
$\displaystyle=-\sum_{\begin{subarray}{c}i,j\in A\\\
[i]_{\mathcal{I}}\neq[j]_{\mathcal{I}}\end{subarray}}\frac{(H_{\Sigma})_{ij}}{2\sqrt{\det\left[(2\pi)^{2}\Sigma\right]}}\int_{\mathbb{R}^{A}}\prod_{a\in
A}\left|(M\underline{x}_{A})_{a}\right|x_{i}x_{j}\exp\left(-\frac{{}^{T}\\!\underline{x}_{A}(\Sigma^{c})^{-1}\underline{x}_{A}}{2}\right)\mathrm{d}\underline{x}_{A}.$

For each $i,j$ of the sum we make the change of variable

$\forall a\in A,\quad
y_{a}=\left\\{\begin{array}[]{ll}-x_{a}\;\text{if}\;[a]_{\mathcal{I}}=[i]_{\mathcal{I}}\\\
\;\;\;x_{a}\;\text{if}\;[a]_{\mathcal{I}}\neq[i]_{\mathcal{I}}\end{array}\right..$

Since $M$ and $\Sigma^{c}$ are block diagonal matrices , one has for $a\in A$
and $\underline{x}_{A}\in\mathbb{R}^{A}$

$|(M\underline{y}_{A})_{a}|=|(M\underline{x}_{A})_{a}|,\quad^{T}\\!\underline{y}_{A}(\Sigma^{c})^{-1}\underline{y}_{A}=^{T}\\!\underline{x}_{A}(\Sigma^{c})^{-1}\underline{x}_{A}\quad\text{but}\quad
y_{i}y_{j}=-x_{i}x_{j}.$

Thus

$\mathrm{d}\widetilde{\rho}_{(M,\Sigma)}.H=-\mathrm{d}\widetilde{\rho}_{(M,\Sigma)}.H=0,$

and the conclusion follows. ∎

We can now state the following proposition that gives the error in Lemma 3.9
when the matrix $\Sigma$ is not block diagonal with respect to the partition
$\mathcal{I}$. Note that the following Lemma gives a quadratic error in the
matrix coefficients of $\Sigma$, where in the somehow analogous Proposition
[1, prop. 6.43] only proves a square root error. The difference resides in
Lemma 3.4, which allows us to bypass the lack of regularity of Gaussian
integrals near the boundary of the cone of symmetric definite matrices.

######  Corollary 3.11.

Let $B$ be a subset of $A$ and $K$ be a compact subset of
$\mathcal{M}_{B}(\mathbb{R})\times\mathcal{M}_{2A,2B}^{*}(\mathbb{R})\times\mathcal{S}_{2A}^{+}(\mathbb{R})$.
Then there is a constant $C_{K}$ such that for all $(M,Q,\Sigma)\in K$ such
that $M=M_{\mathcal{I}}$ and $Q=Q_{\mathcal{I}}$, one has
$\left|\widetilde{\rho}(M,Q\Sigma\,\\!^{T}\\!Q)-\prod_{I\in\mathcal{I}}\widetilde{\rho}(M_{I},Q_{I}\Sigma_{I}\,\\!^{T}\\!Q_{I})\right|\leq
C_{K}\sup_{\begin{subarray}{c}I,J\in\mathcal{I}\\\ I\neq
J\end{subarray}}\|\Sigma_{I,J}\|^{2}.$

###### Proof.

Let $\Pi=Q\Sigma\,\\!^{T}\\!Q$. Lemma 2.6 asserts that the couple $(M,\Pi)$
lives in a compact set of
$\mathcal{M}_{B}(\mathbb{R})\times\mathcal{S}_{2A}^{+}(\mathbb{R})$. From
Lemma 3.10, one has

$\left|\widetilde{\rho}(M,\Pi)-\widetilde{\rho}(M,\Pi_{\mathcal{I}})\right|=O(\|\Pi-\Pi_{\mathcal{I}}\|^{2}).$

By Lemma 3.8, the application $\widetilde{\rho}$ belongs to
$\mathcal{C}^{0,\infty}(\mathcal{M}_{B}(\mathbb{R})\times\mathcal{S}_{2B}^{+}(\mathbb{R}))$.
Lagrange rest formula asserts the existence of a constant $C_{K}$ such that

$\left|\widetilde{\rho}(M,\Pi)-\widetilde{\rho}(M,\Pi_{\mathcal{I}})\right|\leq
C_{K}\|\Pi-\Pi_{\mathcal{I}}\|^{2}\leq
C_{K}\sup_{\begin{subarray}{c}I,J\in\mathcal{I}\\\ I\neq
J\end{subarray}}\|\Pi_{I,J}\|^{2}.$

Since $Q=Q_{\mathcal{I}}$ and $\|Q\|\leq C_{K}$ for some constant $C_{K}$, we
deduce

$\|\Pi_{I,J}\|=\|Q_{I}\Sigma_{I,J}\,\\!^{T}\\!Q_{J}\|\leq
C_{K}\|\Sigma_{I,J}\|.$

Finally the conclusion follows from Lemma 3.9 since

$\widetilde{\rho}(M,\Pi_{\mathcal{I}})=\prod_{I\in\mathcal{I}}\widetilde{\rho}(M_{I},\Pi_{I})=\prod_{I\in\mathcal{I}}\widetilde{\rho}(M_{I},Q_{I}\Sigma_{I}\,\\!^{T}\\!Q_{I}).$

