1) School of Mathematics, University of Minnesota, Minneapolis, MN, USA,
<EMAIL_ADDRESS>
2) Faculty of Mathematics, Bielefeld University, Germany, goetze@math-
uni.bielefeld.de. 4) Research supported by the NSF grant DMS-2154001 and the
GRF – SFB 1283/2 2021 – 317210226

# Strictly subgaussian probability distributions

S. G. Bobkov1,4 missing ,  G. P. Chistyakov2,4 and  F. Götze2,4 missingmissing
Dedicated to the memory of Gennadiy P. Chistyakov * May 1, 1945 $\dagger$
December 30, 2022.

###### Abstract.

We explore the class of probability distributions on the real line whose
Laplace transform admits a strong upper bound of subgaussian type. Using
Hadamard’s factorization theorem, we extend the class $\mathfrak{L}$ of Newman
and propose new sufficient conditions for this property in terms of location
of zeros of the associated characteristic functions in the complex plane. The
second part of this note deals with Laplace transforms of strictly subgaussian
distributions with periodic components. This subclass contains interesting
examples, for which the central limit theorem with respect to the Rényi
entropy divergence of infinite order holds.

###### Key words and phrases:

Subgaussian distributions, zeros, entire functions

###### 2010 Mathematics Subject Classification:

Primary 60E, 60F

##  1\. Introduction

Following Kahane [15], a random variable $X$ is called subgaussian, if
${\mathbb{E}}\,e^{cX^{2}}<\infty$ for some constant $c>0$. Assuming that $X$
has mean zero, this is equivalent to the statement that the moment generating
function (or the two-sided Laplace transform) of $X$ satisfies

${\mathbb{E}}\,e^{tX}\leq e^{\sigma^{2}t^{2}/2},\quad t\in{\mathbb{R}},$ (1.1)

with some constant $\sigma^{2}$. Its optimal value appears in the literature
under different names such as a subgaussian constant or as an optimal proxy
variance. Being deepely connected with logarithmic Sobolev constants and
concentration of measure phenomena, the problem of computation or estimation
of $\sigma$ is of considerable interest (including a similar quantity for a
more general setting of metric spaces, cf. [5]).

For example, in the case of a centered Bernoulli distribution
$p\delta_{q}+q\delta_{-p}$, the subgaussian constant was identified, although
not with a rigorous proof, by Kearns and Saul [17] to be

$\sigma^{2}=\frac{p-q}{2\,(\log p-\log q)}$ (1.2)

(cf. also [6], [2]). A similar expression was obtained by Diaconis and Saloff-
Coste [9] and by Higuchi and Yoshida [13] for the logarithmic Sobolev constant
of a Markov chain on the two point space.

Immediate consequences of inequality (1.1) are the finiteness of moments of
all orders of $X$ and in particular the relations ${\mathbb{E}}X=0$ and
${\mathbb{E}}X^{2}\leq\sigma^{2}$, which follow by an expansion of both sides
of (1.1) around $t=0$. Here the possible case $\sigma^{2}={\rm Var}(X)$ is of
particular interest. The following definition seemed to have appeared first in
the work of Buldygin and Kozachenko [7] who called this property “strongly
subgaussian”.

Definition. The random variable $X$ is called strictly subgaussian, or the
distribution of $X$ is strictly subgaussian, if (1.1) holds with the optimal
constant $\sigma^{2}={\rm Var}(X)$.

Such distributions appear in a natural way in a variety of mathematical
problems, as well as in statistical mechanics and quantum field theory. For
example, under the name “sharp subgaussianity”, this class was recently
considered in the work by Guionnet and Husson [10] as a condition for LDPs for
the largest eigenvalue of Wigner matrices with the same rate function as in
the case of Gaussian entries. Our interest has been motivated by the study of
the central limit theorem with respect to information-theoretic distances. Let
us clarify this connection in the following statement.

Given independent copies $(X_{n})_{n\geq 1}$ of a random variable $X$ with
mean zero and variance one, suppose that the normalized sums
$Z_{n}=\frac{1}{\sqrt{n}}(X_{1}+\dots+X_{n})$ have densities $p_{n}$ for large
$n$. The Rényi divergence of order $\alpha>0$ from the distribution of $Z_{n}$
to the standard normal law with density $\varphi$ (or the relative
$\alpha$-entropy) is defined by

$D_{\alpha}(p_{n}||\varphi)=\frac{1}{\alpha-1}\log\int_{-\infty}^{\infty}\Big{(}\frac{p_{n}(x)}{\varphi(x)}\Big{)}^{\alpha}\,\varphi(x)\,dx.$
(1.3)

It is non-decreasing as a function of $\alpha$, representing a strong
distance-like quantity. Here, the case $\alpha=1$ corresponds to the relative
entropy (Kullback-Leibler’s distance) and another important case $\alpha=2$
leads to the function of the $\chi^{2}$-Pearson distance.

Theorem 1.1. Suppose that $D_{\alpha}(p_{n}||\varphi)<\infty$ for every
$\alpha$ and some $n=n_{\alpha}$. For the convergence
$D_{\alpha}(p_{n}||\varphi)\rightarrow 0$ as $n\rightarrow\infty$ with an
arbitrary $\alpha>0$, it is necessary and sufficient that $X$ is strictly
subgaussian.

This characterization follows from the results of [3], which will be discussed
later.

Of course, the property of being strictly subgaussian does not require that
the distribution of $X$ has a density. From (1.2) it already follows that the
symmeric Bernoulli distribution belongs to this class. More examples are
discussed in Arbel, Marchal and Nguyen [1], where it is also shown that the
distribution of $X$ does not need be symmetric. The problem of
characterization of the whole class of such distributions is still open and
seems to be highly non-trivial. Nevertheless, there is a simple general
sufficient condition for the strict subgaussianity given by Newman [23]
(Theorem 4, see also [8], Chapter 1) in terms of location of zeros of the
characteristic function

$f(z)={\mathbb{E}}\,e^{izX},\quad z\in{\mathbb{C}}.$

Note that the subgaussian property (1.1) ensures that $f$ has an analytic
extension from the real line to the whole complex plane as an entire function
of order at most 2.

Theorem 1.2. Let $X$ be a subgaussian random variable with mean zero. If all
zeros of $f(z)$ are real, then $X$ is strictly subgaussian.

This condition is easily verified for many interesting classes including, for
example, arbitrary Bernoulli sums and (finite or infinite) convolutions of
uniform distributions on bounded symmetric intervals.

The probability distributions of Theorem 1.2 form an important class
$\mathfrak{L}$, introduced and studied by Newman in the mid 1970’s in
connection with the Lee-Yang property which naturally arises in the context of
ferromagnetic Ising models, cf. [23, 24, 25, 26]. We will recall the argument
and several properties of this class in Section 4.

Note that if the characteristic function $f(z)$ of a subgaussian random
variable $X$ does not have any real or complex zeros, a well-known theorem due
to Marcinkiewicz [22] implies that the distribution of $X$ is already
Gaussian. Thus, non-normal subgaussion distributions need to have zeros.
Towards the characterization problem, the main purpose of this note is to
explore two natural subclasses of distributions outside $\mathfrak{L}$ that
are still strictly subgaussian. First, we extend Theorem 1.2 in terms of zeros
of characteristic functions.

Theorem 1.3. Let $X$ be a subgaussian random variable with symmetric
distribution. If all zeros of $f(z)$ with ${\rm Re}(z)\geq 0$ lie in the cone
centered on the real axis defined by

$|{\rm Arg}(z)|\leq\frac{\pi}{8},$ (1.4)

then $X$ is strictly subgaussian.

At the first sight, the condition (1.4) looks artificial. However, it turns
out to be necessary in the following simple situation:

Theorem 1.4. Let $X$ be a random variable with a symmetric subgaussian
distribution. Suppose that $f$ has exactly one zero $z=x+iy$ in the positive
quadrant $x,y\geq 0$. Then $X$ is strictly subgaussian, if and only if $(1.4)$
holds true.

As a consequence of Theorem 1.3, one can partially address the following
question from the theory of entire characteristic functions (which is one of
the central problems in this area): What can one say about the possible
location of zeros of such functions?

Theorem 1.5. Let $(z_{n})$ be a finite or infinite sequence of non-zero
complex numbers in the angle $|{\rm Arg}(z_{n})|\leq\frac{\pi}{8}$ such that

$\sum_{n}\frac{1}{|z_{n}|^{2}}<\infty.$

Then there exists a symmetric strictly subgaussian distribution whose
characteristic function has zeros exactly at the points $\pm z_{n}$,
$\pm\bar{z}_{n}$.

It will be shown that a random variable $X$ with such distribution may be
constructed as the sum $X=\sum_{n}X_{n}$ of independent strictly subgaussian
random variables $X_{n}$ whose characteristic function has zeros at the points
$\pm z_{n}$, $\pm\bar{z}_{n}$ (and only at these points like in Theorem 1.4).
Moreover, one may require that

${\rm Var}(X)=\Lambda\sum_{n}\frac{1}{|z_{n}|^{2}}$

with any prescribed value $\Lambda\geq\Lambda_{0}$ where $\Lambda_{0}$ is a
universal constant ($\Lambda_{0}\sim 5.83$).

Returning to Theorem 1.3, it will actually be shown that, if a strictly
subgaussian random variable $X$ is not normal, the inequality (1.1) may be
further sharpened as follows: For any $t_{0}>0$, there exists $c=c(t_{0})$,
$0<c<\sigma^{2}={\rm Var}(X)$, such that

${\mathbb{E}}\,e^{tX}\leq e^{ct^{2}/2},\quad|t|\geq t_{0}.$ (1.5)

In particular, such a refinement applies to Theorem 1.2. The property (1.5) is
important in the study of rates in the local limit theorems such as CLT for
the Rényi divergence of infinite order. Two results in this direction will be
mentioned in the end of this note.

The sharpening (1.5) raises the question of whether or not this separation-
type property is fulfilled automatically for any non-normal strictly
subgaussian distribution. At least, it looks natural to expect the weaker
relation

$L(t)<e^{\sigma^{2}t^{2}/2},\quad t\neq 0,$ (1.6)

for the Laplace transform $L(t)={\mathbb{E}}\exp\\{tX\\}$. However, the answer
to this is negative, and moreover, (1.6) may turn into an equality for
infinitely many points $t$. In addition, the characteristic function $f(z)$
may have infinitely many zeros approaching the imaginary line ${\rm
Arg}(z)=\frac{\pi}{2}$. To this aim, we introduce the following:

Definition. We say that the distribution $\mu$ of a random variable $X$ is
periodic with respect to the standard normal law $\gamma$, with period $h>0$,
if it has a density $p(x)$ such that the density of $\mu$ with respect to
$\gamma$,

$q(x)=\frac{p(x)}{\varphi(x)}=\frac{d\mu(x)}{d\gamma(x)},\quad
x\in{\mathbb{R}},$

represents a periodic function with period $h$, that is, $q(x+h)=q(x)$ for all
$x\in{\mathbb{R}}$.

We denote the class of all such distributions by $\mathfrak{F}_{h}$ and say
that $X$ belongs to $\mathfrak{F}_{h}$. The following characterization in
terms of Laplace transforms may be useful.

Theorem 1.6. Any random variable $X$ in $\mathfrak{F}_{h}$ is subgaussian, and
the Laplace transform of its distribution is resepresentable as

$L(t)=\Psi(t)\,e^{t^{2}/2},\quad t\in{\mathbb{R}},$ (1.7)

where the function $\Psi$ is periodic with period $h$. Conversely, if
$\Psi(t)$ for a subgaussian random variable $X$ is $h$-periodic, then $X$
belongs to $\mathfrak{F}_{h}$, as long as the characteristic function $f(t)$
of $X$ is integrable.

In this way, we obtain a wide class of strictly subgaussian distributions, by
requiring that $\Psi(t)\leq 1$ for all $t$. As a simple example, for any
sufficiently small $c>0$,

$L(t)=(1-c\sin^{4}t)\,e^{t^{2}/2},\quad f(z)=(1-c\sinh^{4}z)\,e^{-z^{2}/2},$

represent respectively the Laplace transform and the characteristic function
of a strictly subgaussian distribution with mean zero and variance one. In
this case, we have $L(t)=e^{t^{2}/2}$ for all $t=\pi k$, $k\in{\mathbb{Z}}$,
and $f(z_{m})=0$ for $z_{m}=a+2\pi im$, $m\in{\mathbb{Z}}$, where $a>0$
depends on the parameter $c$. Hence ${\rm Arg}(z_{m})\rightarrow\frac{\pi}{2}$
as $m\rightarrow\infty$.

