# Limit Theorems for low dimensional generalized $T,T^{-1}$ transformations

D. Dolgopyat1, C. Dong2, A. Kanigowski1,4, P. Nándori3

(Date:
1University of Maryland, College Park, MD, USA
2Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, China
3Yeshiva University, New York, NY, USA
4Jagiellonian University, Krakow, Poland
)

###### Abstract.

We consider generalized $(T,T^{-1})$ transformations such that the base map
satisfies a multiple mixing local limit theorem and anticoncentration large
deviation bounds and in the fiber we have $\mathbb{R}^{d}$ actions with $d=1$
or $2$ which are exponentially mixing of all orders. If the skewing cocycle
has zero drift, we show that the ergodic sums satisfy the same limit theorems
as the random walks in random scenery studied by Kesten and Spitzer (1979) and
Bolthausen (1989). The proofs rely on the quenched CLT for the fiber action
and the control of the quenched variance. This paper complements our previous
work where the classical central limit theorem is obtained for a large class
of generalized $(T,T^{-1})$ transformations.

###### Key words and phrases:

$(T,T^{-1})$ transformations; Kesten–Spitzer process; quenched and annealed
limit theorems; local time

###### 1991 Mathematics Subject Classification:

Primary: 60F05. Secondary: 37D30, 60J55, 60K35

##  1\. Introduction

This work is a continuation of our study on generalized $T,T^{-1}$
transformations, following previous work [11, 12, 13]. Our main innovation in
this paper is to provide several limit theorems in the low dimensional
setting, complementing to the higher dimensional case in [12].

###  1.1. Results

Let $f$ be a smooth map of a manifold $X$ preserving a measure $\mu,$ and
$G_{t}$ be an $\mathbb{R}^{d}$ action on a manifold $Y$ preserving a measure
$\nu.$

######  Definition 1.1.

$G_{t}$ enjoys multiple exponential mixing of all orders if there is
$\alpha>0$ such that for each $r$ there are constants $C,c>0$ such that for
all zero mean $C^{\alpha}$ functions $A_{1},\dots,A_{r}$, for all $t_{1},\dots
t_{r}\in\mathbb{R}^{d}$

(1.1) $\left|\nu\left(\prod_{j=1}^{r}A(G_{t_{j}}y)\right)\right|\leq
C\left[\prod_{j=1}^{r}\|A\|_{C^{\alpha}}\right]e^{-c\mathfrak{l}}$

where $\mathfrak{l}$ is the gap $\displaystyle\mathfrak{l}=\max_{i\neq
j}\|t_{i}-t_{j}\|.$

######  Remark 1.2.

A simple interpolation shows that if (1.1) holds for some $\alpha$ then it
also holds for all $\alpha$ (see e.g. [14, Appendix A]).

In case $d=1$, there are plenty of examples of multiply exponentially mixing
systems see e.g. the discussion in [11]. In the case $d=2$ our main example is
the following: $Y=SL_{3}(\mathbb{R})/\Gamma$, where $\Gamma$ is a cocompact
lattice, $G_{t}:Y\to Y$ is the Cartan action on $Y$, and $\nu$ is the Haar
measure. More generally one can consider subactions of Cartan actions on
$\mathcal{G}/\Gamma$ where $\mathcal{G}$ is a semisimple Lie group with
compact factors and $\Gamma$ is a cocompact lattice. In particular, we can
take $Y=SL_{d}(\mathbb{R})/\Gamma$, and let $G_{t}$ be an action by a two
dimensional subgroup of the group of diagonal matrices.

Let $\tau:X\to\mathbb{R}^{d}$ be a smooth map. We study the map $F:(X\times
Y)\to(X\times Y)$ given by

(1.2) $F(x,y)=(f(x),G_{\tau(x)}y).$

Note that $F$ preserves the measure $\zeta=\mu\times\nu$ and that

$F^{N}(x,y)=(f^{N}x,G_{\tau_{N}(x)}y)\quad\text{where}\quad\tau_{N}(x)=\sum_{n=0}^{N-1}\tau(f^{n}x).$

Let $H:X\times Y\to\mathbb{R}$ be a sufficiently smooth function with
$\zeta(H)=0$ and

$S_{N}=\sum_{n=0}^{N-1}H(F^{n}(x,y)).$

We want to study the distribution of $S_{N}$ when the initial condition
$(x,y)$ is distributed according to $\zeta.$

We shall assume that $f$ is ergodic and satisfies the CLT for Hölder
functions.

######  Definition 1.3.

We say that $\tau$ satisfies the mixing local limit theorem (MLLT) if for any
sequences $(\delta_{n})_{n\in\mathbb{N}}\in\mathbb{R}$, with
$\displaystyle\lim_{n\to\infty}\delta_{n}=0$ and
$(z_{n})_{n\in\mathbb{N}}\in\mathbb{R}^{d}$ such that
$|\frac{z_{n}}{\sqrt{n}}-z|<\delta_{n}$ for any cube
$\mathcal{C}\subset\mathbb{R}^{d}$ and any continuous functions
$A_{0},A_{1}:X\to\mathbb{R}$

$\lim_{n\to\infty}n^{d/2}\mu\Big{(}A_{0}(\cdot)A_{1}(f^{n}\cdot){\mathbbm{1}}_{\mathcal{C}}(\tau_{n}-z_{n})\Big{)}=\mathfrak{g}(z)\mu(A_{0})\mu(A_{1}){\rm
Vol}(\mathcal{C})$

where $\mathfrak{g}(z)$ is a non-degenerate Gaussian density and the
convergence is uniform once $(\delta_{n})_{n\in\mathbb{N}}$ is fixed and
$A_{0},A_{1},z$ range over compact subsets of $C(X),C(X)$ and
$\mathbb{R}^{d}$.