∎

####  3.2.3 Matrix representation of the cumulant Kac density

Similarly to the Kac density, we can derive a matrix representation for the
cumulant Kac density defined in 40. Note that the divided differences does not
behave well when we consider them on a subfamily of interpolations points
$(\underline{x}_{A})_{a\in A}$. It explains why we introduced in Paragraph
2.3.5 the somehow complicated set
$\widetilde{\mathcal{M}}_{A}^{*}(\mathbb{R})$. We introduce the function
$\widetilde{F}_{A}$ defined by

$\displaystyle\widetilde{F}_{A}:\widetilde{\mathcal{M}}_{A}^{*}(\mathbb{R})\times\mathcal{S}_{2A}^{+}(\mathbb{R})\longrightarrow\mathbb{R}$
$\displaystyle(\widetilde{M},\widetilde{Q})\times\Sigma\longrightarrow\sum_{\mathcal{J}\in\mathcal{P}_{A}}(|\mathcal{J}|-1)!(-1)^{|\mathcal{J}|-1}\prod_{J\in\mathcal{J}}\left|\det
M^{J}\right|\widetilde{\rho}\,(M^{J},Q^{J}\Sigma\,\\!^{T}Q^{J}).$

Let $\mathcal{I}$ be a partition of $A$. The following lemma gives an
alternative expression to the function $F_{A}$ when
$\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$.

######  Lemma 3.12.

For $\underline{x}_{A}\in\Delta_{\mathcal{I}^{+}}$ one has
$F_{A}(\underline{x}_{A})=\widetilde{F}_{A}\left((\widetilde{M}^{\mathcal{I}}(\underline{x}_{A}),\widetilde{Q}^{\mathcal{I}}(\underline{x}_{A})),\Sigma^{\mathcal{I}}(\underline{x}_{A,A})\right).$

###### Proof.

One has

$F_{A}(\underline{x}_{A})=\sum_{\mathcal{J}\in\mathcal{P}_{A}}(|\mathcal{J}|-1)!(-1)^{|\mathcal{J}|-1}\prod_{J\in\mathcal{J}}\rho(\underline{x}_{J}).$

According to lemma 3.7, for a subset $J$ of $A$ one has

$\rho(\underline{x}_{J})=\left|\det
M^{\mathcal{I}_{J}}(\underline{x}_{J})\right|\widetilde{\rho}\left(M^{\mathcal{I}_{J}}(\underline{x}_{J}),Q^{\mathcal{I},2J}(\underline{x}_{A,A})\Sigma^{\mathcal{I}}(\underline{x}_{A,A})\,\\!^{T}\\!Q^{\mathcal{I},2J}(\underline{x}_{A,A})\right).$

The first statement follows from the definition (2.4.2) of
$\widetilde{M}^{\mathcal{I}}(\underline{x}_{A})$ and
$\widetilde{Q}^{\mathcal{I}}(\underline{x}_{A})$. ∎

Corollary 3.11 translates directly into the following bound for the function
$\widetilde{F}_{A}$.

######  Lemma 3.13.

Let $\mathcal{I}$ be a partition of $A$, with $\mathcal{I}\neq\\{A\\}$. Then
there is a constant $C_{K}$ such that for all
$((\widetilde{M},\widetilde{Q}),\Sigma)\in K$ with
$\widetilde{M}=\widetilde{M}_{\mathcal{I}}\quad\widetilde{Q}=\widetilde{Q}_{\mathcal{I}},$
one has
$\left|\widetilde{F}_{A}((\widetilde{M},\widetilde{Q}),\Sigma)\right|\leq
C_{K}\sup_{\begin{subarray}{c}I,J\in\mathcal{I}\\\ I\neq
J\end{subarray}}\|\Sigma_{I,J}\|^{2}.$

###### Proof.

The factorization property of the function $\widetilde{\rho}$ given by Lemma
3.9 and the cumulant cancellation property given by Lemma 2.1 imply that

$\widetilde{F}_{A}((\widetilde{M},\widetilde{Q}),\Sigma_{\mathcal{I}})=0.$

The error term given by Corollary 3.11 translates directly for the function
$\widetilde{F}_{A}$ to the desired estimate. ∎

###  3.3 Decay of the cumulant Kac density

The goal of the following section is to enhance the quadratic bound given by
Lemma 3.13. We will do so, thanks to a refinement of Taylor expansion for
regular functions that cancel on given affine subspaces. The next key Lemma
3.22 bounds the function $\widetilde{F}_{A}$ by a sum over a collection of
graphs. We recall first a few definitions and propositions from graph theory.

####  3.3.1 Graph setting

A graph $G$ is a couple $(E(G),V(G))$, where $E(G)$ is the set of vertices of
the graph $G$ and $V(G)$ the collection of edges of $G$. For our purposes, a
graph $G$ has no loop, but two different edges can have the same endpoints.
The multiplicity of an edge $\\{a,b\\}$ is the number of edges in the graph
that are equal to $\\{a,b\\}$. The edge connectivity of a graph $G$ is the
largest integer $k$ such that the graph $G$ remains connected when $k-1$ edges
are removed from the graph $G$. We define $\mathcal{G}_{A}$ the set of graphs
with set of vertices $A$ and edge connectivity $2$.

Let $\mathcal{I}$ be partition of $A$ and let $G$ be a graph with set of
vertices $A$. We define the graph $G_{\mathcal{I}}$ on the set of vertices
$\mathcal{I}$ to be the quotient graph with respect to the partition
$\mathcal{I}$. That is, the multiplicity of the edge $\\{I,J\\}$ (with $I\neq
J$) of $G_{\mathcal{I}}$ is the number of edges $\\{i,j\\}$ in $G$ (with
multiplicity) such that $\\{I,J\\}=\\{[i]_{\mathcal{I}},[j]_{\mathcal{I}}\\}$.