More examples based on trigonometric polynomials will be described in Section
12. The proof of Theorem 1.6 is given in Sections 10-11. Theorems 1.3 and 1.5
are proved in Sections 8-9, with preliminary steps in Sections 6-7, and
Section 5 is devoted to the proof of Theorem 1.4. In Sections 3-4 we recall
basic definitions and results related to the Hadamard factorization theorem
the class $\mathfrak{L}$. We conclude with some remarks on the central limit
theorem with respect to the Rényi divergences. Thus, our plan is the
following:

1\. Introduction

2\. Basic properties and examples of strictly subgaussian distributions

3\. Hadamard’s and Goldberg-Ostrovskiĭ’s theorems

4\. Characteristic functions with real zeros

5\. More examples of strictly subgaussian distributions

6\. One characterization of characteristic functions

7\. Strictly subgaussian symmetric distributions with

characteristic functions having exactly one non-trivial zero

8\. General case of zeros in the angle $|{\rm Arg}(z)|\leq\frac{\pi}{8}$

9\. Proof of Theorem 1.5

10\. Laplace transforms with periodic components

11\. Proof of Theorem 1.6

12\. Examples involving triginometric series

13\. Examples involving Poisson formula and theta functions

14\. Central limit theorems for Rényi distances

##  2\. Basic Properties and Examples of Strictly Subgaussian Distributions

In addition to the properties ${\mathbb{E}}X=0$ and
${\mathbb{E}}X^{2}\leq\sigma^{2}$, the Taylor expansion of the exponential
function in (1.1) around zero implies as well that necessarily
${\mathbb{E}}X^{3}=0$ and ${\mathbb{E}}X^{4}\leq 3\sigma^{4}$. Here an
equality is attained for symmetric normal distributions (but not exclusively
so).

Turning to other properties and some examples, first let us emphasize the
following two immediate consequences of (1.1).

Proposition 2.1. If the random variables $X_{1},\dots,X_{n}$ are independent
and strictly subgaussian, then their sum $X=X_{1}+\dots+X_{n}$ is strictly
subgaussian, as well.

Proposition 2.2. If strictly subgaussian random variables $(X_{n})_{n\geq 1}$
converge weakly in distribution to a random variable $X$ with finite second
moment, and ${\rm Var}(X_{n})\rightarrow{\rm Var}(X)$ as $n\rightarrow\infty$,
then $X$ is strictly subgaussian.

Proof. By the assumption, putting $\sigma_{n}^{2}={\rm Var}(X_{n})$, we have

${\mathbb{E}}\,e^{tX_{n}}\leq e^{\sigma_{n}^{2}t^{2}/2},\quad
t\in{\mathbb{R}}.$ (2.1)

By the weak convergence,
$\lim_{n\rightarrow\infty}{\mathbb{E}}\,u(X_{n})={\mathbb{E}}\,u(X)$ for any
bounded, continuous function $u$ on the real line. In particular, for any
$c\in{\mathbb{R}}$,

$\lim_{n\rightarrow\infty}{\mathbb{E}}\,e^{t\min(X_{n},c)}={\mathbb{E}}\,e^{t\min(X,c)}.$

Hence, by (2.1), for any $t\in{\mathbb{R}}$,

${\mathbb{E}}\,e^{t\min(X,c)}\leq\liminf_{n\rightarrow\infty}\,{\mathbb{E}}\,e^{tX_{n}}\leq\liminf_{n\rightarrow\infty}\,e^{\sigma_{n}^{2}t^{2}/2}=e^{\sigma^{2}t^{2}/2},$

where $\sigma^{2}={\rm Var}(X)$. Letting $c\rightarrow\infty$, we get (1.1). ∎

Combining Proposition 2.1 with Proposition 2.2, we obtain:

Corollary 2.3. If $\sum_{n=1}^{\infty}{\rm Var}(X_{n})<\infty$ for
independent, strictly subgaussian summands $X_{n}$, then the series
$X=\sum_{n=1}^{\infty}X_{n}$ represents a strictly subgaussian random
variable.

Here, the variance assumption ensures that the series
$\sum_{n=1}^{\infty}X_{n}$ is convergent with probability one (by the
Kolmogorov theorem), so that the partial sums of the series are weakly
convergent to the distribution of $X$. Thus, the class of strictly subgaussian
distributions is closed in the weak topology under infinite convolutions.

Obviously, it is also closed when taking convex mixtures.

Proposition 2.4. If $X_{n}$ are strictly subgaussian random variables with
${\rm Var}(X_{n})=\sigma^{2}$, and $\mu_{n}$ are distributions of $X_{n}$,
then for any sequence $p_{n}\geq 0$ such that $\sum_{n=1}^{\infty}p_{n}=1$,
the random variable with distribution $\mu=\sum_{n=1}^{\infty}p_{n}\mu_{n}$ is
strictly subgaussian as well and has variance ${\rm Var}(X)=\sigma^{2}$.

Note also that, if $X$ is strictly subgaussian, then $\lambda X$ is strictly
subgaussian for any $\lambda\in{\mathbb{R}}$.

Finally, let us give a simple sufficient condition for the property (1.5).
Recall the notation $K(t)=\log{\mathbb{E}}\,e^{tX}$, $t\in{\mathbb{R}}$.

Proposition 2.5. Let $X$ be a non-normal strictly subgaussian random variable.
If the function $K(\sqrt{|t|})$ is concave on the half-axis $t>0$ and is
concave on the half-axis $t<0$, then $(1.5)$ holds true.

Proof. Let ${\rm Var}(X)=\sigma^{2}$. For $t\geq 0$, write

${\mathbb{E}}\,e^{tX}=e^{\frac{1}{2}\,\sigma^{2}t^{2}-W(t^{2})}.$

By the assumption, $W(s)$ is non-negative and convex in $s\geq 0$, with
$W(0)=0$. In addition, it is $C^{\infty}$-smooth on $(0,\infty)$. Since $X$ is
not normal, necessarily $W(s)>0$ and $W^{\prime}(s)>0$ for all $s>0$. Using
that $W^{\prime}(s)\uparrow r$ as $s\rightarrow\infty$ for some
$r\in(0,\infty]$, it follows that

$r(s)\,\equiv\,\frac{1}{s}W(s)\,=\,\int_{0}^{1}W^{\prime}(sv)\,dv\,\uparrow\,r\quad{\rm
as}\ s\rightarrow\infty.$

In particular, given $s_{0}>0$, we have $\frac{1}{s}W(s)\geq r(s_{0})>0$ for
all $s\geq s_{0}$, or equivalently

$K(t)\leq\Big{(}\frac{1}{2}\sigma^{2}-r(s_{0})\Big{)}t^{2},\quad
t\geq\sqrt{s_{0}},$

which is the desired conclusion. A similar argument works for $t<0$ as well. ∎

An application of Corollary 2.3 allows to construct a rather rich family of
probability distributions from the class $\mathfrak{L}$. Recall that
$L(t)={\mathbb{E}}\,e^{tX}$ denotes the Laplace transform.

Example 2.6. First of all, if a random variable $X$ has a normal distribution
with mean zero and variance $\sigma^{2}$, then it is strictly subgaussian with
$L(t)=e^{\sigma^{2}t^{2}/2}$, $t\in{\mathbb{R}}$.

Example 2.7. If $X$ has a symmetric Bernoulli distribution, supported on two
points $\pm 1$, then it is strictly subgaussian with
$L(t)=\cosh(t)=\frac{e^{t}+e^{-t}}{2}$.

Example 2.8. If $X=\sum_{n=1}^{\infty}a_{n}X_{n}$ is a Bernoulli sum,
${\mathbb{P}}\\{X_{n}=\pm 1\\}=\frac{1}{2}$,
$\sum_{n=1}^{\infty}a_{n}^{2}<\infty$, with $X_{n}$ independent, then it is
strictly subgaussian with variance $\sigma^{2}=\sum_{n=1}^{\infty}a_{n}^{2}$.
The Laplace transform and characteristic function $f$ of $X$ are given by

$L(t)=\prod_{n=1}^{\infty}\cosh(a_{n}t),\quad
f(t)=\prod_{n=1}^{\infty}\cos(a_{n}t).$

Example 2.9. If $X$ is uniformly distributed on an interval $[-a,a]$, $a>0$,
it is strictly subgaussian. In this case it may be represented (in the sense
of distributions) as the sum

$X=\sum_{n=1}^{\infty}\frac{a}{2^{n}}\,X_{n},\quad{\mathbb{P}}\\{X_{n}=\pm
1\\}=\frac{1}{2}\quad(X_{n}\ {\rm independent}).$

Hence, this case is covered by the previous example, with
$L(t)=\frac{\sinh(at)}{at}$.

Example 2.10. If the random variables $X_{n}$ are independent and uniformly
distributed on the interval $[-1,1]$, then the infinite sum
$X=\sum_{n=1}^{\infty}a_{n}X_{n}$ with $\sum_{n=1}^{\infty}a_{n}^{2}<\infty$
represents a strictly subgaussian random variable with

$L(t)=\prod_{n=1}^{\infty}\frac{\sinh(a_{n}t)}{a_{n}t}.$

Example 2.11. Suppose that $X$ has density $p(x)=x^{2}\varphi(x)$, where
$\varphi(x)=\frac{1}{\sqrt{2\pi}}\,e^{-x^{2}/2}$ is the standard normal
density. Then ${\mathbb{E}}X=0$, $\sigma^{2}={\mathbb{E}}X^{2}=3$, and

$L(t)=(1+t^{2})\,e^{t^{2}/2}\leq e^{3t^{2}/2}.$

Hence, $X$ is strongly subgaussian.

Example 2.12. More generally, if $X$ has a density of the form

$p(x)=\frac{1}{(2d-1)!!}\,x^{2d}\varphi(x),\quad x\in{\mathbb{R}},\
d=1,2,\dots,$

then ${\mathbb{E}}X=0$, $\sigma^{2}={\mathbb{E}}X^{2}=2d+1$, and

$L(t)=\frac{1}{(2d-1)!!}\,H_{2d}(it)\,e^{t^{2}/2}\leq e^{(2d+1)\,t^{2}/2}.$

Hence, $X$ is strictly subgaussian. The last inequality follows from Theorem
1.2, since the Chebyshev-Hermite polynomials have real zeros, only.

##  3\. Hadamard’s and Goldberg-Ostrovskiĭ’s Theorems

All the previous examples may be included as partial cases of a more general
setup. First, let us recall some basic definitions and notations related to
the Hadamard theorem from the theory of complex variables. Given an entire
function $f(z)$, introduce

$M_{f}(r)=\max_{|z|\leq r}|f(z)|=\max_{|z|=r}|f(z)|,\quad r\geq 0,$

which characterizes the growth of $f$ at infinity. The order of $f$ is defined
by

$\rho=\limsup_{r\rightarrow\infty}\frac{\log\log M_{f}(r)}{\log r}.$

Thus, $\rho$ is an optimal value such that, for any $\varepsilon>0$, we have
$M_{f}(r)<e^{r^{\rho+\varepsilon}}$ for all large $r$. If $f$ is a polynomial,
then $\rho=0$. If $\rho$ is finite, then the type of $f$ is defined by

$\tau=\limsup_{r\rightarrow\infty}\frac{\log M_{f}(r)}{r^{\rho}}.$

Thus, $\tau$ is an optimal value such that, for any $\varepsilon>0$, we have
$M_{f}(r)<e^{(\tau+\varepsilon)\,r^{\rho}}$ for all sufficiently large $r$. If
$0<\tau<\infty$, the function $f$ is said to be of normal type.

For integers $p\geq 0$, introduce the functions

$G_{p}(u)=(1-u)\exp\Big{\\{}u+\frac{u^{2}}{2}+\dots+\frac{u^{p}}{p}\Big{\\}},\quad
u\in{\mathbb{C}},$

called the primary factors, with the convention that $G_{0}(u)=1-u$. Given a
sequence of complex numbers $z_{n}\neq 0$ such that $|z_{n}|\uparrow\infty$,
one considers a function of the form

$\Pi(z)=\prod_{n=1}^{\infty}G_{p}(z/z_{n})$ (3.1)

called a canonical product. An integer $p\geq 0$ is called the genus of this
product, if it is the smallest integer such that

$\sum_{n=1}^{\infty}\frac{1}{|z_{n}|^{p+1}}<\infty.$ (3.2)

There is a simple estimate $\log|G_{p}(u)|\leq A_{p}|u|^{p+1}$ where the
constant $A_{p}$ depends on $p$ only. Therefore, the product in (3.1) is
uniformly convergent as long as (3.2) is fulfilled.

See e.g. Levin [19] for the following classical theorem.

Theorem 3.1 (Hadamard). Any entire function $f$ of a finite order $\rho$ can
be represented in the form

$f(z)=z^{m}\,e^{P(z)}\prod_{n\geq 1}G_{p}(z/z_{n}),\quad z\in{\mathbb{C}}.$
(3.3)

Here $z_{n}$ are the non zero roots of $f(z)$, the genus of the canonical
product satisfies $p\leq\rho$, $P(z)$ is a polynomial of degree $\leq\rho$,
and $m\geq 0$ is the multiplicity of the zero at the origin.