######  Definition 1.4.

$\tau$ satisfies multiple mixing local limit theorem (MMLLT) if for each
$m\in\mathbb{N}$ for any sequence
$(\delta_{n})_{n\in\mathbb{N}}\in\mathbb{R}$, with
$\displaystyle\lim_{n\to\infty}\delta_{n}=0$, for any family of sequences
$(z_{n}^{(1)},\dots,z_{n}^{(m)})_{n\in\mathbb{N}}$ with
$|\frac{z_{n}^{(j)}}{\sqrt{n}}-z^{(j)}|<\delta_{n}$ for any cubes
$\\{\mathcal{C}_{j}\\}_{j\leq m}\subset\mathbb{R}^{d}$ and any continuous
functions $A_{0},A_{1},\dots A_{m}:X\to\mathbb{R}$ for any sequences
$n_{1}\dots n_{m}\in\mathbb{N}$ such that $n_{j+1}-n_{j}\to\infty$ (with
$n_{0}=0$),

$\lim_{\min|n_{j}-n_{j^{\prime}}|\to\infty}\left(\prod_{j=1}^{m}\left(n_{j}-n_{j-1}\right)^{d/2}\right)\mu\left(\prod_{j=0}^{m}A_{j}(f^{n_{j}}\cdot)\prod_{j=1}^{m}{\mathbbm{1}}_{\mathcal{C}_{j}}\left(\tau_{n_{j}}-z_{n_{j}}^{(j)}\right)\right)$
$=\prod_{j=0}^{m}\mu(A_{j})\prod_{j=1}^{m}\mathfrak{g}\left(z^{(j)}-z^{(j-1)}\right)\prod_{j=1}^{m}{\rm
Vol}(\mathcal{C}_{j}),$

where $z^{(0)}=0.$ Moreover, the convergence is uniform once
$(\delta_{n})_{n\in\mathbb{N}}$ is fixed, $A_{0},A_{1},\dots,A_{m}$ range over
compact subsets of $C(X)$ and $z^{(j)}$ range over a compact subset of
$\mathbb{R}^{d}$ for every $j\leq m$.

######  Definition 1.5.

$\tau$ satisfies the anticoncentration large deviation bound of order $s$ if
there exist a constant $K$ and a decreasing function $\Theta$ such that
$\displaystyle\int_{1}^{\infty}\Theta(r)r^{d}\mathrm{d}r<\infty$ and for any
numbers $n_{1},n_{2},\dots,n_{s}$, for any unit cubes
$C_{1},C_{2},\dots,C_{s}$ centered at $c_{1},c_{2},\dots,c_{s}$

(1.3) $\mu\left(x:\tau_{n_{j}}\in C_{j}\text{ for }j=1,\dots,s\right)\leq$
$K\left(\prod_{j=1}^{s}\left(n_{j}-n_{j-1}\right)^{-d/2}\right)\Theta\left(\max_{j}\frac{\|c_{j}-c_{j-1}\|}{\sqrt{n_{j}-n_{j-1}}}\right).$

The class of maps satisfying the MMLLT and the large deviation
anticoncentration bounds includes in particular the maps which admit Young
towers with exponential tails, see [21, 19].

######  Theorem 1.6.

Suppose that $d=2$, $G_{t}$ enjoys multiple exponential mixing of all orders,
$f$ satisfies the CLT for smooth observables and $\tau$ satisfies the MMLLT
and the anticoncentration large deviation bound. Then there is $\Sigma^{2}$
such that $\displaystyle\frac{S_{N}}{\sqrt{N\ln N}}$ converges as $N\to\infty$
to the normal distribution with zero mean and variance $\Sigma^{2}.$

######  Theorem 1.7.

Suppose that $d=1$, $G_{t}$ enjoys multiple exponential mixing of all orders,
$f$ satisfies the CLT for smooth observables and $\tau$ satisfies the MMLLT
and the anticoncentration large deviation bound. Then there is a constant
$\Sigma$ such that

(1.4) $\frac{S_{N}}{N^{3/4}}\quad\text{converges as $N\to\infty$ to a
product}\quad\Sigma\mathcal{L}\mathcal{N},$

where $\mathcal{L}$ and $\mathcal{N}$ are independent, $\mathcal{N}$ has
standard normal distribution and

(1.5) $\mathcal{L}=\sqrt{\int_{-\infty}^{\infty}\ell_{x}^{2}\mathrm{d}x\;\;}$

where $\ell_{x}$ is the local time of the standard Brownian Motion at time $1$
and spatial location $x.$

######  Remark 1.8.

If $d=1$ then the exponential mixing condition can be weakened to sufficiently
fast polynomial mixing, see Theorem 5.1 in Section 5. It seems more difficult
to weaken the mixing assumption in two dimension. We do not pursue this topic
since we do not know examples of smooth $\mathbb{R}^{2}$ actions which are
mixing at a fast polynomial (rather than exponential) speed.

######  Remark 1.9.

The asymptotic variance $\Sigma^{2}$ in Theorems 1.6 and 1.7 has similar form
(see (3.10) and (4.5)). Namely let
$\displaystyle{\tilde{H}}(y)=\int_{X}H(x,y)d\mu(x).$ Then

$\Sigma^{2}=c_{d}\int_{\mathbb{R}^{d}}\int_{Y}{\tilde{H}}(y){\tilde{H}}(G_{t}y)d\nu(y)dt.$

In dimension $1$ if $\Sigma^{2}=0$ then there is an $L^{2}$ function $J(y)$
such that

$\int_{0}^{t}{\tilde{H}}(G_{s}y)ds=J(G_{t}y)-J(y).$

This was shown in the discrete case by Rudolph, [20] (Proposition 2 and Lemma
7), but the proof applies with no changes in the continuous case.

It is shown in [12, Theorem 6.4] that for $\mathbb{R}^{1}$ volume preserving
exponentially mixing flows, the quadratic form $H\mapsto\Sigma^{2}(H)$ is not
identically zero. Therefore the set of functions of vanishing asymptotic
variance is a proper linear subspace. The proof of Theorem 6.4 in [12] works
also for higher dimensional (not necessarily volume preserving actions)
provided that

(i) there are constants $K_{1},\beta_{1},K_{2},\beta_{2}$ such that for all
$y$ and $r$, $\displaystyle
K_{1}r^{\beta_{1}}\\!\\!\leq\\!\\!\nu(B(y,r))\\!\\!\leq\\!\\!K_{2}r^{\beta_{2}}$
and

(ii) there exists a slowly recurrent point, that is, a point $y\in Y$ such
that for all positive constants $K$ and $A$ there exists $r_{0}(K,A)$ so that
for all $r<r_{0}$ and for all $t$ with $|t|<K$,
$\displaystyle\nu(B(y,r)G_{-t}B(y,r))\leq\frac{\nu(B(y,r))}{|\ln r|^{A}}$,
where $B(y,r)$ is the ball of radius $r$ centered at $y$. (We note that for
exponentially mixing volume preserving $\mathbb{R}$ actions, almost all points
are slowly recurrent, see [14].)

We believe that in many cases the vanishing of the asymptotic variance entails
that the ergodic sums of $H$ satisfy the classical CLT but this will be a
subject of a future work.

######  Remark 1.10.

Results analogous to the above theorems can be proved in case $G$ is an action
of $\mathbb{Z}^{d}$ ($d=1,2$) and $\tau:X\to\mathbb{Z}^{d}$ is a piecewise
smooth map satisfying the appropriate assumptions such as the CLT, MMLLT, and
anticoncentration large deviations bounds. Since the results as well as the
proofs are virtually the same, we omit the case of discrete actions. One can
also take $X$ to be a subshift of finite type, see the discussion below.