######  Lemma 3.14.

Let $H\in\mathcal{G}_{\mathcal{I}}$. There is $G\in\mathcal{G}_{A}$ such that
$H=\mathcal{G}_{\mathcal{I}}.$

###### Proof.

For each $I\in\mathcal{I}$, we replace in $H$ the vertex $I$ by the cycle
$(i)_{i\in I}$. The neighbors of $I$ are arbitrary linked to vertices of this
cycle. The obtained graph with set of vertices $A$ satisfies
$G_{\mathcal{I}}=H$ and will also have edge connectivity $2$. ∎

An ear of a graph $G$ is a path in $G$ such that its internal vertices all
have degree two. An ear decomposition of the graph $G$ is a union
$(P_{1},\ldots,P_{k})$ such that $P_{0}$ is a cycle, and for $i\geq 1$,
$P_{i}$ is an ear such that its endpoints belong to $\cup_{j<i}P_{i}$. We
states the following standard fact for $2$-edge connected graphs (see [20]).
The proof is a simple induction on the number of ears.

######  Lemma 3.15.

A $2$-edge connected graph admits an ear decomposition. The number of ears is
necessary the circuit rank of the graph $G$. Moreover, the starting cycle can
be chosen arbitrarily among the cycles of $G$. It implies the following lemma.

######  Lemma 3.16.

Let $G$ be a $2$-edge connected graph. There is a family $(T_{a})_{a\in A}$ of
spanning trees of $G$ such that $a$ is a leaf of the tree $T_{a}$, and for
every edge $e\in E(G)$, one can find an element $a_{e}\in A$ such that $e$ is
not an edge of the spanning tree $T_{a_{e}}$.

###### Proof.

Let $P_{1}$ be a largest cycle in $G$, with vertices $B$, and
$(P_{1},\ldots,P_{k})$ an ear decomposition of $G$. For $i\geq 1$, we define
$E_{i}$ the set of edges of the path $P_{i}$. One has $|B|\geq|E_{i}|$, so
that one can find a surjection $\tau_{i}:B\twoheadrightarrow E_{i}$, and we
can ask that the edge $\tau_{1}(a)$ is adjacent to the vertex $A$.

For $a\notin B$, we define $T_{a}$ to be any spanning tree of $G$ that does
not contain an edge adjacent to $a$. This operation is possible since the
graph $G$ is $2$-edge connected. For $a\in B$, we define $T_{a}$ to be the the
graph $G$ where we removed one edge of the cycle $P_{1}$ adjacent to $a$, and
in each path $E_{i}$ we removed the edge $\tau_{i}(a)$. The number $k$ is the
circuit rank of $G$. Since the graph $T_{a}$ is easily seen to be connected,
it must be a spanning tree of the graph $G$. By construction, for every $a\in
A$, the vertex $a$ is a leaf of $T_{a}$, and every edge $e\in E_{i}$ is the
image of some element $a_{e}\in B$ by the application $\tau_{i}$, so that the
edge $e$ do not belong to the tree $T_{a_{e}}$.

∎

####  3.3.2 Crossed Taylor formula

In this paragraph we prove an enhancement of the Taylor rest estimates for
regular functions that cancel on affine subspaces. A simple observation of
this phenomenon is the following. Assume that $F(x,y)$ is a regular function
such that in a neighborhood of zero,

$|F(x,y)|\leq|x|\quad\text{and}\quad|F(x,y)|\leq|y|.$

Then for some constant $C$, one has in a neighborhood of zero that

$|F(x,y)|\leq C|xy|,$

which improves by a square factor the trivial bound $\sqrt{|xy|}$. We extend
this observation to the more complicated function $\widetilde{F}_{A}$ that
satisfies the bounds given by Lemma 3.13 for several partitions $\mathcal{I}$
of $A$.

Let $\Omega$ be an open subset of a finite dimensional vector space
$V\simeq\mathbb{R}^{E}$. Let $F\in\mathcal{C}^{\infty}(\Omega)$ be a function
bounded for the Frechet topology on $\mathcal{C}^{\infty}(\Omega)$, and
$\underline{y}_{E}$ be a vector in $V$. We fix an integer $d\in\mathbb{N}$.
The following lemma sates equivalent conditions for a regular function $F$ to
cancel on an affine subspace with order of cancellation equal to $d$.

######  Lemma 3.17.

Let $B$ be a subset of $E$. Then the three following conditions are
equivalent. 1. For every closed compact subset $K$ of $\Omega$, there is a
constant $C_{K}$ such that, $\forall\underline{x}_{E}\in
K,\quad|F(\underline{x}_{E})|\leq C_{K}\left(\sup_{b\in
B}|x_{b}-y_{b}|\right)^{d}.$ 2. For every multiset
$\mathsf{C}\in\mathcal{P}^{m}(B)$ with $|\mathsf{C}|=d$, there exists a
function $H_{\mathsf{C}}$ bounded in $\mathcal{C}^{\infty}(\Omega)$, such that
$\forall\underline{x}_{E}\in\Omega,\quad
F(\underline{x}_{E})=\sum_{\begin{subarray}{c}\mathsf{C}\in\mathcal{P}^{m}(B)\\\
|\mathsf{C}|=d\end{subarray}}\prod_{c\in\mathsf{C}}(x_{c}-y_{c})\;H_{\mathsf{C}}(\underline{x}_{E}).$
3. For all $\underline{w}_{E}\in\Omega$ such that
$\underline{w}_{B}=\underline{y}_{B}$, for all multiset
$\mathsf{C}\in\mathcal{P}^{m}(B)$ with $|\mathsf{C}|<d$,
$\partial_{\mathsf{C}}F(\underline{w}_{E})=0.$

###### Proof.

$\bullet$ The implication $(2)\Rightarrow(1)$ follows from the boundedness of
the functions $H_{\mathsf{C}}$ on $K$.