In order to describe the convergence of the canonical product, assume that
$f(z)$ has an infinite sequence of non-zero roots $z_{n}$ arranged in
increasing order of their moduli so that

$0<|z_{1}|\leq|z_{2}|\leq\dots\leq|z_{n}|\rightarrow\infty\quad{\rm as}\
n\rightarrow\infty.$

Define the convergence exponent of the sequence $a_{n}$ by

$\rho_{1}=\inf\Big{\\{}\lambda>0:\sum_{n=1}^{\infty}\frac{1}{|z_{n}|^{\lambda}}<\infty\Big{\\}}.$

A theorem due to Borel asserts that the order $\rho$ of the canonical product
$\Pi(z)$ satisfies $\rho\leq\rho_{1}$. Moreover, Theorem 6 from [19], p.16,
states that the convergence exponent of the zeros of any entire function
$f(z)$ does not exceed its order: $\rho_{1}\leq\rho$. Thus, for canonical
products the convergence exponent of the zeros is equal to the order of the
function: $\rho_{1}=\rho$ (Theorem 7).

There is also the following elementary relation between the convergence
exponent and the genus of the canonical product: $p\leq\rho_{1}\leq p+1$.
Assuming that $\rho_{1}$ is an integer, we have that
$\sum_{n=1}^{\infty}|z_{n}|^{-\rho_{1}}=\infty\Rightarrow p=\rho_{1}$, while
$p=\rho_{1}+1$ means that the latter series is convergent.

The following theorem due to Goldberg and Ostrovskiĭ [11] refines Theorem 3.1
for the class of ridge entire functions whose all zeros are real. Recall that
$f$ is a ridge function, if it satisfies $|f(x+iy)|\leq|f(iy)|$ for all
$x,y\in{\mathbb{R}}$.

Theorem 3.2 (Goldberg-Ostrovskiĭ). Suppose that an entire ridge function $f$
of a finite order has only real roots. Then it can be represented in the form

$f(z)=c\,e^{i\beta z-\gamma z^{2}/2}\prod_{n\geq
1}\Big{(}1-\frac{z^{2}}{z_{n}^{2}}\Big{)},\quad z\in{\mathbb{C}},$ (3.4)

for some $c\in{\mathbb{C}}$, $\beta\in{\mathbb{R}}$, $\gamma\geq 0$, and
$z_{n}>0$ such that $\sum_{n\geq 1}z_{n}^{-2}<\infty$.

We refer to [11]. See also Kamynin [16] for generalizations of Theorem 3.2 to
the case where the zeros of $f$ are not necessarily real.

##  4\. Characteristic Functions with Real Zeros

We are now prepared to prove Theorem 1.2, including the relation (1.5) in the
non-Gaussian case which is stronger than (1.1).

Thus, let $X$ be a subgaussian random variable with mean zero and variance
$\sigma^{2}={\rm Var}(X)$. Then the inequality (1.1) may be extended to the
complex plane in the form

$|f(z)|\leq e^{b\,{\rm Im}(z)^{2}/2},\quad z\in{\mathbb{C}},$

for some constant $b\geq\sigma^{2}$, where $f$ is the characteristic function
of $X$. Hence, $f$ is a ridge entire function of order $\rho\leq 2$. We are
therefore able to apply Theorem 3.2 which yields the representation (3.4) for
some $c\in{\mathbb{C}}$, $\gamma\geq 0$, $\beta\in{\mathbb{R}}$, and for some
finite or infinite sequence $z_{n}>0$ such that $\sum_{n\geq
1}z_{n}^{-2}<\infty$. Note that $f(z_{n})=f(-z_{n})=0$, so that
$\\{z_{n},-z_{n}\\}$ are all zero of $f$ (this set may be empty). Since
$f(0)=1$ and $f^{\prime}(0)=0$, we necessarily have $c=1$ and $\beta=0$.
Hence, this representation is simplified to

$f(z)=e^{-\gamma z^{2}/2}\prod_{n\geq
1}\Big{(}1-\frac{z^{2}}{z_{n}^{2}}\Big{)}.$ (4.1)

Since $f^{\prime\prime}(0)=-\sigma^{2}$, we also have

$\frac{1}{2}\sigma^{2}=\frac{1}{2}\gamma+\sum_{n\geq 1}\frac{1}{z_{n}^{2}},$
(4.2)

so that $\gamma\leq\sigma^{2}$. Applying (4.1) with $z=-it$,
$t\in{\mathbb{R}}$, we get a similar representation for the Laplace transform

${\mathbb{E}}\,e^{tX}=e^{\gamma t^{2}/2}\prod_{n\geq
1}\Big{(}1+\frac{t^{2}}{z_{n}^{2}}\Big{)}.$ (4.3)

Using $1+x\leq e^{x}$ ($x\in{\mathbb{R}}$), we see that the right-hand side
above does not exceed $e^{\sigma^{2}t^{2}/2}$, where we used (4.2). Hence
(4.3) leads to the desired bound (1.1), and Theorem 1.2 is proved.

Let us also verify the property (1.5) in the case where the random variable
$X$ is not normal. Then the product in (4.3) is not empty and therefore
$\gamma<\sigma^{2}$. Let us rewrite (4.3) as

${\mathbb{E}}\,e^{tX}=e^{V(t^{2})},\quad V(s)=\gamma s+\sum_{n\geq
1}\log\Big{(}1+\frac{s}{z_{n}^{2}}\Big{)}.$

Since the function $V$ is concave, it remains to refer to Proposition 2.5. ∎

Remark. Using (4.2), let us rewrite (4.1) with $z=t\in{\mathbb{R}}$ in the
form

$f(t)=e^{-(3\gamma-\sigma^{2})\,t^{2}/4}\prod_{n\geq
1}\Big{(}1-\frac{t^{2}}{z_{n}^{2}}\Big{)}\,e^{-\frac{t^{2}}{2z_{n}^{2}}}.$
(4.4)

Here, the terms in the product represent characteristic funtions of random
variables $\frac{1}{z_{n}}X_{n}$ such that all $X_{n}$ have density
$p(x)=x^{2}\varphi(x)$ which we discussed in Example 2.11. Hence, if
$\frac{1}{3}\sigma^{2}\leq\gamma\leq\sigma^{2}$, the function $f(t)$ in (4.4)
represents the characteristic function of

$X=cZ+\sum_{n\geq 1}\frac{1}{z_{n}}X_{n},\quad
c^{2}=\frac{3}{2}\gamma-\frac{1}{2}\sigma^{2},$

assuming that $X_{n}$ are independent and $Z\sim N(0,1)$ is independent of all
$X_{n}$.

Note that (4.1) does not always define a characteristic function. For example,
when there is only one term in the product, we have $f(t)=e^{-\gamma
t^{2}/2}(1-\frac{t^{2}}{z_{1}^{2}})$. It is a characteristic function, if and
only if $\gamma\geq\frac{1}{z_{1}^{2}}$ (cf. e.g. [20], p. 34). We will return
to this question in Section 8.

Properties of the class $\mathfrak{L}$. Following Newman [23], let us
emphasize several remarkable properties of strictly subgaussian distributions
whose characteristic functions have real zeros, only. Starting from (4.3), one
can represent the log-Laplace transform of $X$ as

$K(t)=\log{\mathbb{E}}\,e^{tX}=\frac{\gamma t^{2}}{2}+\sum_{n\geq
1}\log\Big{(}1+\frac{t^{2}}{z_{n}^{2}}\Big{)}.$

Hence the cumulants of even order $2m$ of $X$ are given for $m\geq 2$ by

$\gamma_{2m}=K^{(2m)}(0)=(-1)^{m-1}\,\frac{(2m)!}{m}\sum_{n\geq
1}\frac{1}{z_{n}^{2m}},$

while $\gamma_{2m-1}=0$. In particular, the distribution of $X$ has to be
symmetric about the origin, with $(-1)^{m-1}\gamma_{2m}\geq 0$. As was also
shown in [23], the cumulants and the moments of $X$ admit the bounds

$(-1)^{m-1}\gamma_{2m}\leq\frac{(2m)!}{2^{m}\,m}\,\sigma^{2m},\quad{\mathbb{E}}X^{2m}\leq\frac{(2m)!}{2^{m}\,m!}\,\sigma^{2m}.$
(4.5)

In addition, for all integers $k\geq 0$ and $t\in{\mathbb{R}}$,

$\sum_{m=1}^{2k}\frac{\gamma_{2m}}{(2m)!}\,t^{2m}\leq
K(t)\leq\sum_{m=1}^{2k+2}\frac{\gamma_{2m}}{(2m)!}\,t^{2m}.$

Since the class $\mathfrak{L}$ is closed under convolutions, the second
inequality in (4.5) continues to hold for weighted sums of independent,
strictly subgaussian random variables. This provides a natural extension of
Khinchine’s inequality for Bernoulli sums, as noticed in [24] (cf. also a
recent work [12]).

##  5\. More Examples of Strictly Subgaussian Distributions

In connection with the problem of location of zeros, we now examine
probability distributions with characteristic functions of the form

$f(t)=e^{-t^{2}/2}\,(1-\alpha t^{2}+\beta t^{4}),$ (5.1)

where $\alpha,\beta\in{\mathbb{R}}$ are parameters. It was already mentioned
that when $\beta=0$, we obtain a characteristic function

$f(t)=e^{-t^{2}/2}\,(1-\alpha t^{2}),$

if and only if $0\leq\alpha\leq 1$. As we will see, in the general case, it is
necessary that $\beta\geq 0$ for $f(t)$ to be a characteristic function
(although negative values of $\alpha$ are possible for small $\beta$). Before
deriving a full characterization, first let us emphasize the following.

Proposition 5.1. Given $\beta\geq 0$, a random variable $X$ with
characteristic function of the form $(5.1)$ is strongly subgaussian, if and
only if $\alpha$ satisfies $\alpha\geq\sqrt{2\beta}$.

Proof. Recall that $X$ is strongly subgaussian, if and only if, for all
$t\in{\mathbb{R}}$,

${\mathbb{E}}\,e^{tX}\leq
e^{\sigma^{2}t^{2}/2},\quad\sigma^{2}=-f^{\prime\prime}(0).$ (5.2)

Near zero, the characteristic function in (5.1) behaves like a quadratic
polynomial $f(t)=1-\frac{1}{2}\,t^{2}-\alpha t^{2}+O(t^{4}),$ so that
$\sigma^{2}=1+2\alpha$ (in particular, $\alpha\geq-\frac{1}{2}$). Hence,
applying (5.1) to the values $-it$, one may rewrite (5.2) equivalently
(multiplying both sides by $\exp(-t^{2}/2)$) as

$1+\alpha t^{2}+\beta t^{4}\leq e^{\alpha t^{2}}=1+\alpha
t^{2}+\frac{1}{2}\,\alpha^{2}t^{4}+\frac{1}{6}\,\alpha^{3}t^{6}+\dots$

If $\alpha\geq 0$, this inequality holds for all $t\in{\mathbb{R}}$, if and
only if $\alpha^{2}\geq 2\beta$. As for the case $\alpha<0$, this is
impossible, since then $e^{\alpha t^{2}}\rightarrow 0$ as $t\rightarrow\infty$
exponentially fast. ∎

As already emphasized, if a random variable $X$ is subgaussian (even if it is
not strictly subgaussian), its characteristic function $f(t)$ may be extended
to the complex plane as an entire function $f(z)={\mathbb{E}}\,e^{izX}$ of
order $\rho\leq 2$ and of finite type like in the strictly subgaussian case
(5.2). Since in general $f(-\bar{z})=\bar{f}(z)$, any zero $z=x+iy$ of $f$
($x,y\in{\mathbb{R}}$) is complemented with zero $-\bar{z}=-y-ix$. If in
addition the distribution of $X$ is symmetric about zero, then $-z$ and
$\bar{z}$ will also be zeros of $f$. Thus, in this case with every non-real
zero $z$, the characteristic function has 3 more distinct zeros, and hence we
have 4 distinct zeros $\pm x\pm iy$, $x,y>0$. One can now apply Proposition
5.1 to prove Theorem 1.4.

Proof of Theorem 1.4. Given a random variable $X$ with a symmetric subgaussian
distribution, suppose that its characteristic function has exactly one zero
$z=x+iy$ in the positive quadrant $x,y\geq 0$. We need to show that $X$ is
strongly subgaussian, if and only if

$0\leq{\rm Arg}(z)\leq\frac{\pi}{8}.$ (5.3)

The case where $z=x$ is real is covered by Theorem 1.2. The argument below
also works in this case, but for definiteness let us assume that $z$ is
complex, so that $x,y>0$ (the case $x=0$ and $y>0$ is impossible, since then
$f(z)=f(iy)\geq 1$).