###  1.2. Discussion

Here, we discuss previous results related to Theorems 1.6, 1.7.

The first results about $T,T^{-1}$ transformations pertain to so called random
walks in random scenery. In this model we are given a sequence
$\\{\xi_{z}\\}_{z\in\mathbb{Z}^{d}}$ of i.i.d. random variables. Let
$\tau_{n}$ be a simple random walk on $\mathbb{Z}^{d}$ independent of $\xi$s.
We are interested in $\displaystyle S_{N}=\sum_{n=1}^{N}\xi_{\tau_{n}}.$ This
model could be put in the present framework as follows. Let $X$ be a set of
sequences $\\{v_{n}\\}_{n\in\mathbb{Z}},$ where $v_{n}\in\\{\pm e_{1},\pm
e_{2},\dots\pm e_{d}\\}$ where $e_{j}$ are basis vectors in $\mathbb{Z}^{d},$
$\mu$ is the Bernoulli measure with ${\mathbb{P}}(v=\pm e_{j})=\frac{1}{2d}$
for all $j\in 1,\dots,d,$ $Y$ is the space of sequences
$\\{\xi_{z}\\}_{z\in\mathbb{Z}^{d}}$, $\nu$ is the product of distribution
functions of $\xi$, $f$ and $G_{t}$ are shifts and $\tau(\\{v\\})=v_{0}.$ For
random walks in random scenery, Theorem 1.7 is due to [15], and Theorem 1.6
are due to [4]. The results of [15] and [4] are extended to more general
actions in the fiber (still assuming the random walk in the base) in [7, 8].

In the context of dynamical systems, Theorem 1.7 was proven in [17] under the
assumption that we have a good rate of convergence in the Central Limit
Theorem for $f$. In the present paper we follow a method of [4] which seems
more flexible and allows a larger class of base systems.

We also note that if $d\geq 3$ or $\tau$ has a drift, one has a classical CLT
(that is, $\dfrac{S_{N}-{\mathbb{E}}(S_{N})}{\sqrt{N}}$ converges to a
Gaussian distributions). These results are proven in [11, 12]. The paper [11]
also studies mixing properties of $(T,T^{-1})$ transformation defined by
(1.2). In particular [11] shows that in many cases the mixing of the whole
product $F$ is exponential. We observe that in the case where $F$ enjoys
multiple exponential mixing, one can obtain the classical CLT by applying the
results of [3]. The CLT of Björklund and Gorodnik [3] also plays a key role in
our proof and we review it in Section 2.

In the case where the base map $f$ is uniformly hyperbolic the skew product
map $F$ is partially hyperbolic. It is possible that generic partially
hyperbolic maps enjoy strong statistical properties including the multiple
exponential mixing and the Central Limit Thereom. However, such a result seems
currently beyond reach (some special cases are considered in [1, 6]). In
present $(T,T^{-1})$ setting the exotic limit theorems such as our Theorems
1.7 and 1.6 require the zero drift assumption, which is a codimension $d$
condition.

##  2\. Björklund - Gorodnik CLT

In order to prove our results we use the strategy of [4] replacing the Feller
Lindenberg CLT for iid random variables by a CLT for exponentially mixing
systems due to [3]. More precisely we need the following fact.

######  Proposition 2.1.

Let $\mathfrak{m}_{N}$, $N\in\mathbb{N}$ be a sequence of measures on
$\mathbb{R}^{d}$ and let $\\{A_{t,N}\\}_{t\in\mathbb{R}^{d},N\in\mathbb{N}}$
be a family of real valued functions on $Y$ so that $\|A_{t,N}\|_{C^{1}(Y)}$
is uniformly bounded and $\nu(A_{t,N})\equiv 0$. Set
$\mathcal{S}_{N}:=\int_{\mathbb{R}^{d}}A_{t,N}(G_{t}y)\mathrm{d}\mathfrak{m}_{N}(t).$
Suppose that

(a) $\displaystyle\lim_{N\to\infty}\|\mathfrak{m}_{N}\|=\infty$ where
$\displaystyle\|\mathfrak{m}\|=\mathfrak{m}(\mathbb{R}^{d}).$

(b) For each $r\in\mathbb{N},$ $r\geq 3$ for each $K$

$\int\mathfrak{m}_{N}^{r-1}[B(t,K\ln\|\mathfrak{m}_{N}\|)]\mathrm{d}\mathfrak{m}_{N}(t)=0,$

where $B(t,R)\subset\mathbb{R}^{d}$ is the ball of radius $R$ around
$t\in\mathbb{R}^{d}$.

(c) $\displaystyle\lim_{N\to\infty}V_{N}=\sigma^{2}$ where

$V_{N}:=\int\mathcal{S}_{N}^{2}(y)\mathrm{d}\nu(y)=\iiint
A_{t_{1},N}(G_{t_{1}}y)A_{t_{2},N}(G_{t_{2}}y)\mathrm{d}\mathfrak{m}_{N}(t_{1})\mathrm{d}\mathfrak{m}_{N}(t_{2})\mathrm{d}\nu(y).$

Then, as $N\to\infty$, $\mathcal{S}_{N}$ converges weakly to the normal
distribution with zero mean and variance $\sigma^{2}.$

This proposition is proven in [3] in case $A_{t,T}$ does not depend on $t$ and
$T$, however the proof easily extends to the case of $t,T$-dependent $A$, see
[13].

##  3\. Dimension two

###  3.1. Reduction to quenched CLT

Here we prove Theorem 1.6.

Consider a function ${\tilde{H}}$ satisfying

(3.1) $\int{\tilde{H}}(x,y)\mathrm{d}\nu(y)=0$

for all $x\in X.$

Given $x\in X$ define the measures $\mathfrak{m}_{N}$ on $\mathbb{R}^{2}$ by

(3.2) $\mathfrak{m}_{N}(x)=\frac{1}{\sqrt{N\ln
N}}\sum_{n=0}^{N-1}\delta_{\tau_{n}(x)}$

and functions $A_{t,N,x}(y)$ by

(3.3) $A_{t,N,x}(y)=\frac{1}{{\rm Card}(n\leq N:\tau_{n}(x)=t)}\sum_{n\leq
N:\tau_{n}(x)=t}{\tilde{H}}({f^{n}}x,y).$

######  Proposition 3.1.

Under the assumptions of Theorem 1.6, there exists $\sigma^{2}$ and subsets
$X_{N}\subset X$ such that $\displaystyle\lim_{N\to\infty}\mu(X_{N})=1$ and
for any sequence $x_{N}\in X_{N}$ the measures $\\{\mathfrak{m}_{N}(x_{N})\\}$
satisfy the conditions of Proposition 2.1.