$\bullet$ The implication $(1)\Rightarrow(3)$ is a consequence of the
definition of partial derivatives.

$\bullet$ As for the implication $(3)\Rightarrow(2)$, It suffices to prove
show that $(2)$ holds when $\underline{x}_{E}$ belongs to a compact subset $K$
of $\mathbb{R}^{E}$, where $K$ is a product of compact intervals. The general
case follows from an exhaustion of open and relatively compacts set covering
$\Omega$ and a partition of unity. We prove this implication by induction on
$d$. If $d=0$, this is trivial : we take $H_{\text{\O}}=F$. Now assume that
the proposition is true for some $d\in\mathbb{N}$, and let us prove it for
$d+1$. Let $\underline{w}_{E}\in K$ such that
$\underline{w}_{B}=\underline{y}_{B}$. Then for a multiset
$\mathsf{D}\in\mathcal{P}^{m}(B)$ with $|\mathsf{D}|=d$,

$\displaystyle\partial_{\mathsf{D}}F(\underline{w}_{E})$
$\displaystyle=\sum_{\begin{subarray}{c}\mathsf{C}\subset B\\\
|\mathsf{C}|=d\end{subarray}}\partial_{\mathsf{D}}\left(\prod_{c\in\mathsf{C}}(\,.\,-y_{c})\;H_{\mathsf{C}}(\,.\,)\right)(\underline{w}_{E})$
$\displaystyle=\partial_{\mathsf{D}}\left(\prod_{c\in\mathsf{D}}(\,.\,-\underline{y}_{c})\right)(\underline{w}_{E})H_{\mathsf{D}}(\underline{w}_{E})$
$\displaystyle=M_{\mathsf{D}}H_{\mathsf{D}}(\underline{w}_{E})$
$\displaystyle=0,$

where $M_{\mathsf{D}}$ is a positive constant involving the number of
repetitions of elements in $\mathsf{D}$. We deduce that for all
$\underline{w}_{E}\in K$ such that $\underline{w}_{B}=\underline{y}_{B}$,

$H_{\mathsf{D}}(\underline{w}_{E})=0.$

Let $\underline{x}_{E}\in K$ and $\mathsf{C}\in\mathcal{P}^{m}(B)$. There are
two possibilities.

  * •

If the compact cube $K$ contains an element $\underline{w}_{E}$ such that
$\underline{w}_{B}=\underline{y}_{B}$, then it also contains the point
$(\underline{w}_{B},\underline{x}_{E\setminus B})$. Taylor development with
integral rest implies the existence of functions
$(H_{\mathsf{C}\sqcup\\{\\!\\!\\{b\\}\\!\\!\\}})_{b\in B}$ bounded in
$\mathcal{C}^{\infty}(\Omega)$ such that for all $\underline{x}_{E}\in K$

$H_{\mathsf{C}}(\underline{x}_{E})=\sum_{b\in
B}(x_{b}-y_{b})H_{\mathsf{C}\sqcup\\{\\!\\!\\{b\\}\\!\\!\\}}(\underline{x}_{E}).$

  * •

If the compact cube $K$ does not contain any such element $\underline{w}_{E}$
such that $\underline{w}_{B}=\underline{y}_{B}$, then there is $c\in B$ such
that for all $\underline{x}_{E}\in K$,
$\underline{x}_{c}\neq\underline{y}_{c}$. By compactness one can find a
positive constant $\varepsilon_{K}$ such that for all $\underline{x}_{E}\in
K$, $|\underline{x}_{c}-y_{c}|\geq\varepsilon_{K}$. In that case we define

$\forall b\in B,\forall\underline{x}_{E}\in\Omega,\quad
H_{\mathsf{C}\sqcup\\{\\!\\!\\{b\\}\\!\\!\\}}(\underline{x}_{E})=\left\\{\begin{array}[]{ll}(x_{c}-y_{c})^{-1}H_{\mathsf{C}}(\underline{x}_{E})\chi_{\varepsilon_{K}}(\underline{x}_{E})\;\;\text{if}\;b=c\\\
0\;\;\text{else}\end{array}\right.,$

where $\chi_{\varepsilon_{K}}$ is a function in $\mathcal{C}^{\infty}(\Omega)$
that is constant equal to one on $K$, and is zero outside
$K+B(0,\varepsilon_{K}/2)$. The functions
$(H_{\mathsf{C}\sqcup\\{\\!\\!\\{b\\}\\!\\!\\}})_{b\in B}$ are bounded in
$\mathcal{C}^{\infty}(\Omega)$, and satisfy

$H_{\mathsf{C}}(\underline{x}_{E})=\sum_{b\in
B}(x_{b}-y_{b})H_{\mathsf{C}\sqcup\\{\\!\\!\\{b\\}\\!\\!\\}}(\underline{x}_{E}).$