Thus, let $f(z)$ have four distinct roots $z_{1}=z$, $z_{2}=-z=-x-iy$,
$z_{3}=\bar{z}=x-iy$, $z_{4}=-\bar{z}=-x+iy$. Applying Hadamard’s theorem, we
get a representation

$f(z)=e^{P(z)}\,\Big{(}1-\frac{z}{z_{1}}\Big{)}\Big{(}1-\frac{z}{z_{2}}\Big{)}\Big{(}1-\frac{z}{z_{3}}\Big{)}\Big{(}1-\frac{z}{z_{4}}\Big{)},$

where $P(z)$ is a quadratic polynomial. Since $f(0)=1$, necessarily $P(0)=0$.
Also, by the symmetry of the distribution of $X$, we have $f(z)=f(-z)$, which
implies $P(z)=P(-z)$ for all $z\in{\mathbb{C}}$. It follows that $P(z)$ has no
linear term, so that $P(z)=-\frac{1}{2}\gamma z^{2}$ for some
$\gamma\in{\mathbb{C}}$. Thus, putting $w=a+bi=\frac{1}{x+iy}$, we have

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle e^{-\gamma
t^{2}/2}\,(1-wt)(1+wt)(1-\bar{w}t)(t+\bar{w}t)$ (5.4) $\displaystyle=$
$\displaystyle e^{-\gamma
t^{2}/2}\,\big{(}1-(w^{2}+\bar{w}^{2})\,t^{2}+|w|^{4}t^{4}\big{)}$
$\displaystyle=$ $\displaystyle e^{-\gamma
t^{2}/2}\,\big{(}1-2(a^{2}-b^{2})\,t^{2}+(a^{2}+b^{2})^{2}\,t^{4}\big{)}.$

Comparing both sides of (5.4) near zero according to Taylor’s expansion, we
get that

$\gamma+4(a^{2}-b^{2})=\sigma^{2}.$ (5.5)

In particular, $\gamma$ must be a real number, necessarily positive (since
otherwise $f(t)$ would not be bounded on the real axis). Moreover, the case
$a=|b|$ is impossible, since then
$f(t)=e^{-\sigma^{2}t^{2}/2}\,(1+2b^{4}t^{4})$. Rescaling the variable and
applying Proposition 5.1 with $\alpha=0$, we would conclude that the random
variable $X$ is not strictly subgaussian.

Thus, let $a\neq|b|$ (as we will see, necessarily $\gamma>\sigma^{2}$). Again
rescaling of the $t$-variable, one may assume that $\gamma=1$ in which case
the representation (5.4) becomes

$f(t)=e^{-t^{2}/2}\,\big{(}1-2(a^{2}-b^{2})\,t^{2}+(a^{2}+b^{2})^{2}\,t^{4}\big{)}.$

One can now apply Proposition 5.1 with parameters $\alpha=2(A-B)$,
$\beta=(A+B)^{2}$, where $A=a^{2},\ B=b^{2}$. Since the condition $\alpha\geq
0$ is necessary for $f(t)$ to be a characteristic function of a strictly
subgaussian distribution, we may assume that $A\geq B$ (in fact, we have
$A>B$, since $a\neq|b|$). The condition $\beta\leq\frac{1}{2}\,\alpha^{2}$,
that is, $2(A-B)^{2}\geq(A+B)^{2}$ is equivalent to

$(A+B)^{2}\geq 8AB\ \Longleftrightarrow\ (a^{2}+b^{2})^{2}\geq 8a^{2}b^{2}.$

To express this in polar coordinates, put $a=r\cos\theta$, $b=r\sin\theta$
with $r^{2}=a^{2}+b^{2}$ and $|\theta|\leq\frac{\pi}{2}$. Since $A\geq B$,
that is $a\geq|b|$, necessarily $|\theta|\leq\frac{\pi}{4}$, and the above
turns out to be the same as

$\cos^{2}(\theta)\,\sin^{2}(\theta)\leq\frac{1}{8}\ \Longleftrightarrow\
\sin^{2}(2\theta)\leq\frac{1}{2}\ \Longleftrightarrow\
|\theta|\leq\frac{\pi}{8}.$

Since $\theta={\rm Arg}(a+bi)=-{\rm Arg}(z)$, the desired characterization
(5.3) follows. ∎

##  6\. One Characterization of Characteristic Functions

It remains to decide whether or not the characteristic functions in
Proposition 5.1 with non-real zeros do exist. Therefore, we now turn to the
characterization of the property that the functions of the form

$f(t)=e^{-t^{2}/2}\,(1-\alpha t^{2}+\beta t^{4})$ (6.1)

are positive definite. The more general class of functions $f(t)=e^{-\gamma
t^{2}/2}\,(1-\alpha t^{2}+\beta t^{4})$, $\gamma>0$, is reduced to (6.1) by
rescaling the $t$-variable.

Proposition 6.1. The equality $(6.1)$ defines a characteristic function, if
and only if the point $(\alpha,\beta)$ belongs to one of the following two
regions:

$4\beta-2\sqrt{\beta(1-2\beta)}\leq\alpha\leq 3\beta+1,\qquad
0\leq\beta\leq\frac{1}{3},$ (6.2)

or

$4\beta-2\sqrt{\beta(1-2\beta)}\leq\alpha\leq
4\beta+2\sqrt{\beta(1-2\beta)},\quad\frac{1}{3}\leq\beta\leq\frac{1}{2}.$
(6.3)

The expression on the left-hand sides in (6.2)-(6.3) is negative, if and only
if $\beta<\frac{1}{6}$. Hence, for such values of $\beta$, the parameter
$\alpha$ may be negative.

Combining Propositions 5.1 and 6.1, we obtain a full characterization of
strictly subgaussian distributions with characteristic functions of the form
(6.1). To this aim, one should complement (6.2)-(6.3) with the bound
$\alpha\geq\sqrt{2\beta}$. To describe the full region, we need to solve the
corresponding inequalities. First, it should be clear that $\sqrt{2\beta}$ is
smaller than the right-hand sides of (6.2)-(6.3) for all
$0\leq\beta\leq\frac{1}{2}$. In this $\beta$-interval, we also have

$4\beta-2\sqrt{\beta(1-2\beta)}\leq\sqrt{2\beta}\,\Longleftrightarrow\,12\beta-3\leq
2\sqrt{2(1-2\beta)}.$

The latter is fulfilled automatically for $\beta\leq\frac{1}{4}$. For
$\frac{1}{4}\leq\beta\leq\frac{1}{2}$, squaring the above inequality, we
arrive at the quadratic inequality

$144\beta^{2}-56\beta+1\leq 0.$

The corresponding quadratic equation has two real roots, one of which
$0.0188...$ is out of our interval, while the other one

$\beta_{0}=\frac{1}{36}\,(7+2\sqrt{10})\sim 0.3701...$

belongs to the interval $(\frac{1}{3},\frac{1}{2})$. Therefore, the left-hand
side in (6.2) should be replaced with $\sqrt{2\beta}$ on the whole interval
$0\leq\beta\leq\frac{1}{3}$, while the lower bounds in (6.3) should be
properly changed for $\beta\leq\beta_{0}$ and $\beta\geq\beta_{0}$. That is,
we obtain:

Proposition 6.2. The equality $(6.1)$ defines a characteristic function of a
strictly subgaussian distribution, if and only if

$\displaystyle\sqrt{2\beta}\leq$ $\displaystyle\hskip-5.69054pt\alpha$
$\displaystyle\hskip-5.69054pt\leq 3\beta+1,\qquad\qquad\qquad
0\leq\beta\leq\frac{1}{3},$ (6.4) $\displaystyle\sqrt{2\beta}\leq$
$\displaystyle\hskip-5.69054pt\alpha$ $\displaystyle\hskip-5.69054pt\leq
4\beta+2\sqrt{\beta(1-2\beta)},\quad\frac{1}{3}\leq\beta\leq\beta_{0},$ (6.5)
$\displaystyle 4\beta-2\sqrt{\beta(1-2\beta)}\leq$
$\displaystyle\hskip-5.69054pt\alpha$ $\displaystyle\hskip-5.69054pt\leq
4\beta+2\sqrt{\beta(1-2\beta)},\quad\beta_{0}\leq\beta\leq\frac{1}{2}.$ (6.6)

Proof of Proposition 6.1. Recall that the Chebyshev-Hermite polynomial
$H_{k}(x)$ of degree $k=0,1,2,\dots$ is defined via the identity
$\varphi^{(k)}(x)=(-1)^{k}H_{k}(x)\varphi(x)$. In particular,

$H_{0}(x)=1,\quad H_{2}(x)=x^{2}-1,\quad H_{4}(x)=x^{4}-6x^{2}+3.$

Equivalently, for even orders

$t^{2k}\,e^{-t^{2}/2}=\int_{-\infty}^{\infty}(-1)^{k}H_{2k}(x)\varphi(x)\,e^{itx}\,dx.$

Therefore, the function in (6.1) represents the Fourier transform of the
function

$\displaystyle p(x)$ $\displaystyle=$ $\displaystyle\big{(}1+\alpha
H_{2}(x)+\beta H_{4}(x)\big{)}\,\varphi(x)$ $\displaystyle=$
$\displaystyle\big{(}(1-\alpha+3\beta)+(\alpha-6\beta)x^{2}+\beta
x^{4}\big{)}\,\varphi(x),$

whose total integral is $f(0)=1$. Hence, $p(x)$ represents a probability
density, if and only if

$\psi(y)\equiv(1-\alpha+3\beta)+(\alpha-6\beta)y+\beta y^{2}\geq 0\quad{\rm
for\ all}\ y\geq 0.$

Choosing $y=0$ and $y\rightarrow\infty$, we obtain necessary conditions

$\alpha\leq 3\beta+1,\quad\beta\geq 0.$ (6.7)

Assuming this, a sufficient condition for the inequality $\psi(y)\geq 0$ to
hold for all $y\geq 0$ is $\alpha\geq 6\beta$. As a result, we obtain a
natural region for the parameters, namely

$6\beta\leq\alpha\leq 3\beta+1,\quad 0\leq\beta\leq\frac{1}{3},$ (6.8)

for which $f(t)$ in (6.1) is a characteristic function.

In the case $\alpha<6\beta$, we obtain a second region. Note that the
quadratic function $\psi(y)=c_{0}+2c_{1}y+c_{2}y^{2}$ with $c_{0},c_{2}\geq 0$
and $c_{1}<0$ is non-negative in $y\geq 0$, if and only if $c_{1}^{2}\leq
c_{0}c_{2}$. For the coefficients $c_{2}=\beta>0$ and
$2c_{1}=\alpha-6\beta<0$, the condition $c_{1}^{2}\leq c_{0}c_{2}$ means that

$\Big{(}\frac{\alpha-6\beta}{2}\Big{)}^{2}\leq(1-\alpha+3\beta)\beta\,\Longleftrightarrow\,(\alpha-4\beta)^{2}\leq
4\beta(1-2\beta).$

Thus, necessarily $\beta\leq\frac{1}{2}$, and then admissible values of
$\alpha$ are described by the relations

$4\beta-2\sqrt{\beta(1-2\beta)}\leq\alpha\leq 4\beta+2\sqrt{\beta(1-2\beta)}$
(6.9)

in addition to the assumption $\alpha<6\beta$ and the necessary conditions in
(6.7).

If $\frac{1}{3}\leq\beta\leq\frac{1}{2}$, we arrive at the desired relations
in (6.3), since

$4\beta+2\sqrt{\beta(1-2\beta)}\leq 3\beta+1\leq 6\beta.$

If $\beta\leq\frac{1}{3}$, then $6\beta\leq 3\beta+1$. In the case
$\alpha<6\beta$, the upper bound in (6.9) will hold automatically, since

$6\beta\leq 4\beta+2\sqrt{\beta(1-2\beta)}\quad{\rm for\ all}\
0\leq\beta\leq\frac{1}{3}.$

So, for the values $\alpha<6\beta$ and $\beta\leq\frac{1}{3}$, (6.9) is
simplified to

$4\beta-2\sqrt{\beta(1-2\beta)}\leq\alpha\leq 6\beta,\quad
0<\beta\leq\frac{1}{3}.$ (6.10)

It remains to take the union of the two regions described by (6.10) with
(6.8), and then we arrive at (6.2). ∎

##  7\. Strictly Subgaussian Symmetric Distributions with Characteristic
Functions Having Exactly One Non-trivial Zero

One may illustrate Proposition 6.2 by the following simple example. For
$\beta=\frac{1}{3}$, admissible values of $\alpha$ cover the interval
$\sqrt{2/3}\leq\alpha\leq 2$, following both (6.4) and (6.5). Choosing
$\alpha=\sqrt{2/3}$, we obtain the characteristic function

$f(t)=e^{-t^{2}/2}\,\Big{(}1-\sqrt{\frac{2}{3}}\,t^{2}+\frac{1}{3}t^{4}\Big{)}$

of a strictly subgaussian random variable. It has four distinct complex zeros
$z_{k}$ defined by $z^{2}=r^{2}\,(1\pm i)$ with $r^{2}=\frac{1}{3}\sqrt{2/3}$,
so

$z_{1}=(2r)^{1/4}\,e^{i\pi/8},\quad z_{2}=(2r)^{1/4}\,e^{-i\pi/8},\quad
z_{3}=(2r)^{1/4}\,e^{7i\pi/8},\quad z_{4}=(2r)^{1/4}\,e^{-7i\pi/8}.$

Note that $|{\rm Arg}(z_{1,2})|=\frac{\pi}{8}$. As already mentioned, it was
necessary that $|{\rm Arg}(z)|\leq\frac{\pi}{8}$ for all zeros with ${\rm
Re}(z)>0$ in the class of all strictly subgaussian probability distributions
with characteristic functions of the form (6.1).