The proposition will be proven later. Now we shall show how to obtain Theorem
1.6 from the proposition.

###### Proof of Theorem 1.6 assuming Proposition 3.1.

Split

(3.4)
$H(x,y)={\tilde{H}}(x,y)+{\bar{H}}(x)\quad\text{where}\quad{\bar{H}}(x)=\int
H(x,y)\mathrm{d}\nu(y).$

Denote

$\mathcal{S}_{N}(x,y)=\frac{1}{\sqrt{N\ln
N}}\sum_{n=0}^{N-1}{\tilde{H}}(F^{n}(x,y)).$

Note that ${\tilde{H}}$ satisfies (3.1) and hence, by Proposition 3.1,
$\mathcal{S}_{N}(x,y)$ is asymptotically normal and is asymptotically
independent from $x$. Finally the contribution of ${\bar{H}}$ is negligible.
Indeed, the CLT for smooth observables gives $\displaystyle\frac{1}{\sqrt{N\ln
N}}\sum_{n=0}^{N-1}{\bar{H}}(f^{n}(x))\Rightarrow 0.$ This completes the proof
of the theorem (modulo Proposition 3.1). $\square$

The remaining part of this section is devoted to the proof of Proposition 3.1.
It suffices to verify the conditions of Proposition 2.1.

Property (a) is clear, since $\|\mathfrak{m}_{N}(x)\|=\sqrt{N/\ln N}.$

Verifying properties (b) and (c) requires longer computations that are
presented in §3.2 and §3.3.

###  3.2. Property (b)

Let

(3.5) $X_{K,N}=\left\\{x:{\rm Card}(n:|n|<N\text{ and }\|\tau_{n}(x)\|\leq
K\ln N)\geq N^{1/5}\right\\}.$

######  Lemma 3.2.

If for each $K$, $\displaystyle\lim_{N\to\infty}N\mu(X_{K,N})=0$, then there
are sets ${\hat{X}}_{N}$ such that for all $x_{N}\in{\hat{X}}_{N}$ the
measures $\mathfrak{m}_{N}(x_{N})$ satisfy property (b) and
$\mu({\hat{X}}_{N})\to 1$

###### Proof.

Given $K$ let $\displaystyle{\hat{X}}_{N}=\\{x:f^{n}x\not\in X_{K,N}\text{ for
}n<N\\}.$ By the assumption of the lemma, $\mu({\hat{X}}_{N})\to 1.$ On the
other hand, for $x\in{\hat{X}}_{N}$,

$\int\mathfrak{m}_{N}^{r-1}(x)[B(t,K\ln
N)]\mathrm{d}\mathfrak{m}_{N}(x)=\frac{1}{(N\ln N)^{r/2}}\sum_{n=0}^{N-1}{\rm
Card}^{r-1}(j<N:\;\|\tau_{j}-\tau_{n}\|\leq K\ln N)$ $\leq
N^{-\frac{r}{2}+1+\frac{r-1}{5}}\leq N^{-1/10},$

where the first inequality holds since $x\in{\hat{X}}_{N}$ and the second one
holds because $r\geq 3$. Since $K$ is arbitrary, the result follows. $\square$

Let

(3.6) $\ell(x,t,N)={\rm Card}(n\leq N:|\tau_{n}(x)-t|\leq 1).$

######  Lemma 3.3.

For each $p$ there is a constant $C_{p}$ such that for each
$t\in\mathbb{R}^{d}$ and $N\in\mathbb{N}$,

(3.7) $\mu\left(\ell^{p}(\cdot,t,N)\right)\leq C_{p}\ln^{p}N.$

###### Proof.

Since $\displaystyle\ell(x,t,N)=\sum_{n<N}{\mathbbm{1}}_{B(t,1)}(\tau_{n}(x))$
we have

$\mu\left(\ell^{p}(\cdot,t,n)\right)=\sum_{n_{1},\dots
n_{p}}\mu\left(x:|\tau_{n_{j}}(x)-t|\leq 1\text{ for }j=1,\dots p\right)$
$\leq\sum_{n_{1},\dots
n_{p}}K_{p}\frac{1}{n_{1}}\prod_{j=2}^{p}\left(\frac{1}{n_{j}-n_{j-1}+1}\right),$

where the last step uses the anticoncentration large deviation bound.
$\square$

Lemma 3.3 and Markov inequality imply that for each ${\varepsilon},t,p$ we
have

$\mu\left(x:\ell(x,t,N)\geq\frac{N^{1/5}}{(K\ln
N)^{2}}\right)\leq\frac{C_{p}(K\ln N)^{2p}}{N^{p/5}}$

for $N$ large enough. It follows that

$\mu\left(x:\exists t:\|t\|\leq K\ln N\text{ and
}\ell(x,t,N)\geq\frac{N^{1/5}}{(K\ln N)^{2}}\right)\leq\frac{C_{p}(K\ln
N)^{2p+2}}{N^{p/5}}.$

Taking $p=6$ we verify the conditions of Lemma 3.2.

###  3.3. Property (c)

Theorem 4.7 in [11] implies that

(3.8)
$\int{\tilde{H}}(x,y){\tilde{H}}(F^{k}(x,y))\mathrm{d}\mu(x)\mathrm{d}\nu(y)=\frac{\mathfrak{g}(0)}{k}\varsigma_{2}^{2}+o\left(\frac{1}{k}\right),$

where

(3.9)
$\varsigma_{2}^{2}=\iiint_{-\infty}^{\infty}\int\tilde{H}(x_{1},y)\tilde{H}(x_{2},G_{t}(y))\mathrm{d}\nu(y)\mathrm{d}t\mathrm{d}\mu(x_{1})\mathrm{d}\mu(x_{2})$

and $\mathfrak{g}$ is the limit Gaussian density of $\tau$ (that is,
$\tau_{N}/\sqrt{N}$ converges as $N\to\infty$ to the normal distribution with
density $\mathfrak{g}$). We note that the integral (3.9) converges by the
exponential mixing of $G$ and (3.1). Furthermore, $\varsigma_{2}^{2}\geq 0$,
which can be seen from the following formula (whose proof is standard and so
is omitted)

$\varsigma_{2}^{2}=\lim_{T\to\infty}\frac{1}{T}\int\left[\int\int_{0}^{T}{\tilde{H}}(x,G_{t}(y))\mathrm{d}t\mathrm{d}\mu\right]^{2}\mathrm{d}\nu.$