In both cases we deduce that

$\displaystyle F(\underline{x}_{E})$
$\displaystyle=\sum_{\begin{subarray}{c}\mathsf{C}\in\mathcal{P}^{m}(B)\\\
|\mathsf{C}|=d\end{subarray}}\prod_{c\in\mathsf{C}}(x_{c}-y_{c})\left(\sum_{b\in
B}(x_{b}-y_{b})H_{\mathsf{C}\sqcup\\{\\!\\!\\{b\\}\\!\\!\\}}(\underline{x}_{E})\right)$
$\displaystyle=\sum_{\begin{subarray}{c}\mathsf{C}\in\mathcal{P}^{m}(B)\\\
|\mathsf{C}|=d+1\end{subarray}}\prod_{c\in\mathsf{C}}(x_{c}-y_{c})\;H_{\mathsf{C}}(\underline{x}_{E}),$

which concludes the induction. ∎

Now we extend the previous Lemma 3.17 to a collection $\mathcal{B}$ of (not
necessarily disjoints) subsets of $E$. We use the notations of multisets
defined in Paragraph 2.1.2. For a multiset $\mathsf{C}\in\mathcal{P}^{m}(E)$
we define

$I_{\mathsf{C}}(\mathcal{B})=\min_{B\in\mathcal{B}}|\mathsf{C}_{B}|\quad\text{and}\quad\mathcal{C}_{\mathcal{B}}=\left\\{\mathsf{C}\subset
E\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
I_{\mathsf{C}}(\mathcal{B})=d\quad\text{and}\quad\forall
c\in\mathsf{C},\;I_{\mathsf{C}\setminus\\{c\\}}(\mathcal{B})=d-1\right\\}.$

The set $\mathcal{C}_{\mathcal{B}}$ comprises of all the possible multisets
$\mathsf{C}\in\mathcal{P}^{m}(E)$ such that for every $B\in\mathcal{B}$, one
can find at least $d$ elements in $\mathsf{C}$ that are also in $B$, with the
additional requirement that this collection is minimal. If we remove any
element of $\mathsf{C}$, one can find a $B\in\mathcal{B}$ such that
$|\mathsf{C}_{B}|=d-1$.

######  Remark 3.18.

This last minimality assumption implies in particular that for a multiset
$\mathsf{C}\in\mathcal{C}_{\mathcal{B}}$, the multiplicity of an element in
$\mathsf{C}$ is at most $d$. In particular, the set $\mathsf{C}$ has finite
cardinal, as well as the collection $\mathcal{C}_{\mathcal{B}}$.

For instance, if $\mathcal{B}=\\{\\{1,2\\},\\{2,3\\},\\{1,3\\}\\}$ and $d=2$,
then

$\mathcal{C}_{\mathcal{B}}=\left\\{\\{\\!\\!\\{1,1,2,2\\}\\!\\!\\},\\{\\!\\!\\{2,2,3,3\\}\\!\\!\\},\\{\\!\\!\\{1,1,3,3\\}\\!\\!\\},\\{\\!\\!\\{1,2,3\\}\\!\\!\\}\right\\}.$

######  Proposition 3.19.

Assume that for every $B\in\mathcal{B}$, the function $F$ satisfies the
equivalent propositions of Lemma 3.17, Then there exists functions
$(H_{\mathsf{C}})_{\mathsf{C}\in\mathcal{C}_{\mathcal{B}}}$ bounded in
$\mathcal{C}^{\infty}(\Omega)$ such that
$F(\underline{x}_{E})=\sum_{\mathsf{C}\in\mathcal{C}_{\mathcal{B}}}\prod_{c\in\mathsf{C}}(x_{c}-y_{c})H_{\mathsf{C}}(\underline{x}_{E}).$

###### Proof.

The proof is a induction on the size of the set $\mathcal{B}$. If
$\mathcal{B}=\\{B\\}$, this exactly the hypothesis on $F$ (second
characterization), since in that case $\mathcal{C}_{\mathcal{B}}$ comprises of
all the multisets in $\mathcal{P}^{m}(B)$ of size $d$. Now let
$B\in\mathcal{B}$ and suppose that the lemma is true for the family
$\mathcal{B}\setminus\\{B\\}$. We have

$F(\underline{x}_{E})=\sum_{\mathsf{C}\in\mathcal{C}_{\mathcal{B}\setminus\\{B\\}}}\prod_{c\,\in\mathsf{C}}(x_{c}-y_{c})H_{\mathsf{C}}(\underline{x}_{E}).$

Let $\mathsf{C}\in\mathcal{C}_{\mathcal{B}\setminus\\{B\\}}$. If
$|\mathsf{C}_{B}|\geq d$ then $\mathsf{C}\in\mathcal{C}_{\mathcal{B}}$. If
$|\mathsf{C}_{B}|=d-k$ with some $k\in\\{1,\ldots,d\\}$, observe that the
function $H_{\mathsf{C}}$ satisfies Lemma 3.17 with integer $d-k$ and set $B$.
In implies that

$H_{\mathsf{C}}(\underline{x}_{E})=\sum_{\begin{subarray}{c}\mathsf{D}\in\mathcal{P}^{m}(B)\\\
|\mathsf{D}|=k\end{subarray}}\prod_{d\in\mathsf{D}}(x_{d}-y_{d})H_{\mathsf{C}\sqcup\mathsf{D}}(\underline{x}_{E}),$

where for $\mathsf{D}\in\mathcal{P}^{m}(B)$ with $|\mathsf{D}|=k$, the
function $H_{\mathsf{C}\sqcup\mathsf{D}}$ is bounded in
$\mathcal{C}^{\infty}(\Omega)$.Moreover, we have
$\mathsf{C}\sqcup\mathsf{D}\in\mathcal{C}_{\mathcal{B}}$, from which we deduce