In order to describe the possible location of zeros, let us see what
Proposition 6.2 is telling us about the class of functions

$f(t)=e^{-t^{2}/2}\,(1-wt)(1+wt)(1-\bar{w}t)(t+\bar{w}t),\quad
t\in{\mathbb{R}},$ (7.1)

with $w=a+bi$. Thus, in the complex plane $f(z)$ has two or four distinct
zeros $z=\pm 1/w$, $z=\pm 1/\bar{w}$ depending on whether $b=0$ or $b\neq 0$.
Note that

$|{\rm Arg}(z)|=|{\rm Arg}(w)|$

when $z$ and $w$ are taken from the half-plane ${\rm Re}(z)>0$ and ${\rm
Re}(w)>0$.

Proposition 7.1. Let $w=a+bi$ with $a>0$. The function $f(t)$ in $(7.1)$
represents a characteristic function of a strictly subgaussian random
variable, if and only if

$a\leq 2^{-1/4}\sim 0.8409,$

while $|b|$ is sufficiently small. More precisely, this is the case whenever
$|b|\leq b(a)$ with a certain function $b(a)\geq 0$ such that $b(2^{-1/4})=0$
and $b(a)>0$ for $0<a<2^{-1/4}$.

Moreover, there exists a universal constant $0<a_{0}<2^{-1/4}$, $a_{0}\sim
0.7391$, such that for $0\leq a\leq a_{0}$ and only for these $a$-values, the
property $|b|\leq b(a)$ is equivalent to the angle requirement ${\rm
Arg}(w)\leq\frac{\pi}{8}$. As for the values $a_{0}<a\leq 2^{-1/4}$, this
angle must be smaller.

Proof. We may assume that $b\geq 0$. The function in (7.1) may be expressed in
the form

$f(t)=e^{-t^{2}/2}\,(1-\alpha t^{2}+\beta t^{4})$ (7.2)

with parameters $\alpha=2(A-B)$, $\beta=(A+B)^{2}$, where $A=a^{2}$,
$B=b^{2}$. Since the condition $\alpha\geq 0$ is necessary for $f(t)$ to be a
characteristic function of a strictly subgaussian distribution, we may require
that $a\geq b$, that is, $A\geq B$. Recall that

${\rm
Arg}(w)\leq\frac{\pi}{8}\,\Longleftrightarrow\,\alpha\geq\sqrt{2\beta}\,\Longleftrightarrow\,b\leq\frac{1}{\sqrt{2}+1}\,a.$
(7.3)

In fact, as easy to check, if $w=re^{i\theta}$, then

$\alpha^{2}-2\beta=2\beta\cos(4\theta).$

In order to apply Proposition 6.2, first note that the above parameters
satisfy $\alpha\leq 2\sqrt{\beta}$. In this case, the upper bounds in
(6.4)-(6.6) are fulfilled automatically. Therefore, we only need to take into
account the lower bounds in (6.4)-(6.6). Thus, $f(t)$ in (7.2) represents the
characteristic function of a strongly subgaussian distribution, if and only if

$\frac{1}{\sqrt{2}}\,(A+B)\leq A-B\quad{\rm for}\ \ 0<A+B<\sqrt{\beta_{0}}$
(7.4)

or

$2(A+B)^{2}-(A+B)\sqrt{1-2(A+B)^{2}}\leq A-B\quad{\rm for}\ \
\sqrt{\beta_{0}}\leq A+B\leq\frac{1}{\sqrt{2}},$ (7.5)

where $\beta_{0}=\frac{1}{36}\,(7+2\sqrt{10})\sim 0.3701...$ Since the
condition $A+B\leq 2^{-1/2}$ is necessary, we should require that $a\leq
2^{-1/4}$. Moreover, for $a=2^{-1/4}$, there is only one admissible value
$b=0$, when $w$ is a real number, $w=2^{-1/4}$.

Let us recall that

$\sqrt{2\beta}<4\beta-2\sqrt{\beta(1-2\beta)}\quad{\rm for}\ \
\beta_{0}<\beta\leq\frac{1}{2},$

in which case there is a strict inequality $\alpha>\sqrt{2\beta}$ for
admissible values of $\alpha$ in (6.6). Hence, ${\rm Arg}(w)<\frac{\pi}{8}$
according to (6.3). Thus, ${\rm Arg}(w)<\frac{\pi}{8}$ for the region
described in (7.5).

Turning to the region of couples $(A,B)$ as in (7.4), let us fix a value
$0<A<\sqrt{\beta_{0}}$. The first inequality in (7.4) is equivalent to

$B\leq\frac{\sqrt{2}-1}{\sqrt{2}+1}\,A=\frac{1}{(\sqrt{2}+1)^{2}}\,A,$

which is the same as (7.3). The value $B=\frac{1}{(\sqrt{2}+1)^{2}}\,A$
satisfies the second constraint, if and only if
$(1+\frac{1}{(\sqrt{2}+1)^{2}})\,A\leq\sqrt{\beta_{0}}$, which is equivalent
to $0<a\leq a_{0}$ with

$a_{0}=\beta_{0}^{1/4}\frac{\sqrt{2}+1}{\sqrt{4+2\sqrt{2}}}\sim 0.7391.$ (7.6)

Therefore, in this $a$-interval Proposition 7.1 holds true with
$b(a)=\frac{1}{\sqrt{2}+1}\,a$.

Now, let $a_{0}<a<2^{-1/4}$. Since $A<\frac{1}{2}$, both (7.4) and (7.5) are
fulfilled for all $B$ small enough. Indeed, if $A\geq\sqrt{\beta_{0}}$ and
$B=0$, (7.5) becomes

$2A^{2}-A\sqrt{1-2A^{2}}\leq A,$

which holds with a strict inequality sign. To show that (7.5) is solved as
$B\leq B(A)$ for a certain positive function $B(A)$, it is sufficient to
verify that the left-hand side of (7.5) is increasing in $B$ (since the right-
hand side is decreasing in $B$). Consider the function

$u(x)=2x^{2}-x\sqrt{1-2x^{2}},\quad\sqrt{\beta_{0}}\leq x<\frac{1}{\sqrt{2}}.$

We have

$u^{\prime}(x)=4x-\sqrt{1-2x^{2}}+\frac{2x^{2}}{\sqrt{1-2x^{2}}}\geq 0$

for $x\geq\frac{1}{2}$, hence for $x\geq\sqrt{\beta_{0}}$. Thus, $u(A+B)$ is
increasing in $B$, proving the claim. ∎

##  8\. General Case of Zeros in the Angle $|{\rm Arg}(z)|\leq\frac{\pi}{8}$

We are now prepared to prove Theorem 1.3, which covers the case where the
zeros of the characteristic function

$f(z)={\mathbb{E}}\,e^{izX},\quad z\in{\mathbb{C}},$

of the subgaussian random variable $X$ are not necessarily real, but belong to
the angle $|{\rm Arg}(z)|\leq\frac{\pi}{8}$. Let us state it once more
together with the stronger property (1.5).

Theorem 8.1. Let $X$ be a subgaussian random variable with a symmetric
distribution. If all zeros of $f(z)$ with ${\rm Re}(z)\geq 0$ lie in the angle
$|{\rm Arg}(z)|\leq\frac{\pi}{8}$, then $X$ is strictly subgaussian. Moreover,
if $X$ is not normal, then for any $t_{0}>0$, there exists $c=c(t_{0})$,
$0<c<\sigma^{2}={\rm Var}(X)$, such that

${\mathbb{E}}\,e^{tX}\leq e^{ct^{2}/2},\quad|t|\geq t_{0}.$ (8.1)

In the proof of (8.1) we employ Proposition 2.5, which asserts that (8.1)
would follow from the property that the function
$t\rightarrow\log{\mathbb{E}}\,e^{\sqrt{t}X}$ is concave on the positive half-
axis $t\geq 0$ (in the symmetric case). In this connection recall Proposition
5.1: A random variable $\xi$ with characteristic function

$f_{\xi}(t)=e^{-t^{2}/2}\,(1-\alpha t^{2}+\beta t^{4})$

is strictly subgaussian, if and only if $\beta\geq 0$ and
$\alpha\geq\sqrt{2\beta}$. In fact, the latter description is also equivalent
to the concavity of the function

$t\rightarrow\log{\mathbb{E}}\,e^{\sqrt{t}\xi}=-\frac{1}{2}\,t+\log(1+\alpha
t+\beta t^{2}),\quad t\geq 0.$

That is, we have:

Lemma 8.2. Given $\alpha,\beta\geq 0$, the function $Q(t)=\log(1+\alpha
t+\beta t^{2})$ is concave in $t\geq 0$, if and only if
$\alpha\geq\sqrt{2\beta}$, and then the function $R(t)=\alpha t-Q(t)$ is
convex and non-decreasing.

Indeed, by the direct differentiation,

$R^{\prime}(t)=\frac{(\alpha^{2}-2\beta)t+\alpha\beta t}{1+\alpha t+\beta
t^{2}},\quad Q^{\prime\prime}(t)=-\frac{(\alpha^{2}-2\beta)+2\alpha\beta
t+2\beta^{2}t^{2}}{(1+\alpha t+\beta t^{2})^{2}},$

from which the claim readily follows.

Proof of Theorem 8.1. We may assume that $X$ is not normal. By the symmetry
assumption, with every zero $z=x+iy$, we have more zeros $\pm x\pm iy$. So,
one may arrange all zeros in increasing order of their moduli and by coupling
$\pm z_{1},\pm\bar{z}_{1},\dots$. Let us enumerate only the zeros
$z_{n}=x_{n}+iy_{n}$ lying in the quadrant $x_{n}\geq 0$, $y_{n}\leq 0$ and
deal with $-z_{n},\bar{z}_{n},-\bar{z}_{n}$ as associated zeros. If $z_{n}$ is
real, then we have only one associated zero $-z_{n}$. For simplicity of
notations, let us assume that all zeros are complex.

Since $X$ is subgaussian, the characteristic function $f(t)$ may be extended
from the real line to the complex plane as an entire function satisfying

$|f(z)|\leq e^{b\,{\rm Im}(z)^{2}/2},\quad z\in{\mathbb{C}},$

for some constant $b\geq 0$. Therefore, $f$ is a ridge entire function of
order $\rho\leq 2$ and of a finite type like in the strongly subgaussian case.
Thus, Hadamard’s theorem is applicable, with parameters $\rho\leq 2$ and
$p\leq 2$. In this case, the representation (3.3) takes the form

$f(z)=e^{P(z)}\prod_{n\geq
1}G_{p}(z/z_{n})\,G_{p}(z/\bar{z}_{n})G_{p}(-z/z_{n})\,G_{p}(-z/\bar{z}_{n}).$

Here, the genus of the canonical product satisfies $p\leq 2$, and $P(z)$ is a
polynomial of degree at most 2 such that $P(0)=0$. Thus, putting in the sequel
$w_{n}=\frac{1}{z_{n}}=a_{n}+b_{n}i$, we have

$f(z)=e^{i\beta z-\gamma z^{2}/2}\prod_{n\geq 1}\pi_{p,n}(z)$ (8.2)

for some $\beta,\gamma\in{\mathbb{C}}$, where

$\pi_{p,n}(z)=G_{p}(w_{n}z)\,G_{p}(-w_{n}z)G_{p}(\bar{w}_{n}z)\,G_{p}(-\bar{w}_{n}z).$

By the symmetry assumption, $f(-z)=f(z)$ for all $z\in{\mathbb{C}}$. Since
also $\pi_{p,n}(-z)=\pi_{p,n}(z)$, we conclude that $\beta=0$. Put

$\alpha_{n}=w_{n}^{2}+\bar{w}_{n}^{2}=2(a_{n}^{2}-b_{n}^{2}),\quad\beta_{n}=|w_{n}|^{4}=(a_{n}^{2}+b_{n}^{2})^{2}.$

There are three cases for the values of the genus, $p=0$, $p=1$, and $p=2$,
for which

$G_{0}(u)=1-u,\quad G_{1}(u)=(1-u)\,e^{u},\quad
G_{2}(u)=(1-u)\,e^{u+\frac{u^{2}}{2}}.$

Since

$G_{1}(u)G_{1}(-u)=1-u^{2}=G_{0}(u)G_{0}(-u)\quad{\rm and}\quad
G_{2}(-u)G_{2}(u)=(1-u^{2})\,e^{u^{2}},$

(8.2) is simplified to

$f(z)=e^{-\gamma z^{2}/2}\prod_{n\geq 1}Q_{p,n}(z),$ (8.3)

where

$Q_{0,n}(z)=Q_{1,n}(z)=(1-w_{n}^{2}z^{2})(1-\bar{w}_{n}^{2}z^{2})=1-\alpha_{n}z^{2}+\beta_{n}z^{4}$

and

$Q_{2,n}(z)=(1-w_{n}^{2}z^{2})(1-\bar{w}_{n}^{2}z^{2})\,e^{(w_{n}^{2}+\bar{w}_{n}^{2})z^{2}}=(1-\alpha_{n}z^{2}+\beta_{n}z^{4})\,e^{\alpha_{n}z^{2}}.$

These functions are real-valued for $z=t\in{\mathbb{R}}$, as well as $f(t)$,
by the symmetry assumption on the distribution of $X$. Hence, necessarily
$\gamma\in{\mathbb{R}}$. Moreover, we have $\gamma\geq 0$, since otherwise
$f(t)$ would not be bounded on the real axis $t\in{\mathbb{R}}$.