Defining

$V_{N}=\frac{1}{N\ln
N}\int\left[\sum_{n=0}^{N-1}{\tilde{H}}(F^{n}(x,y))\right]^{2}\mathrm{d}\nu(y)\,,$

we compute

$\displaystyle V_{N}$ $\displaystyle=\frac{1}{N\ln
N}\sum_{n_{1}=0}^{N-1}\sum_{n_{2}=0}^{N-1}\int{\tilde{H}}(F^{n_{1}}(x,y)){\tilde{H}}(F^{n_{2}}(x,y))\mathrm{d}\nu(y)$
$\displaystyle=\frac{1}{N\ln N}\sum_{k=0}^{N-1}(N-k)(1+{\mathbbm{1}}_{k\neq
0})\int{\tilde{H}}(x,y){\tilde{H}}(F^{k}(x,y))\mathrm{d}\nu(y)\,,$

whence from (3.8) we obtain

(3.10)
$\Sigma^{2}:=\lim_{N\to\infty}\mu\left(V_{N}(x)\right)=2\mathfrak{g}(0)\iiiint{\tilde{H}}(x,y){\tilde{H}}({\bar{x}},G_{t}y)\mathrm{d}\mu(x)\mathrm{d}\mu({\bar{x}})\mathrm{d}\nu(y)\mathrm{d}t.$

By Chebyshev’s inequality, to establish property (c), it suffices to show that

(3.11) $\lim_{N\to\infty}\mu(V_{N}^{2})=\Sigma^{4}.$

Note that

$\mu(V_{N}^{2})=\frac{1}{N^{2}\ln^{2}N}\sum_{n_{1},n_{2},n_{3},n_{4}}\mu(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}),$

where

$\sigma_{n_{1},n_{2}}(x)=\int
H(f^{n_{1}}x,G_{\tau_{n_{1}}x}y)H(f^{n_{2}}x,G_{\tau_{n_{2}}(x)}y)\mathrm{d}\nu(y).$

Fix a large $L$ and let $\mathbf{k}=\mathbf{k}(n_{1},n_{2},n_{3},n_{4})$ be
the number of indices $j$ such that $|n_{i}-n_{j}|\geq L$ for all $i\neq j.$
The number of terms where $\mathbf{k}\leq 1$ is $O(L^{2}N^{2})$. Using the
trivial bound
$|\mu(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}})|\leq\|{\tilde{H}}\|_{\infty}^{2}$,
we see that the contribution of terms with $\mathbf{k}\leq 1$ is
$O(N^{2}L^{2})$ which is negligible.

Next, we consider the contribution of the terms with $\mathbf{k}=4$. Without
loss of generality, we will assume that

(3.12) $n_{1}<n_{2},\quad n_{3}<n_{4},\quad n_{1}<n_{3}.$

We distniguish $3$ cases based on the relative position of $n_{2}$ with
respect to $n_{3}$ and $n_{4}.$

Case 1. We claim that for each ${\varepsilon}>0$ there is $L$ such that if

(3.13) $n_{1}<n_{1}+L\leq n_{2}<n_{2}+L\leq n_{3}<n_{3}+L\leq n_{4},$

then

(3.14)
$\left|\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)-\frac{\Sigma^{4}}{4|n_{2}-n_{1}||n_{4}-n_{3}|}\right|\leq\frac{{\varepsilon}}{|n_{2}-n_{1}||n_{4}-n_{3}|}.$

The proof of (3.14) is based on [11]. First, by [11, Lemma 3.3], we can assume
that ${\tilde{H}}(x,y)=A(x)B(y)$ and without loss of generality we assume
$\nu(B)$ = 0. Then we have

$\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)=\int\left[\prod_{i=1}^{4}A(f^{n_{i}}x)\right]\rho(\tau_{n_{2}-n_{1}}(f^{n_{1}}x))\rho(\tau_{n_{3}-n_{4}}(f^{n_{4}}x))\mathrm{d}\mu(x),$

where $\rho(t)=\int B(y)B(G_{t}y)\mathrm{d}\nu(y)$. Note that for
${\tilde{H}}$ as above (3.11) simplifies to

(3.15) $\Sigma^{2}=2\mathfrak{g}(0)\mu(A)^{2}\int\rho(t)\mathrm{d}t.$

The remaining part of the proof of (3.14) closely follows the lines of [11,
Theorems 4.6, 4.7] and so we only give a sketch. Decompose $\mathbb{R}^{2}$
into a countable disjoint family of small squares $\mathcal{C}_{i}$. Let
$z_{i}$ be the center of square $\mathcal{C}_{i}$. Then

$\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)\approx\sum_{i,j}S_{i,j},$

where

$S_{i,j}=\rho(z_{i})\rho(z_{j})\int\prod_{k=1}^{4}A(f^{n_{k}}x){\mathbbm{1}}_{\mathcal{C}_{i}}(\tau_{n_{2}-n_{1}}(f^{n_{1}}x)){\mathbbm{1}}_{\mathcal{C}_{j}}(\tau_{n_{3}-n_{4}}(f^{n_{4}}x))\mathrm{d}\mu(x).$

Fixing $i,j$, letting $L\to\infty$ and using the MMLLT and (3.13), we find

$S_{i,j}\approx\frac{1}{|n_{2}-n_{1}|}\frac{1}{|n_{4}-n_{3}|}\int_{\mathcal{C}_{i}}\rho(t)\mathrm{d}t\int_{\mathcal{C}_{j}}\rho(t)\mathrm{d}t\mathfrak{g}(0)^{2}[\mu(A)]^{4}.$

Summing over $i,j$ and using (3.15), we obtain (3.14).

Case 2. We claim that

(3.16)
$\sum_{(n_{1},...,n_{4}):n_{1}<n_{3}<n_{4}<n_{2}}\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)\leq
CN^{2}\ln N.$

To prove (3.16) take $(n_{1},...,n_{4})$ as in (3.16) and write

$m_{0}=0,\quad m_{1}=n_{3}-n_{1},\quad m_{2}=n_{4}-n_{1},\quad
m_{3}=n_{2}-n_{1}.$

Then we have

$\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)=\int\left[\prod_{i=0}^{3}A(f^{m_{i}}(x))\right]\rho(\tau_{m_{3}}(x))\rho(\tau_{m_{2}-m_{1}}(f^{m_{1}}x))\mathrm{d}\mu(x).$

Decompose $\mathbb{R}^{2}$ into unit boxes $\mathcal{C}_{i}$ with center
$z_{i}$. Then, we find