$\displaystyle F(\underline{x}_{E})$
$\displaystyle=\sum_{\mathsf{C}\in\mathcal{C}_{\mathcal{B}\setminus\\{B\\}}\cap\,\mathcal{C}_{\mathcal{B}}}\prod_{c\,\in\mathsf{C}}(x_{c}-y_{c})H_{\mathsf{C}}(\underline{x}_{E})+\sum_{k=1}^{d}\sum_{\begin{subarray}{c}\mathsf{C}\in\mathcal{C}_{\mathcal{B}\setminus\\{B\\}}\\\
|\mathsf{C}\cap
B|=d-k\end{subarray}}\sum_{\begin{subarray}{c}\mathsf{D}\subset B\\\
|\mathsf{D}|=k\end{subarray}}\prod_{c\,\in\mathsf{C}\cup\mathsf{D}}(x_{c}-y_{c})H_{\mathsf{C}\sqcup\mathsf{D}}(\underline{x}_{E})$
$\displaystyle=\sum_{\mathsf{C}\in\mathcal{C}_{\mathcal{B}}}\prod_{c\in\mathsf{C}}(x_{c}-y_{c})H_{\mathsf{C}}(\underline{x}_{E}).$

∎

The previous Proposition 3.19 directly implies the following corollary.

######  Corollary 3.20.

If the function $F$ satisfies the hypotheses of Proposition 3.19, then one can
find a constant $C_{K}$ such that for all $\underline{x}_{E}$ in $K$,
$|F(\underline{x}_{E})|=C_{K}\sum_{\mathsf{C}\in\mathcal{C}_{\mathcal{B}}}\prod_{c\,\in\mathsf{C}}\left|x_{c}-y_{c}\right|.$

For instance, let $E=\\{1,2,3\\}$. Let $F$ be an infinitely differentiable
function such that for $(x,y,z)$ in a compact set $K$ of $\mathbb{R}^{3}$,

$|F(x,y,z)|\leq x^{2}+y^{2},\quad|F(x,y,z)|\leq
y^{2}+z^{2}\quad\text{and}\quad|F(x,y,z)|\leq x^{2}+z^{2}.$

Then the function $F$ satisfies the hypotheses of Proposition 3.19 with
$\mathcal{B}=\\{\\{1,2\\},\\{2,3\\},\\{1,3\\}\\}$ and $d=2$. It implies the
existence of a constant $C_{K}$ such that

$|F(x,y,z)|\leq C_{K}\left(x^{2}y^{2}+y^{2}z^{2}+x^{2}z^{2}+xyz\right).$

######  Remark 3.21.

Let $\Omega^{1}$ be an open subset of a finite dimensional vector space and
assume that $F\in\mathcal{C}^{0,\infty}(\Omega^{1},\Omega)$ (where this
function space is defined in (3.2.1)). Then Proposition 3.19 remains true if
one replace $\mathcal{C}^{\infty}(\Omega)$ by
$\mathcal{C}^{0,\infty}(\Omega^{1},\Omega)$ and the proof is in all points
similar.

We now apply the previous Corollary 3.20 to the function $\widetilde{F}_{A}$.

######  Lemma 3.22.

Let $\mathcal{I}$ be a partition of $A$ and $K$ be a compact subset of
$\widetilde{\mathcal{M}}_{A}(\mathbb{R})\times\mathcal{S}_{2A}^{+}(\mathbb{R})$.
Then there is a constant $C_{K}$ such that for all
$((\widetilde{M},\widetilde{Q}),\Sigma)\in K$ such that
$\widetilde{M}=\widetilde{M}_{\mathcal{I}}$ and
$\widetilde{Q}=\widetilde{Q}_{\mathcal{I}}$, one has
$\left|\widetilde{F}_{A}((\widetilde{M},\widetilde{Q}),\Sigma)\right|\leq
C_{K}\sum_{G\in\mathcal{G}_{\mathcal{I}}}\prod_{\\{I,J\\}\in
E(G)}\|\Sigma_{I,J}\|.$

###### Proof.

The proposition is trivial if $\mathcal{I}=\\{A\\}$, and we can assume that
$\mathcal{I}\neq\\{A\\}$. The proof is an application of Proposition 3.19 and
its Corollary 3.20. To this end, we set $V=\mathcal{S}_{2A}(\mathbb{R})$,
endowed with its canonical basis. Then $\mathcal{S}_{2A}(\mathbb{R})$ can be
identified with $\mathbb{R}^{E}$, where

$E=\left\\{\vphantom{\int^{a}}\\{(k,a),(l,b)\\}\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ k,l\in\\{1,2\\}\quad\text{and}\quad a,b\in
A\right\\}.$

For $\mathcal{J}\in\mathcal{P}_{A}$ with $\mathcal{J}\succeq\mathcal{I}$ we
define

$B_{\mathcal{J}}=\left\\{\vphantom{\int^{a}}\\{(k,a),(l,b)\\}\in
E\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
[a]_{\mathcal{J}}\neq[b]_{\mathcal{J}}\right\\},$

and

$\mathcal{B}_{\mathcal{I}}=\left\\{\vphantom{\int^{a}}B_{\mathcal{J}}\leavevmode\nobreak\
\middle|\leavevmode\nobreak\ \mathcal{J}\succeq\mathcal{I}\right\\}.$