Since ${\rm Arg}(z_{n})=-{\rm Arg}(w_{n})$, we have ${\rm
Arg}(w_{n})\leq\frac{\pi}{8}$, by the main angle hypothesis. In particular,
$a_{n}>b_{n}>0$ so that $\alpha_{n}>0$ (since $x_{n}>0$, $y_{n}<0$). As
already noticed in the proof of Theorem 1.4, the angle hypothesis is
equivalent to the relation $\alpha_{n}^{2}\geq 2\beta_{n}$.

Applying (8.3) with $z=it$, $t\in{\mathbb{R}}$, we get that

${\mathbb{E}}\,e^{tX}=e^{\gamma t^{2}/2}\prod_{n\geq 1}Q_{p,n}(it)$ (8.4)

with positive factors given by

$Q_{0,n}(it)=Q_{1,n}(it)=1+\alpha_{n}t^{2}+\beta_{n}t^{4},\qquad
Q_{2,n}(it)=(1+\alpha_{n}t^{2}+\beta_{n}t^{4})\,e^{-\alpha_{n}t^{2}}.$

We have already observed in the proof of Proposition 5.1 that, by the angle
hypothesis,

$1+\alpha_{n}t^{2}+\beta_{n}t^{4}<e^{\alpha_{n}t^{2}},\quad t>0,$ (8.5)

so that $Q_{2,n}(it)<1$. Moreover, this inequality was strengthened by
improving the constant $\alpha_{n}$ in the exponent, provided that $t$ is
bounded away from zero. We will thus repeat some steps from the proof of
Proposition 5.1. However, formally, we need to consider the three cases
separately according to the three possible values of $p$.

Genus $p=2$. By the very definition of the genus,

$\sum_{n\geq 1}|w_{n}|^{3}=\sum_{n\geq
1}(a_{n}^{2}+b_{n}^{2})^{3/2}=\sum_{n\geq 1}\beta_{n}^{3/4}<\infty.$

Since $Q_{2,n}(it)=1+O(\beta_{n}t^{4})$ as $t\rightarrow 0$, the product in
(8.4) is absolutely convergent. Moreover, the right-hand side of (8.4) near
zero is $1+\gamma t^{2}+O(t^{3})$. Hence, necessarily $\gamma=\sigma^{2}$, and
(8.4) becomes

${\mathbb{E}}\,e^{tX}=e^{\sigma^{2}t^{2}/2}\prod_{n\geq 1}Q_{2,n}(it).$ (8.6)

Recalling the bound $Q_{2,n}(it)\leq 1$, we conclude that

${\mathbb{E}}\,e^{tX}\leq e^{\sigma^{2}t^{2}/2},\quad t\in{\mathbb{R}},$ (8.7)

which means that $X$ is strictly subgaussian. For the second claim of the
theorem, write

${\mathbb{E}}\,e^{tX}=e^{V(t^{2})},$ (8.8)

where

$V(s)=\frac{1}{2}\gamma s+\sum_{n\geq 1}\log
Q_{n}(it)=\frac{1}{2}\sigma^{2}s+\sum_{n\geq
1}\big{[}\log(1+\alpha_{n}s+\beta_{n}s^{2})-\alpha_{n}s\big{]},\quad s\geq 0,$

and define

$W(s)=\frac{1}{2}\sigma^{2}s-V(s)=\sum_{n\geq 1}R_{n}(s),\quad
R_{n}(s)=\alpha_{n}s-\log(1+\alpha_{n}s+\beta_{n}s^{2}).$

By Lemma 8.2, and using the assumption $\alpha_{n}^{2}\geq 2\beta_{n}$, all
$R_{n}(s)>0$ for $s>0$, representing convex increasing functions. Hence, $W$
is a convex increasing function with $W(0)=0$. It remains to apply Proposition
2.5, and we obtain the property (8.1).

Genus $p=1$. By definition, the following sum converges

$\sum_{n\geq 1}|w_{n}|^{2}=\sum_{n\geq 1}(a_{n}^{2}+b_{n}^{2})=\sum_{n\geq
1}\beta_{n}^{1/2}<\infty.$

Since

$\alpha_{n}=2(a_{n}^{2}-b_{n}^{2})\leq
2(a_{n}^{2}+b_{n}^{2})=2\beta_{n}^{1/2},$

the product in (8.4) is convergent. Moreover, the right-hand side of (8.4)
near zero is

$1+\frac{1}{2}\gamma t^{2}+t^{2}\sum_{n\geq 1}\alpha_{n}+O(t^{3}).$

Hence, necessarily $\frac{1}{2}\sigma^{2}=\frac{1}{2}\gamma+\sum_{n\geq
1}\alpha_{n}$, so that the characteristic function and the Laplace transform
admit the same representation (8.6). As a result, since the summation property
defining the genus became stronger, we immediately obtain (8.7) and its
improvement (8.1) using the previous step.

Genus $p=0$. By definition, the following sum converges

$\sum_{n\geq 1}|w_{n}|=\sum_{n\geq 1}(a_{n}^{2}+b_{n}^{2})^{1/2}=\sum_{n\geq
1}\beta_{n}^{1/4}<\infty.$

Since this assumption is stronger than the one of the previous step, while
$Q_{0,n}=Q_{1,n}$, we are reduced to the previous step. ∎

##  9\. Proof of Theorem 1.5

As in the proof of Theorem 8.1, let us enumerate the points
$z_{n}=x_{n}+iy_{n}$ lying in the quadrant $x_{n}\geq 0$, $y_{n}\leq 0$ and
deal with $-z_{n},\bar{z}_{n},-\bar{z}_{n}$ as associated zeros. For
simplicity of notations, we assume that all these numbers are complex. Put
$w_{n}=\frac{1}{z_{n}}=a_{n}+b_{n}i$ and define

$\displaystyle f_{n}(z)$ $\displaystyle=$ $\displaystyle
e^{-\gamma_{n}z^{2}/2}\,(1-w_{n}z)(1+w_{n}z)(1-\bar{w}_{n}z)(1+\bar{w}_{n}z)$
$\displaystyle=$ $\displaystyle
e^{-\gamma_{n}z^{2}/2}\,(1-\alpha_{n}z^{2}+\beta_{n}z^{4}),\quad
z\in{\mathbb{C}},$

for a given sequence $\gamma_{n}>0$ (to be precised later on) with
$\alpha_{n}=2(a_{n}^{2}-b_{n}^{2})$ and $\beta_{n}=(a_{n}^{2}+b_{n}^{2})^{2}$
as before. By the assumption, $a_{n},b_{n}>0$. Moreover, the angle assumption
$|{\rm Arg}(z_{n})|={\rm Arg}(w_{n})\leq\frac{\pi}{8}$ is equivalent to
$\alpha_{n}^{2}\geq 2\beta_{n}$, which may also be written as

$b_{n}\leq\frac{1}{\sqrt{2}+1}\,a_{n}.$ (9.1)

Now, if $\gamma_{n}$ is sufficiently large, $f_{n}(t)$, $t\in{\mathbb{R}}$,
will be the characteristic function of a strictly subgaussian distribution. A
full description of the minimal possible value of $\gamma_{n}$ is provided in
Proposition 7.1. More precisely, consider the function

$\displaystyle g_{n}(t)\,=\,f_{n}\Big{(}\frac{t}{\sqrt{\gamma_{n}}}\Big{)}$
$\displaystyle=$ $\displaystyle
e^{-t^{2}/2}\,(1-w_{n}^{\prime}t)(1+w_{n}^{\prime}t)(1-\bar{w}_{n}^{\prime}t)(1+\bar{w}_{n}^{\prime}t)$
$\displaystyle=$ $\displaystyle
e^{-t^{2}/2}\,(1-\alpha_{n}^{\prime}t^{2}+\beta_{n}^{\prime}t^{4})$

with

$w_{n}^{\prime}=a_{n}^{\prime}+b_{n}^{\prime}i,\quad
a_{n}^{\prime}=\frac{a_{n}}{\sqrt{\gamma_{n}}},\quad
b_{n}^{\prime}=\frac{b_{n}}{\sqrt{\gamma_{n}}},\quad\alpha_{n}^{\prime}=\frac{2(a_{n}^{2}-b_{n}^{2})}{\gamma_{n}},\quad\beta_{n}^{\prime}=\frac{(a_{n}^{2}+b_{n}^{2})^{2}}{\gamma_{n}^{2}}.$

As we know, $g_{n}(t)$ represents the characteristic function of a strictly
subgaussian random variable $X_{n}^{\prime}$, as long as

$b_{n}^{\prime}\leq\frac{1}{\sqrt{2}+1}\,a_{n}^{\prime},\quad
a_{n}^{\prime}\leq a_{0},$

where the universal constant $a_{0}$ was explicitly identified in (7.6),
$a_{0}\sim 0.7391$. Here, the first condition is satisfied in view of (9.1),
while the second one is equivalent to

$\gamma_{n}\geq\frac{a_{n}^{2}}{a_{0}^{2}}.$ (9.2)

Moreover, $X_{n}^{\prime}$ has variance

${\rm
Var}(X_{n}^{\prime})=-g_{n}^{\prime\prime}(0)=2\alpha_{n}^{\prime}+1=\frac{4(a_{n}^{2}-b_{n}^{2})}{\gamma_{n}}+1.$

Thus, subject to (9.2), $f_{n}(t)$ will be the characteristic function of the
strictly subgaussian random variable
$X_{n}=\sqrt{\gamma_{n}}\,X_{n}^{\prime}$, whose variance is given by

${\rm Var}(X_{n})=4(a_{n}^{2}-b_{n}^{2})+\gamma_{n}.$ (9.3)

Now, assuming that $\Lambda\geq 4+\frac{1}{a_{0}^{3}}\sim 5.83$, let us choose

$\gamma_{n}=(\Lambda-4)a_{n}^{2}+(\Lambda+4)b_{n}^{2},$

so that the expression in (9.3) would be equal to
$\Lambda(a_{n}^{2}+b_{n}^{2})=\Lambda|w_{n}|^{2}$. Then the condition (9.2) is
satisfied, and also $\sum_{n}\gamma_{n}<\infty$. As a result, the series
$\sum_{n}X_{n}$ is convergent with probability one, and the sum of the series,
call it $X$, represents a strictly subgaussian random variable with
characteristic function

$f(z)=\prod_{n}f_{n}(z)$

(cf. Proposition 2.2). By the construction, all $f_{n}(z)$ have exactly
prescribed zeros, and

${\rm Var}(X)=\sum_{n}{\rm Var}(X_{n})=\Lambda\sum_{n}|w_{n}|^{2}.$

∎

##  10\. Laplace Transforms with Periodic Components

We now turn to a second class of Laplace transforms – the ones that contain
periodic components. Recall that a random variable $X$ belongs to the class
$\mathfrak{F}_{h}$, $h>0$, if it has a density $p(x)$ such that the function

$q(x)=\frac{p(x)}{\varphi(x)},\quad x\in{\mathbb{R}},$

is periodic with period $h$. This section is devoted to basic properties of
this class (some of them will be used in the proof of Theorem 1.6).

Proposition 10.1. If $X$ belongs to the class $\mathfrak{F}_{h}$, then for all
integers $m$,

${\mathbb{E}}\,e^{mhX}=e^{(mh)^{2}/2}.$ (10.1)

In particular, the random variable $X$ is subgaussian.

Proof. By the periodicity of $q$, the random variable $X+mh$ has density

$\displaystyle p(x-mh)$ $\displaystyle=$ $\displaystyle q(x-mh)\varphi(x-mh)$
$\displaystyle=$ $\displaystyle
q(x)\varphi(x)\,e^{mhx-(mh)^{2}/2}\,=\,p(x)\,e^{mhx-(mh)^{2}/2}.$

It remains to integrate this equality over $x$, which leads to (10.1).

Next, starting from (10.1), it is easy to see that
${\mathbb{E}}\,e^{cX^{2}}<\infty$ for some $c>0$. ∎

As a consequence, the Laplace transform $L(t)={\mathbb{E}}\,e^{tX}$,
$t\in{\mathbb{R}}$, is finite and may be extended to the complex plane as an
entire function. This property may be refined.

Proposition 10.2. If $X$ belongs to $\mathfrak{F}_{h}$, then its Laplace
transforms is an entire function of order 2. Moreover, if ${\mathbb{E}}X=0$,
it satisfies

$|L(z)|\leq e^{(|t|+h)^{2}/2},\quad z=t+iy\in{\mathbb{C}}.$ (10.2)

Proof. We may assume that ${\mathbb{E}}X=0$. In this case, by Jensen’s
inequality, $L(t)\geq 1$ for all $t\in{\mathbb{R}}$, so that $t=0$ is the
point of miminum of $L$ on the real line. Since $L(t)$ is convex (and
moreover, $\log L(t)$ is convex), $L(t)$ is decreasing for $t<0$ and is
increasing for $t>0$.