$\left|\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)\right|\leq
C\sum_{i_{1},i_{2},i_{3}}\tilde{\rho}(z_{i_{3}})\tilde{\rho}(z_{i_{2}}-z_{i_{1}})\int{\prod_{j=1}^{3}}{\mathbbm{1}}_{\tau_{m_{j}}\in{\mathcal{C}_{i_{j}}}}\mathrm{d}\mu(x),$

where $\displaystyle\tilde{\rho}(t)=\sup_{t^{\prime}:\|t^{\prime}-t\|\leq
1}\left|\rho(t)\right|.$ Applying (1.3), we find

(3.17)
$\displaystyle\left|\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)\right|\leq$
$\displaystyle
C\sum_{i_{1},i_{2},i_{3}}\frac{\tilde{\rho}(z_{i_{3}})\tilde{\rho}(z_{i_{2}}-z_{i_{1}})}{m_{1}(m_{2}-m_{1})(m_{3}-m_{2})}\Theta\left(\max_{1\leq
j\leq 3}\frac{\|z_{i_{j}}-z_{i_{j-1}}\|}{\sqrt{m_{j}-m_{j-1}}}\right)$
$\displaystyle=:\sum_{i_{1},i_{2},i_{3}}\mathcal{S}_{i_{1},i_{2},i_{3}}(m_{1},m_{2},m_{3}).$

Let us consider the special case when $i_{2}=i_{1}$ and $z_{i_{3}}=0$ (without
loss of generality we say that in this case $i_{3}=0$). Then we have

(3.18) $\sum_{i_{1}}\mathcal{S}_{i_{1},i_{1},0}(m_{1},m_{2},m_{3})\leq$
$C\sum_{i_{1}}\frac{1}{m_{1}}\frac{1}{m_{2}-m_{1}}\frac{1}{m_{3}-m_{2}}\Theta\left(\frac{\|z_{i_{1}}\|}{\min\\{\sqrt{m_{1}},\sqrt{m_{3}-m_{2}}\\}}\right).$

Note that the last expression is symmetric in $m_{1}=n_{3}-n_{1}$ and
$m_{3}-m_{2}=n_{2}-n_{4}$. Thus without loss of generality we can assume
$n_{3}-n_{1}\leq n_{2}-n_{4}$ (indeed, the multiplier $2$ can be incorporated
into the constant $C$). Denoting $a=n_{3}-n_{1}$, $b=n_{4}-n_{3}$,
$c=n_{2}-n_{4}$, we obtain

$\sum_{(n_{1},...,n_{4}):n_{1}<n_{3}<n_{4}<n_{2}<N}\sum_{i_{1}}\mathcal{S}_{i_{1},i_{1},0}(m_{1},m_{2},m_{3})\leq
CN\sum_{i}\sum_{a=1}^{N}\sum_{b=1}^{N}\sum_{c=a}^{N}\frac{1}{abc}\Theta\left(\frac{\|z_{i}\|}{\sqrt{a}}\right).$

Note that the multiplier $N$ accounts for all choices of $n_{1}$. Thus

$\sum_{(n_{1},...,n_{4}):n_{1}<n_{3}<n_{4}<n_{2}<N}\sum_{i_{1}}\mathcal{S}_{i_{1},i_{1},0}(m_{1},m_{2},m_{3})\leq
CN\ln
N\sum_{i}\sum_{a=1}^{N}\sum_{c=a}^{N}\frac{1}{ac}\Theta\left(\frac{\|z_{i}\|}{\sqrt{a}}\right)$
$\leq CN\ln
N\int_{\mathbb{R}^{2}}\int_{1}^{N}\int_{a}^{N}\frac{1}{ac}\Theta\left(\frac{\|z\|}{\sqrt{a}}\right)\mathrm{d}c\mathrm{d}a\mathrm{d}z\leq
CN\ln N\int_{1}^{N}\int_{a}^{N}\frac{1}{c}\mathrm{d}c\mathrm{d}a\leq CN^{2}\ln
N$

where in the third inequality we used that
$\displaystyle{\int_{1}^{\infty}\Theta(r)rdr}<\infty$ and the last inequality
follows from direct integration. Thus we have verified that the terms
corresponding to $i_{1}=i_{2}$ and $i_{3}=0$ are in $O(N^{2}\ln N)$.

To complete the proof of (3.16), we need to estimate the contribution of the
sum corresponding to all $i_{2}$ and $i_{3}$. To this end, we will distinguish
two cases based on whether the following assumption holds or not:

(3.19) $\|z_{i_{2}}-z_{i_{1}}\|<\frac{1}{4}\|z_{i_{1}}\|,\text{ and
}\|z_{i_{3}}-z_{i_{2}}\|<\frac{1}{4}\|z_{i_{1}}\|.$

For any given fixed $i_{1}$, let $\widehat{\sum}_{i_{2},i_{3}}$ denote the sum
over $i_{2},i_{3}$ that do not satisfy (3.19) and let
$\widetilde{\sum}_{i_{2},i_{3}}$ denote the sum for $i_{2},i_{3}$ satisfying
(3.19).

First, we assume that (3.19) fails. Then, we have

$\sum_{i_{1}}\widehat{\sum_{i_{2},i_{3}}}\mathcal{S}_{i_{1},i_{2},i_{3}}(m_{1},m_{2},m_{3})\leq
C\sum_{i_{1}}\widehat{\sum_{i_{2},i_{3}}}\tilde{\rho}(z_{i_{2}}-z_{i_{1}})\tilde{\rho}(z_{i_{3}})\frac{1}{m_{1}}\frac{1}{m_{2}-m_{1}}\frac{1}{m_{3}-m_{2}}\\\
\times\left[\Theta\left(\frac{\|z_{i_{1}}\|}{4\min\\{\sqrt{m_{1}},\sqrt{m_{2}-m_{1}}\\}}\right)+\Theta\left(\frac{\|z_{i_{1}}\|}{4\min\\{\sqrt{m_{1}},\sqrt{m_{3}-m_{2}}\\}}\right)\right].$

Now we can perform the summation with respect to $i_{2},i_{3}$ first. Denote
$a=m_{1}$, $b=m_{2}-m_{1}$, $c=m_{3}-m_{2}$. By the exponential decay of
$\tilde{\rho}$ we get

$\sum_{i_{1}}\widehat{\sum_{i_{2},i_{3}}}\mathcal{S}_{i_{1},i_{2},i_{3}}(m_{1},m_{2},m_{3})\leq
C\sum_{i_{1}}\frac{1}{abc}\left[\Theta\\!\left(\frac{\|z_{i_{1}}\|}{4\min\\{\sqrt{a},\sqrt{b}\\}}\right)\\!+\\!\Theta\\!\left(\frac{\|z_{i_{1}}\|}{4\min\\{\sqrt{a},\sqrt{c}\\}}\right)\right]\\!.$

Summing the last displayed formula for $n_{1},n_{2},n_{3},n_{4}$, we obtain a
term $O(N^{2}\ln N)$ exactly as in the estimate of (3.18).