As a consequence of Lemma 3.8 the function $\widetilde{F}_{A}$ is in
$\mathcal{C}^{0,\infty}(\widetilde{\mathcal{M}}_{A}^{*}(\mathbb{R}),\mathcal{S}_{2A}^{+}(\mathbb{R}))$.
Let $\mathcal{J}\succeq\mathcal{I}$. According to Lemma 3.13 we have

$\displaystyle\left|\widetilde{F}_{A}((\widetilde{M},\widetilde{Q}),\Sigma)\right|$
$\displaystyle\leq C_{K}\sup_{\begin{subarray}{c}I,J\in\mathcal{I}\\\
[I]_{\mathcal{J}}\neq[J]_{\mathcal{J}}\end{subarray}}\|\Sigma_{I,J}\|^{2}$
$\displaystyle\leq C_{K}\sup_{\\{(k,a),(l,b)\\}\in
B_{\mathcal{J}}}|\Sigma_{ab}^{kl}|^{2}.$

The function $\widetilde{F}$ then satisfies the hypotheses of Proposition 3.19
with $d=2$ and family of subsets $\mathcal{B}_{\mathcal{I}}$, from which we
deduce the existence of a constant $C_{K}$ such that

$\displaystyle\left|\widetilde{F}_{A}((\widetilde{M},\widetilde{Q}),\Sigma)\right|$
$\displaystyle\leq
C_{K}\sum_{\mathsf{C}\in\mathcal{C}_{\mathcal{B}_{\mathcal{I}}}}\prod_{\\{(k,a),(l,b)\\}\in\mathsf{C}}|\Sigma_{ab}^{kl}|$
$\displaystyle\leq
C_{K}\sum_{\mathsf{C}\in\mathcal{C}_{\mathcal{B}_{\mathcal{I}}}}\prod_{\\{(i,a),(j,b)\\}\in\mathsf{C}}\|\Sigma_{\\{a\\},[b]_{\mathcal{I}}}\|.$
(44)

To every multiset $\mathsf{C}$ in $\mathcal{C}_{\mathcal{B}_{\mathcal{I}}}$,
we can associate a graph $G^{\mathsf{C}}$ with set of vertices $\mathcal{I}$.
The edge $\\{I,J\\}$ is an edge of the graph $G^{\mathsf{C}}$ with
multiplicity one (resp. two) if one can find one (resp. two) elements
$\\{(k,a),(l,b)\\}\in\mathsf{C}$ such that
$\\{[a]_{\mathcal{I}},[b]_{\mathcal{I}}\\}=\\{I,J\\}$. It follows directly
from the definition of the set $\mathcal{C}_{\mathcal{B}_{\mathcal{I}}}$ and
the set $\mathcal{G}_{\mathcal{I}}$ that the graph $G^{\mathsf{C}}$ belongs to
the set $\mathcal{G}_{\mathcal{I}}$. Following inequality (44), one has

$\left|\widetilde{F}_{A}((\widetilde{M},\widetilde{Q}),\Sigma)\right|\leq
C_{K}\sum_{G\in\mathcal{G}_{\mathcal{I}}}\prod_{\\{I,J\\}\in
E(G)}\|\Sigma_{I,J}\|.$

∎

##  4 Asymptotics of the cumulants of the zeros counting measure

We are now in position to study the asymptotics of the cumulants of the zeros
counting measure associated with a sequence of processes
$(f_{n})_{n\in\mathfrak{N}}$. We first prove that the non degeneracy condition
(3) holds uniformly for $n\in\mathfrak{N}$, which allows us to translate the
previous Lemma 3.22 to the cumulant Kac density $F_{A,n}$ associated with the
sequence $(f_{n})_{n\in\mathfrak{N}}$.

###  4.1 Uniform non degeneracy of the covariance matrix

Up to now, we assumed that the generic process $f$ that we considered
satisfied the non degeneracy condition (3). For stationary Gaussian processes,
this non-degeneracy condition is true on very mild assumptions on the process.
For non-stationary processes there seems to be no simple conditions that
ensures the validity of (3). Nevertheless in our case of interest, we consider
a sequence of Gaussian processes that converges in distribution towards a
stationary Gaussian process and we are able to prove some uniform non-
degeneracy condition in this setting.

In this subsection, $A$ denotes a finite set and $\mathcal{I}$ a partition of
$A$. For $n\in\overline{\mathfrak{N}}$, we consider $f_{n}$ a Gaussian process
defined on $nU$. We will use the notations introduced in Section 3. In
particular we consider the quantities $\rho_{A,n}$, $F_{A,n}$,
$\Sigma^{\mathcal{I}}_{n}(\underline{x}_{A})$, etc. associated with the
process $f_{n}$.

We assume for now that the sequence $(f_{n})_{n\in\overline{\mathfrak{N}}}$
satisfies hypotheses $H_{1}(q)$ and $H_{2}(q)$ defined in ($\bullet$ ‣ 1.1.2)
and ($\bullet$ ‣ 1.1.2), with $q=|A|-1$. In particular the quantity
$\Sigma^{\mathcal{I}}_{n}(\underline{x}_{A})=\operatorname{Cov}((f_{n})_{\mathcal{I}}[\underline{x}_{A}])$
is well defined. Since the function $g$ of hypothesis $H_{2}(q)$ decreases to
zero, then for $\varepsilon>0$, one can find a constant $T_{\varepsilon}$ such
that

$\left\\{x\in\mathbb{R}\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
g(x)\geq\varepsilon\right\\}\subset[-T_{\varepsilon},T_{\varepsilon}].$

The main proposition of this section is the following.