Given $t\geq 0$, take an integer number $m\geq 1$ such that $(m-1)h\leq t<mh$.
Then, by (10.1), and using the monotonicity of $L$, we get

$L(t)\leq L(mh)=e^{(mh)^{2}/2}\leq e^{(t+h)^{2}/2}.$ (10.3)

By a similar argument, $L(-t)\leq e^{(t+h)^{2}/2}$. Thus, we obtain (10.2) for
real values of $z$ (when $y=0$). In the general case, it remains just to note
that $|L(z)|\leq L(t)$, and we obtain (10.2). This bound shows that $L(z)$ is
an entire function of order at most 2.

On the other hand, (10.1) shows that $L(z)$ is an entire function of order at
least 2. ∎

Proposition 10.3. If $X$ belongs to $\mathfrak{F}_{h}$, then the function

$\Psi(t)=L(t)\,e^{-t^{2}/2},\quad t\in{\mathbb{R}},$ (10.4)

is periodic with period $h$. It can be extended to the complex plane as an
entire function. Moreover, if ${\mathbb{E}}X=0$, it satisfies

$|\Psi(z)|\leq C_{h,y}\,e^{h|t|},\quad z=t+iy\in{\mathbb{C}},$ (10.5)

with $C_{h,y}=e^{(h^{2}+y^{2})/2}$.

The inequality (10.5) shows that $\Psi$ is an entire function of order at most
2.

By analyticity and periodicity on the real line,

$\Psi(z+h)=\Psi(z)\quad{\rm for\ all}\ z\in{\mathbb{C}}.$ (10.6)

Proof. By periodicity of $q$, changing the variable $x=y+h$, we have

$\displaystyle L(t+h)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}e^{(t+h)\,x}\,q(x)\,\varphi(x)\,dx\,=\,\int_{-\infty}^{\infty}e^{(t+h)\,(y+h)}\,q(y+h)\,\varphi(y+h)\,dy$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}e^{(t+h)\,(y+h)}\,q(y)\,\varphi(y)\,e^{-yh-h^{2}/2}\,dy\,=\,L(t)\,e^{th+h^{2}/2}.$

Hence

$L(t+h)\,e^{-(t+h)^{2}/2}=L(t)\,e^{-t^{2}/2},$

which was the first claim. Since $L(z)$ is an entire function, $\Psi(z)$ is
entire as well.

Next, assuming that ${\mathbb{E}}X=0$, one may apply (10.2) which gives
$\Psi(t)\leq C_{h}\,e^{h|t|}$ with $C_{h}=e^{h^{2}/2}$. Thus, we obtain (10.5)
for real values of $z$. In the general case, for simplicity let $t={\rm
Re}(z)\geq 0$. By the previous step,

$|L(z)|\leq L(t)\leq C_{h}\,e^{ht}.$

Hence

$|\Psi(z)|\leq L(t)\,e^{{\rm Re}(z^{2})/2}\leq
e^{(t+h)^{2}/2}\,e^{(y^{2}-t^{2})/2}=C_{h,y}\,e^{ht}.$

∎

Let us also examine the periodicity property for convolutions. Here, the basic
observation concerns the normalized sums
$Z_{n}=\frac{X_{1}+\dots+X_{n}}{\sqrt{n}}$, where $X_{k}$’s are independent
copies of the random variable $X$.

Proposition 10.4. If $X$ belongs to $\mathfrak{F}_{h}$, then $Z_{n}$ belongs
to $\mathfrak{F}_{h\sqrt{n}}$.

Proof. Let $n=2$ for simplicity of notations. Let $p$ be the density of $X$
such that $q=p/\varphi$ is $h$-periodic. Since $X_{1}+X_{2}$ has density

$\int_{-\infty}^{\infty}p(x-y)\,p(y)\,dy=\int_{-\infty}^{\infty}p\Big{(}\frac{x}{2}+z\Big{)}\,p\Big{(}\frac{x}{2}-z\Big{)}\,dz,$

the density of $Z_{2}=\frac{X_{1}+X_{2}}{\sqrt{2}}$ may be written as

$\displaystyle p_{2}(x)$ $\displaystyle=$
$\displaystyle\sqrt{2}\int_{-\infty}^{\infty}q\Big{(}\frac{x}{\sqrt{2}}+z\Big{)}\,q\Big{(}\frac{x}{\sqrt{2}}-z\Big{)}\,\varphi\Big{(}\frac{x}{\sqrt{2}}+z\Big{)}\,\varphi\Big{(}\frac{x}{\sqrt{2}}-z\Big{)}\,dz$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\pi}}\,\varphi(x)\int_{-\infty}^{\infty}q\Big{(}\frac{x}{\sqrt{2}}+z\Big{)}\,q\Big{(}\frac{x}{\sqrt{2}}-z\Big{)}\,e^{-z^{2}}\,dz.$

Thus, the correspondong $q$-function for $Z_{2}$ is given by

$q_{2}(x)=\frac{p_{2}(x)}{\varphi(x)}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}q\Big{(}\frac{x}{\sqrt{2}}+z\Big{)}\,q\Big{(}\frac{x}{\sqrt{2}}-z\Big{)}\,e^{-z^{2}}\,dz.$

As $q$ is $h$-periodic, the last integrand is periodic with respect to the
variable $x$, with period $h\sqrt{2}$. Consequently,
$q_{2}(x+h\sqrt{2})=q_{2}(x)$ for all $x\in{\mathbb{R}}$. ∎

##  11\. Proof of Theorem 1.6

In view of the previous observations, we only need to consider the necessity
part in the statement of Theorem 1.6 and prove the periodicity of the density
$q$.

Since $X$ is subgaussian, its Laplace transform is an entire function of order
at most 2. Hence $\Psi(z)$ is also entire and satisfies (10.6). Thus, the
characteristic function of $X$ is an entire function representable in the
complex plane as $f(z)=\Psi(iz)\,e^{-z^{2}/2}$. Hence, by (10.6),

$f(t+ih)\,e^{(t+ih)^{2}/2}=f(t)\,e^{t^{2}/2}$

for all $t\in{\mathbb{R}}$, that is,

$f(t+ih)=f(t)\,e^{-ith+h^{2}/2}.$ (11.1)

By the integrability assumption, the random variable $X$ has a continuous
density $p(x)$ given by the Fourier inversion formula

$p(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-itx}f(t)\,dt,\quad
x\in{\mathbb{R}}.$

This yields

$q(x)=\frac{p(x)}{\varphi(x)}=\frac{1}{\sqrt{2\pi}}\,e^{x^{2}/2}\int_{-\infty}^{\infty}e^{-itx}f(t)\,dt$

and

$q(x+h)=\frac{1}{\sqrt{2\pi}}\,e^{x^{2}/2}\,e^{xh+h^{2}/2}\int_{-\infty}^{\infty}e^{-itx-
ith}f(t)\,dt.$

Hence, we need to show that

$\int_{-\infty}^{\infty}e^{-itx}f(t)\,dt=e^{xh+h^{2}/2}\int_{-\infty}^{\infty}e^{-itx-
ith}f(t)\,dt.$ (11.2)

Using contour integration, one may rewrite the first integral in a different
way. Given $T>0$, consider the rectangle contour with sides

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle[-T,T],\qquad\qquad\quad
C_{2}=[T,T+ih],$ $\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle[T+ih,-T+ih],\quad C_{4}=[-T+ih,-T],$

so that to apply Cauchy’s theorem and write down

$\int_{C_{1}}e^{-izx}f(z)\,dz+\int_{C_{2}}e^{-izx}f(z)\,dz+\int_{C_{3}}e^{-izx}f(z)\,dz+\int_{C_{4}}e^{-izx}f(z)\,dz=0.$

For points $z=t+iy$ on the contour, we have $|e^{-izx}|=e^{xy}\leq e^{|x|h}$.
In addition, $f(z)\rightarrow 0$ as $|t|\rightarrow\infty$ uniformly over all
$y$ such that $|y|\leq h$. This follows from the fact that the functions
$t\rightarrow f(t+iy)$ represent the Fourier transform of the functions
$p_{y}(x)=e^{-xy}p(x)$. Indeed, by the subgaussian assumption, the family
$\\{p_{y}:|y|\leq h\\}$ is pre-compact in $L^{1}({\mathbb{R}}^{n})$, so that
the Riemeann-Lebesgue lemma is applicable to the whole family. As a
consequence,

$\displaystyle\int_{-\infty}^{\infty}e^{-itx}f(t)\,dt$ $\displaystyle=$
$\displaystyle\lim_{T\rightarrow\infty}\int_{C_{1}}e^{-izx}f(z)\,dz$
$\displaystyle\hskip-28.45274pt=\
-\lim_{T\rightarrow\infty}\int_{C_{3}}e^{-izx}f(z)\,dz\,=\,\int_{-\infty}^{\infty}e^{-i(t+ih)x}f(t+ih)\,dt.$

where the last integral is convergent due to (11.1). Moreover, by (11.1), the
last integrand is equal to

$e^{-i(t+ih)x}\,\,e^{-ith+h^{2}/2}\,f(t),$

which coincides with the integrand on the right-hand side of (11.2) multiplied
by the indicated factor. This proves (11.2). ∎

Remark 11.1. Since $f(t)=L(it)=\Psi(it)\,e^{-t^{2}/2}$, the integrability
assumption in Theorem 1.6 is fulfilled, as long as $\Psi(z)$ has order smaller
than 2.

##  12\. Examples Involving Triginometric Series

Theorem 1.6 is applicable to a variety of interesting examples including the
underlying distributions whose Laplace transform has the form

$L(t)=\Psi(t)\,e^{t^{2}/2},\quad t\in{\mathbb{R}},$

where $\Psi$ is a $2\pi$-periodic functions of the form

$\Psi(t)=1-cP(t),\quad
P(t)=a_{0}+\sum_{k=1}^{\infty}(a_{k}\cos(kt)+b_{k}\sin(kt)).$ (12.1)

Here $a_{k},b_{k}$ are real coefficients which are supposed to satisfy

$\sum_{k=1}^{\infty}e^{k^{2}/2}\,(|a_{k}|+|b_{k}|)<\infty,$ (12.2)

and $c\in{\mathbb{R}}$ is a non-zero parameter.

Proposition 12.1. If $P(0)=P^{\prime}(0)=P^{\prime\prime}(0)=0$ and $|c|$ is
small enough, then $L(t)$ represents the Laplace transform of a subgaussian
random variable $X$ with ${\mathbb{E}}X=0$, ${\mathbb{E}}X^{2}=1$, and with
density $p=q\varphi$, where $q$ is a bounded, $2\pi$-periodic function. This
random variable is strictly subgaussian, if $P(t)\geq 0$ for all
$t\in{\mathbb{R}}$ and if $c>0$ is small enough.

Proof. The functions of the form $u_{\lambda}(x)=\cos(\lambda x)\,\varphi(x)$
and $v_{\lambda}(x)=\sin(\lambda x)\,\varphi(x)$ have respectively the Laplace
transforms

$\displaystyle\int_{-\infty}^{\infty}e^{tx}u_{\lambda}(x)\,dx$
$\displaystyle=$ $\displaystyle e^{-\lambda^{2}/2}\,\cos(\lambda
t)\,e^{t^{2}/2},$
$\displaystyle\int_{-\infty}^{\infty}e^{tx}v_{\lambda}(x)\,dx$
$\displaystyle=$ $\displaystyle e^{-\lambda^{2}/2}\,\sin(\lambda
t)\,e^{t^{2}/2}.$

Define

$q(x)=\varphi(x)-c\varphi(x)\Big{(}a_{0}+\sum_{k=1}^{\infty}e^{k^{2}/2}\,(a_{k}\cos(kx)+b_{k}\sin(kx))\Big{)}.$
(12.3)

In this case, the Laplace transform of the function $p(x)=q(x)\varphi(x)$ is
exactly

$\int_{-\infty}^{\infty}e^{tx}p(x)\,dx=(1-cP(t))\,e^{-t^{2}/2}.$

The requirement $P(0)=0$ guarantees that $\int_{-\infty}^{\infty}p(x)\,dx=1$.
Moreover, according to (12.3), the condition on the parameter $c$ which
ensures that the function $p$ is indeed a probability density may be stated as

$|a_{0}|+\sum_{k=1}^{\infty}e^{k^{2}/2}\,(|a_{k}|+|b_{k}|)\leq\frac{1}{|c|}.$

This is fulfilled due to (12.1) when $|c|$ is small enough. Finally, the
properties ${\mathbb{E}}X=0$, ${\mathbb{E}}X^{2}=1$ are equivalent to
$P^{\prime}(0)=P^{\prime\prime}(0)=0$. ∎

Note that in terms of the coeficients in the series (12.1), the condition
$P(0)=P^{\prime}(0)=P^{\prime\prime}(0)=0$ has the form

$a_{0}+\sum_{k=1}^{\infty}a_{k}=\sum_{k=1}^{\infty}kb_{k}=\sum_{k=1}^{\infty}k^{2}a_{k}=0.$

It should also be mentioned that, when $P$ is a trigonometric polynomial of
degree $N$, the function $q(x)$ in (12.3) will be a trigonometric polynomial
of degree $N$ as well.