Now we assume that (3.19) holds. Then observing that
$\|z_{i_{3}}\|\geq\|z_{i_{1}}\|/2,$ and using the exponential decay of
$\tilde{\rho}$, we find that

$\tilde{\rho}(z_{i_{2}}-z_{i_{1}})\tilde{\rho}(z_{i_{3}})\leq
Ce^{-c\|z_{i_{1}}\|}$

with a fixed constant $c$. Thus we conclude

$\sum_{m_{1}\leq m_{2}\leq m_{3}\leq
N}\sum_{i_{1}}\widetilde{\sum_{i_{2},i_{3}}}\mathcal{S}_{i_{1},i_{2},i_{3}}\\\
\leq C\sum_{m_{1}\leq m_{2}\leq m_{3}\leq
N}\frac{1}{m_{1}}\frac{1}{m_{2}-m_{1}}\frac{1}{m_{3}-m_{2}}\sum_{i_{1}}\widetilde{\sum_{i_{2},i_{3}}}e^{-c\|z_{i_{1}}\|}\leq
C\ln^{3}N.$

Summing over all choices of $n_{1}$, we obtain a term in $O(N\ln^{3}N)$. This
completes the proof of (3.16).

Case 3. We claim that

(3.20)
$\sum_{(n_{1},...,n_{4}):n_{1}<n_{3}<n_{2}<n_{4}}\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)\leq
CN^{2}\ln N.$

The proof of (3.20) is similar to that of (3.16). Indeed, the right hand side
of (3.17) is now replaced by

$C\sum_{i_{1},i_{2},i_{3}}\frac{\tilde{\rho}(z_{i_{2}})\tilde{\rho}(z_{i_{3}}-z_{i_{1}})}{m_{1}(m_{2}-m_{1})(m_{3}-m_{2})}\Theta\left(\max_{j}\frac{\|z_{i_{j}}-z_{i_{j-1}}\|}{\sqrt{m_{j}-m_{j-1}}}\right).$

We consider the special case $i_{2}=0,i_{3}=i_{1}$ first. Then (3.18) is
replaced by

(3.21) $\sum_{i_{1}}\mathcal{S}_{i_{1},0,i_{1}}(m_{1},m_{2},m_{3})\\\ \leq
C\sum_{i_{1}}\frac{1}{m_{1}}\frac{1}{m_{2}-m_{1}}\frac{1}{m_{3}-m_{2}}\Theta\left(\frac{\|z_{i_{1}}\|}{\min\\{\sqrt{m_{1}},\sqrt{m_{2}-m_{1}},\sqrt{m_{3}-m_{2}}\\}}\right).$

which is bounded by the right hand side of (3.18) (perhaps with a different
constant $C$) and hence is in $O(N^{2}\ln N)$. The contribution of general
$i_{2}$, $i_{3}$ can be bounded as before. (3.20) follows.

The upshot of the above three cases is that the leading term only comes from
indices $n_{1},...,n_{4}$ satisfying (3.13). Summing
$\mu(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}})$ for all indices
$n_{1},...,n_{4}$ satisfying (3.13), we obtain
$N^{2}\ln^{2}N[\Sigma^{4}/8+o_{L\to\infty}(1)]$ where we have used (3.14) and
the fact that
$\displaystyle\sum_{n_{1}<n_{3}<N}\ln(n_{3}-n_{1})\ln(N-n_{3})=\frac{N^{2}\ln^{2}N}{2}(1+o_{N\to\infty}(1)).$

Next we claim that

(3.22)
$\sum_{n_{1},...,n_{4}:\mathbf{k}=4}\mu(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}})=[\Sigma^{2}+o_{L\to\infty}(1)]N^{2}\ln^{2}N.$

Indeed similarly to the case of (3.13) considered above we see that the main
contribution to our sums comes from the terms where the pairs $(n_{1},n_{2})$
and $(n_{3},n_{4})$ are not intertwined. Note that if we choose a random
permutation of $(n_{1},n_{2},n_{3},n_{4})$ then the probability of non
intertwining is 1/3 (since the second element should come from the same pair
as the first one). Therefore there are 24/3=8 permutations 111These eight
permutations corresponds to all possible choices of signs
($*\in\\{\leq,\geq\\}$) in the inequalities  $n_{1}*n_{2},\quad
n_{3}*n_{4},\quad\min(n_{1},n_{2})*\min(n_{3},n_{4})$ contributing to our sum
which explains the additional factor of 8 in (3.22).

To complete the proof of property (c), it remains to verify that the
contribution of terms with $\mathbf{k}=2,3$ is negligible. First note that
$\mathbf{k}=3$ is impossible by the definition of $\mathbf{k}$. Finally, if
$\mathbf{k}=2$, the we can repeat a simplified version of the proof for
$\mathbf{k}=4$ using the MLLT instead of the MMLLT. For example, let us
consider the case $n_{1}\leq n_{2}\leq n_{3}\leq n_{4}$. Since $\mathbf{k}=2$,
we must have $|n_{2}-n_{1}|\geq L$ or $|n_{4}-n_{3}|\geq L$. Without loss of
generality assume $|n_{4}-n_{3}|\geq L$. Then, using the fact that due to the
exponential mixing of $G$, $\sigma_{n_{3},n_{4}}\leq Ce^{-c\ell}$ on the set
$\|\tau_{n_{4}}-\tau_{n_{3}}\|\in[\ell,\ell+1]$, we obtain from the MLLT that

$\mu\left(\sigma_{n_{1},n_{2}}\sigma_{n_{3},n_{4}}\right)=O\left(\frac{1}{|n_{4}-n_{3}|}\right).$

Hence the total contribution terms with $\mathbf{k}=2$ is in $O\left(LN^{2}\ln
N\right)$. This completes the proof of property (c). We have finished the
proof of Proposition 3.1 and hence the proof of Theorem 1.6.

##  4\. Dimension one

###  4.1. Reduction to the limit theorem for the variance.

Here we prove Theorem 1.7.

We use the decomposition (3.4). Since $f$ satisfies the CLT for smooth
observables, $\displaystyle\frac{1}{\sqrt{N}}\sum_{n=1}^{N}\bar{H}(f^{n}(x))$
converges weakly, and so this sum is negligible after rescaling by $N^{3/4}$.
It remains to study the ergodic sum of ${\tilde{H}}$.