######  Proposition 4.1.

Under the above setting, there is a compact set $K_{\eta}$ of
$\mathcal{S}_{A}^{+}(\mathbb{R})$ such that for all
$n\in\overline{\mathfrak{N}}$ and
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$, the matrix
$\Sigma^{\mathcal{I}}_{n}(\underline{x}_{A})$ lives in $K_{\eta}$.

We prove first Proposition 4.1 for the limit stationary process $f_{\infty}$.

######  Lemma 4.2.

Under the above setting, there is a compact set $K_{\eta}$ of
$\mathcal{S}_{A}^{+}(\mathbb{R})$ such that for all
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$, the matrix
$\Sigma^{\mathcal{I}}_{\infty}(\underline{x}_{A})$ lives in $K_{\eta}$.

###### Proof.

According to Lemma 2.4, one must show the existence of positive constants
$C_{\eta}$ and $c_{\eta}$ such that

$\forall\underline{x}_{A}\in\Delta_{\mathcal{I},\eta},\quad\|\Sigma_{\infty}^{\mathcal{I}}(\underline{x}_{A})\|\leq
C_{\eta}\quad\text{and}\quad\det\Sigma_{\infty}^{\mathcal{I}}(\underline{x}_{A})\geq
c_{\eta}.$

From the Hermite Genocchi formula 2.8 and Lemma 2.18, we observe that the
coefficients of the matrix $\Sigma_{\infty}^{\mathcal{I}}(\underline{x}_{A})$
are bounded by $\|g\|_{\infty}$. It remains to prove the uniform positiveness
of $\det\Sigma_{\infty}^{\mathcal{I}}(\underline{x}_{A})$ on
$\Delta_{\mathcal{I},\eta}$.

The covariance function of $f_{\infty}$ decreases to zero by assumption. Since
for Gaussian vectors, decorrelation implies independence, one see that the
process $f_{\infty}$ is weakly mixing, which in turn implies ergodicity. By
Maruyama theorem (see [16]), the spectral measure $\mu_{\infty}$ of of
$f_{\infty}$ has no atoms. It is then a standard fact that this observation
implies the non-degeneracy condition (3), and Lemma 3.1 implies that the
Gaussian vector $(f_{\infty})_{\mathcal{I}}[\underline{x}_{A}]$ is also non-
degenerate for $\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$.

We prove the uniform lower bound for
$\underline{x}_{A}\in\Delta_{\mathcal{I},\eta}$ by induction on the size of
the set $A$. If $|A|=1$ it reduces to the fact that the process $f_{\infty}$
is non-degenerate. Assume that the property is true for every strict subset
$B$ of $A$. For another partition $\mathcal{J}$ of $A$ such that
$\mathcal{J}\succeq\mathcal{I}$, and $\varepsilon>0$, we define the subset

$K_{\mathcal{J},\varepsilon}=\left\\{\underline{x}_{A}\in
U^{A}\leavevmode\nobreak\ \middle|\leavevmode\nobreak\
[x_{a}]_{\mathcal{J}}=[\underline{x}_{b}]_{\mathcal{J}}\;\Leftrightarrow\;|x_{a}-x_{b}|\leq
T_{\varepsilon}\right\\}.$

We can assume that $T_{\varepsilon}\geq|A|\eta$. In that case, one has

$\Delta_{\mathcal{I},\eta}\subset\bigsqcup_{\mathcal{J}\succeq\mathcal{I}}K_{\mathcal{J},\varepsilon}.$

In the case $\mathcal{J}=\\{A\\}$, one has for all $a,b\in A$

$|x_{a}-x_{b}|\leq T_{\varepsilon}.$

Stationarity implies that the matrix
$\Sigma^{\mathcal{I}}(\underline{x}_{A,A})$ depends only on the differences
$x_{a}-x_{b}$. By compactness of the set $\Delta_{\mathcal{I},\eta}\cap
K_{\\{A\\},\varepsilon}$, one can find a positive constant
$c_{\eta,\varepsilon}$ such that

$\forall\underline{x}_{J}\in\Delta_{\mathcal{I},\eta}\cap
K_{\\{A\\},\varepsilon},\quad\det\Sigma^{\mathcal{I}}(\underline{x}_{A})\geq
c_{\eta,\varepsilon}.$

Now assume that $\mathcal{J}\neq\\{A\\}$. If $\underline{x}_{A}\in
K_{J,\varepsilon}$ then for $a,b\in A$ such that
$[a]_{\mathcal{J}}\neq[b]_{\mathcal{J}}$, and $u,v\leq|A|-1$,

$|r_{\infty}^{(u,v)}(x_{a}-x_{b})|\leq\varepsilon.$

It implies that,

$\sup_{\begin{subarray}{c}I,J\in\mathcal{J}\\\ I\neq
J\end{subarray}}\|\Sigma_{\infty}^{\mathcal{I}}(\underline{x}_{A})_{I,J}\|\leq\varepsilon,$

and thus

$\|\Sigma^{\mathcal{I}}(\underline{x}_{A})-(\Sigma^{\mathcal{I}}(\underline{x}_{A}))_{\mathcal{J}}\|\leq\varepsilon.$

Since the determinant is a smooth function of the matrix coefficients and the
matrix $\Sigma^{\mathcal{I}}(\underline{x}_{A,A})$ is bounded, we deduce the