Example 12.2. As a particular case, one may consider the transforms

$L(t)=(1-c\sin^{m}(t))\,e^{t^{2}/2}$ (12.4)

with an arbitrary integer $m\geq 3$, where $c$ is small enough. Then
${\mathbb{E}}X=0$, ${\mathbb{E}}X^{2}=1$, and the cumulants of $X$ satisfy

$\gamma_{k}(X)=0,\quad 3\leq k\leq m-1.$

Moreover, if $m\geq 4$ is even and $c>0$, the random variable $X$ with the
Laplace transform (12.3) is strictly subgaussian. In the case $m=4$, (12.4)
corresonds to the $\pi$-periodic polynomial
$P(t)=\sin^{4}t=\frac{1}{8}\,(3-4\cos(2t)+\cos(4t))$.

##  13\. Examples Involving Poisson Formula and Theta Functions

Often, the periodic functions $\Psi(t)$ in (12.1) appear naturally by means of
the Poisson formula, rather than as a trigonometric series. Let $w(t)\geq 0$
be an integrable, even, absolutely continuous function on the real line with
Fourier transform

$\hat{w}(x)=\int_{-\infty}^{\infty}e^{itx}w(t)\,dt,\quad x\in{\mathbb{R}}.$

As a natrural generalization of Example 12.2 with $m=4$, we have the following
corollary from Proposition 12.1 assuming that

$\sum_{k=1}^{\infty}e^{(k+4)^{2}/2}\,|\hat{w}(k)|<\infty.$ (13.1)

Corollary 13.1. For all $c>0$ small enough,

$L(t)=\Psi(t)\,e^{t^{2}/2},\quad\Psi(t)=1-c\,(\sin
t)^{4}\sum_{m\in{\mathbb{Z}}}w(t+2\pi m),$

represents the Laplace transform of a strictly subgaussian random variable $X$
with ${\mathbb{E}}X=0$, ${\mathbb{E}}X^{2}=1$, which has density
$p(x)=q(x)\varphi(x)$, where $q(x)$ is a $2\pi$-periodic function.

Proof. The function $Q(t)=\sum_{m\in{\mathbb{Z}}}w(t+2\pi m)$ is well-defined
(since the series is absolutely convergent), $2\pi$-periodic, and admits a
Fourier series expansion

$Q(t)=\frac{1}{2\pi}\sum_{k\in{\mathbb{Z}}}\hat{w}(k)\,e^{-ikt}.$

This is a well-known Poisson formula, in which the series is understood as a
limit of symmetric partial sums, cf. e.g. [27], p. 68. Under (13.1), this
series is absolutely convergent and defines a smooth function. By the symmetry
$\hat{w}(-k)=\hat{w}(k)$, $k\in{\mathbb{Z}}$, this formula takes the form

$Q(t)=\frac{1}{2\pi}\,\Big{[}\hat{w}(0)+2\sum_{k=1}^{\infty}\hat{w}(k)\cos(kt)\Big{]}.$

Using $\sin^{4}t=\frac{1}{8}\,(3-4\cos(2t)+\cos(4t))$, we have

$\displaystyle 16\pi\,(\sin t)^{4}\,Q(t)$ $\displaystyle=$ $\displaystyle
3\hat{w}(0)-4\hat{w}(0)\cos(2t)+\hat{w}(0)\cos(4t)$
$\displaystyle\hskip-71.13188pt+\
6\sum_{k=1}^{\infty}\hat{w}(k)\cos(kt)-8\sum_{k=1}^{\infty}\hat{w}(k)\cos(kt)\cos(2t)+2\sum_{k=1}^{\infty}\hat{w}(k)\cos(kt)\cos(4t).$

Applying the identity $\cos a\cos
b=\frac{1}{2}\,\cos(a+b)+\frac{1}{2}\,\cos(a-b)$, one may rewrite the last
line as

$6\sum_{k=1}^{\infty}\hat{w}(k)\cos(kt)-4\sum_{k=1}^{\infty}\hat{w}(k)(\cos((k+2)t)+\cos((k-2)\,t))$
$+\sum_{k=1}^{\infty}\hat{w}(k)(\cos((k+4)t)+\cos((k-4)\,t)).$

Hence for $k\geq 5$ the coefficients $a_{k}$ in the Fourier series for
$P(t)=(\sin t)^{4}\,Q(t)$ are given by

$16\pi\,a_{k}=6\hat{w}(k)-4\hat{w}(k+2)-4\hat{w}(k-2)+\hat{w}(k+4)+\hat{w}(k-4).$

Hence, the condition (12.2) is fulfilled under (13.1), and one may apply
Proposition 12.1. ∎

Example 13.2. One may further apply Corollary 13.1 to the theta functions
$Q(t)$ corresponding to

$w(t)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-t^{2}/2\sigma^{2}},\quad\hat{w}(x)=e^{-\sigma^{2}x^{2}/2}$

with an arbitrary parameter $\sigma>1$.

##  14\. Central Limit Theorem for Rényi Distances

Finally, let us describe the role of subgaussian distributions in the central
limit theorem with respect to the Rényi divergences $D_{\alpha}$ defined in
(1.3). Consider the normalized sums

$Z_{n}=\frac{X_{1}+\dots+X_{n}}{\sqrt{n}},$

where $X_{k}$’ are independent copies of a random variable $X$ with mean zero
and variance one. Assuming that $Z_{n}$ have densities $p_{n}$ for some or
equivalently for all sufficiently large $n$, the following characterization
was obtained in [3], which we state in dimension one.

Theorem 14.1. Fix $1<\alpha<\infty$. For the convergence

$D_{\alpha}(p_{n}||\varphi)\rightarrow 0\ {\rm as}\ n\rightarrow\infty,$
(14.1)

it is necessary and sufficient that $D_{\alpha}(p_{n}||\varphi)<\infty$ for
some $n=n_{0}$, and

${\mathbb{E}}\,e^{tX}<e^{\beta t^{2}/2}\quad{\sl for\ all}\ t\neq 0,$ (14.2)

where $\beta=\frac{\alpha}{\alpha-1}$ is the conjugate index.

Thus, for the CLT as in (14.1), the random variable $X$ has to be subgaussian.
In order to obtain this convergence for all $\alpha$ simultaneously, the
condition (14.2) on the Laplace transform should be fulfilled for all
$\beta>1$. But this is equivalent to saying that $X$ is strictly subgaussian,
thus proving Theorem 1.1.

In this connection, it is natural to raise the question of whether or not
(14.1) may hold for the critical index $\alpha=\infty$, which corresponds to
the strongest distance in this hierarchy. Note that in the limit case it is
defined to be

$D_{\infty}(p_{n}||\varphi)=\lim_{\alpha\rightarrow\infty}D_{\alpha}(p_{n}||\varphi)=\log\,{\rm
ess\,sup}_{x\in{\mathbb{R}}}\ \frac{p_{n}(x)}{\varphi(x)}.$

As an equivalent quantity, one may also consider the limit Tsallis distance

$T_{\infty}(p_{n}||\varphi)={\rm ess\,sup}_{x\in{\mathbb{R}}}\
\frac{p_{n}(x)-\varphi(x)}{\varphi(x)}.$

Suppose it is finite for some $n=n_{0}$. The following two theorems can be
obtained using recent results on the sharpened Richter-type local limit
theorem, cf. [4].

Theorem 14.2. Suppose that, for every $t_{0}>0$,

${\mathbb{E}}\,e^{tX}\leq\delta e^{t^{2}/2}\quad{\sl for\ all}\ |t|\geq t_{0}$
(14.3)

with some $\delta=\delta(t_{0})\in(0,1)$. Then

$T_{\infty}(p_{n}||\varphi)=O\Big{(}\frac{(\log n)^{3}}{n}\Big{)}\quad{\sl
as}\ n\rightarrow\infty.$ (14.4)

Note that (14.3) is a weakened form of the separation property (1.5), which in
turn is a sharpening of strict subgaussianity. In particular, this rate for
the convergence in $D_{\infty}$ holds true for all distributions from the
class $\mathfrak{L}$ whose densities $p(x)$ are dominated by $\varphi(x)$.

A similar assertion holds true in the period case.

Theorem 14.3. Suppose that $X$ is strictly subgaussian, with an $h$-periodic
function $\Psi(t)=L(t)\,e^{-t^{2}/2}$, $h>0$. If $\Psi(t)<1$ in the interval
$0<t<h$, then $(14.4)$ is true as well.

## References

  * [1] Arbel, J.; Marchal, O.; Nguyen, H. D. On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables. ESAIM Probab. Stat. 24 (2020), 39–55.
  * [2] Berend, D.; Kontorovich, A. On the concentration of the missing mass. Electron. Commun. Probab. 18 (2013), no. 3, 7 pp.
  * [3] Bobkov, S. G.; Chistyakov, G. P.; Götze, F. Rényi divergence and the central limit theorem. Ann. Probab. 47 (2019), no. 1, 270–323.
  * [4] Bobkov, S. G.; Chistyakov, G. P.; Götze, F. Richter’s local limit theorem, its refinement, and related results. Preprint (2023). To appear in: Lithuanian J. Math.
  * [5] Bobkov, S. G.; Götze, F. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999), no. 1, 1–28.
  * [6] Bobkov, S. G.; Houdré, C.; Tetali, P. The subgaussian constant and concentration inequalities. Israel J. Math. 156 (2006), 255–283.
  * [7] Buldygin, V. V.; Kozachenko, Yu. V. Sub-Gaussian random variables. (Russian) Ukrain. Mat. Zh. 32 (1980), no. 6, 723–730.
  * [8] Buldygin, V. V.; Kozachenko, Yu. V. Metric characterization of random variables and random processes. Translated from the 1998 Russian original by V. Zaiats. Transl. Math. Monogr., 188 American Mathematical Society, Providence, RI, 2000. xii+257 pp.
  * [9] Diaconis, P.; Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996), no. 3, 695–750.
  * [10] Guionnet, A.; Husson, J. Large deviations for the largest eigenvalue of Rademacher matrices. Ann. Probab. 48 (2020), no. 3, 1436–1465.
  * [11] Goldberg, A. A.; Ostrovskiĭ, I. V. The growth of entire ridge functions with real zeros. (Russian) Mathematical physics and functional analysis, No. V (Russian), pp. 3–10, 156. Akad. Nauk Ukrain. SSR Fiz.-Tehn. Inst. Nizkih Temperatur, Kharkov, 1974.
  * [12] Havrilla, A.; Nayar, P.; Tkocz, T. Khinchin-type inequalities via Hadamard’s factorisation. Int. Math. Res. Not., no. 3 (2023), 2429–2445.
  * [13] Higuchi, Y.; Yoshida, N. Analytic conditions and phase transition for Ising models. Lecture notes (in Japanese), 1995.
  * [14] Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman. Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp.
  * [15] Kahane, J.-P. Propriétés locales des fonctions à séries de Fourier aléatoires. Studia Math. 19 (1960), 1–25.
  * [16] Kamynin, I. P. A generalization of Marcinkiewicz’s theorem on entire characteristic functions of probability distributions. (Russian) Investigations in the theory of probability distributions, IV. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 (1979), 94–103, 239, 245.
  * [17] Kearns, M.; Saul, L. Large Deviation Methods for Approximate Probabilistic Inference. UAI’98: Proceedings of the Fourteenth conference on Uncertainty in artificial intelligence. July 1998, Pages 311–319.
  * [18] Lee, T.-D.; Yang, C.-N. Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87 (3), 410 (1952).
  * [19] Levin, B. Ja. Distribution of zeros of entire functions. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. Revised edition. Translations of Mathematical Monographs, 5. American Mathematical Society, Providence, R.I., 1980. xii+523 pp.
  * [20] Linnik, Ju. V.; Ostrovskiĭ, Ĭ. V. Decomposition of random variables and vectors. Translated from the Russian. Translations of Mathematical Monographs, Vol. 48. American Mathematical Society, Providence, R. I., 1977. ix+380 pp.
  * [21] Marchal, O.; Arbel, J. On the sub-Gaussianity of the beta and Dirichlet distributions. Electron. Commun. Probab. 22 (2017), Paper No. 54, 14 pp.
  * [22] Marcinkiewicz, J. Sur une propriété de la loi de Gauß. (French) Math. Z. 44 (1939), no. 1, 612–618.
  * [23] Newman, C. M. Inequalities for Ising models and field theories which obey the Lee-Yang theorem. Comm. Math. Phys. 41 (1975), 1–9.
  * [24] Newman, C. M. Moment inequalities for ferromagnetic Gibbs distributions. J. Mathematical Phys. 16 (1975), no. 9, 1956–1959.
  * [25] Newman, C. M. An extension of Khintchine’s inequality. Bull. Amer. Math. Soc. 81 (1975), no. 5, 913–915.
  * [26] Newman, C.; Wu, W. Lee-Yang property and Gaussian multiplicative chaos. Comm. Math. Phys. 369 (2019), no. 1 , 153–170.
  * [27] Zygmund, A. Trigonometric series. Vol. I, II. Third edition. With a foreword by Robert A. Fefferman Cambridge University Press, Cambridge, 2002. xii; Vol. I: xiv+383 pp.; Vol. II: viii+364 pp.