Following the ideas of the previous section, we let

(4.1)
$\mathfrak{m}_{N}(x)=\frac{1}{\sqrt{V_{N}(x)}}\sum_{n=0}^{N-1}\delta_{\tau_{n}(x)},$
$A_{t,N,x}(y)=\frac{1}{\mbox{Card}({n\leq N:\tau_{n}(x)=t})}\sum_{n\leq
N:\tau_{n}(x)=t}{\tilde{H}}(f^{n}(x),y),$

where

$V_{N}(x)=\int S_{N}^{2}(x,y)\mathrm{d}\nu(y),\quad
S_{N}(x,y)=\sum_{n=0}^{N-1}{\tilde{H}}\circ F^{n}(x,y).$

Thus in the notation of Proposition 2.1, we have
$\displaystyle\mathcal{S}_{N}=\frac{S_{N}}{\sqrt{V_{N}(x)}}.$

In contrast with Theorem 1.6, $V_{N}(x)$ does not satisfy a weak law of large
numbers. Instead we have

######  Proposition 4.1.

There is a constant $\Lambda$ so that $\frac{V_{N}}{\Lambda N^{3/2}}$
converges in law as $N\to\infty$ to the random variable $\mathcal{L}^{2}$
given by (1.5).

Before proving Proposition 4.1, let us use it to derive Theorem 1.7. We start
with the following analogue of Proposition 3.1:

######  Lemma 4.2.

Under the assumptions of Theorems 1.7, there are subsets $X_{N}\subset X$ such
that $\displaystyle\lim_{N\to\infty}\mu(X_{N})=1$ and for any sequence
$x_{N}\in X_{N}$ the measures $\\{\mathfrak{m}_{N}(x_{N})\\}$ satisfy the
conditions of Proposition 2.1 with $\sigma=1$.

###### Proof of Lemma 4.2 assuming Proposition 4.1.

Clearly, $\|\mathfrak{m}_{N}\|\to\infty$ for a large set of $x$’s and so (a)
holds. Recalling that $\mathcal{S}_{N}(x,y)=S_{N}(x,y)/\sqrt{V_{N}(x)}$,
property (c) holds trivially for all $x\in X$. It remains to show property
(b).

Note that $\mathcal{L}^{2}$ is non-negative and non-atomic at zero, i.e. for
any ${\varepsilon}>0$ there is ${\varepsilon}^{\prime}>0$ so that
$\mathbb{P}(\mathcal{L}^{2}<{\varepsilon}^{\prime})<{\varepsilon}$ and so by
Proposition 4.1,

$\lim_{N\to\infty}\mu(x:N^{1.49}<V_{N}(x)<N^{1.51})=1.$

Thus we can assume that $x$ is such that $N^{1.49}<V_{N}(x)<N^{1.51}$. Next,
we define

$\tilde{\ell}(x,t,N)=\frac{{\rm Card}(n\leq N:|\tau_{n}(x)-t|<1)}{\sqrt{N}}.$

As in the proof of Lemma 3.3, we find that for every $p\in\mathbb{Z}_{+}$,
every $t\in\mathbb{R}$, and every $N\in\mathbb{N}$ we have
$\mu(\tilde{\ell}^{p}(.,t,N))<C_{p}$. Then choosing $p=1000$, the Markov
inequality gives $\mu(x:\tilde{\ell}(x,t,N)>N^{1/100})<CN^{-10}$. Thus we can
assume that for all $t\in\mathbb{Z}$ with $|t|<N^{0.6}$,
$\tilde{\ell}(x,t,N)<N^{1/100}$ holds. By the anticoncentration large
deviation bounds, we can also assume that $\displaystyle\max_{n\leq
N}|\tau_{n}(x)|\leq N^{0.6}$ since the set of points where this condition
fails has negligible measure. In summary, the measure of the set of $x$’s
satisfying that

$N^{1.49}<V_{N}(x)<N^{1.51}\quad\text{and}\quad\tilde{\ell}(x,t,N)\leq\begin{cases}N^{0.01}&|t|\leq
N^{0.6},\\\ 0&|t|\geq N^{0.6}.\end{cases}$

tends to $1$ as $N\to\infty.$ For any such $x$, we have

$\int\mathfrak{m}_{N}^{r-1}(x)(t,K\ln
N)\mathrm{d}\mathfrak{m}_{N}(x)=\frac{1}{(V_{N}(x))^{r/2}}\sum_{n=0}^{N-1}{\rm
Card}^{r-1}(j<N:\;|\tau_{j}-\tau_{n}|\leq K\ln\|\mathfrak{m}_{N}\|)$
$\leq\frac{1}{(N^{1.49})^{r/2}}\sum_{t\in\mathbb{Z}:|t|\leq
N^{0.6}}\ell^{r}(x,t,N)\leq 2N^{0.51r+0.6-0.745r}=o(1).$

if $r\geq 3$. Property (b) and the lemma follow. $\square$

###### Proof of Theorem 1.7 assuming Proposition 4.1.

Recall that
$\displaystyle\mathcal{S}_{N}(x,y)=\frac{S_{N}(x,y)}{\sqrt{V_{N}(x)}}.$ By
Lemma 4.2 and Proposition 2.1, there is a sequence of subsets $X_{N}\subset X$
with $\mu(X_{N})\to 1$ such that for every sequence $x_{N}\in X_{N}$, the
distribution of $\mathcal{S}_{N}(x,y)$ w.r.t. $\nu(y)$ converges to the
standard normal (note that this time properties (a) and (c) are immediate).

In fact, we have the stronger statement that

(4.2) $\left(\frac{V_{N}}{\Lambda
N^{3/2}},\mathcal{S}_{N}\right)\Rightarrow(\mathcal{L}^{2},\mathcal{N})\quad\text{as}\quad
N\to\infty,$

where $\mathcal{L}$ is given by (1.5), $\mathcal{N}$ is standard normal and
$\mathcal{L}^{2}$ and $\mathcal{N}$ are independent. This follows from
Proposition 4.1, Lemma 4.2 and the asymptotic independence of
$\mathcal{S}_{N}$ and $V_{N}$ which comes from the fact that $V_{N}$ depends
only on $x$ while $\mathcal{S}_{N}$ is asymptotically independent of $x$. More
precisely, let $\phi$ and $\psi$ be two continuous compactly supported test
functions. Then

$\lim_{N\to\infty}\zeta\left(\phi\left(\frac{V_{N}}{\Lambda
N^{3/2}}\right)\psi(\mathcal{S}_{N})\right)=\lim_{N\to\infty}\zeta\left(\phi\left(\frac{V_{N}}{\Lambda
N^{3/2}}\right)\psi(\mathcal{S}_{N}){\mathbbm{1}}_{X_{N}}\right)$
$=\lim_{N\to\infty}\int_{X_{N}}\left[\phi\left(\frac{V_{N}}{\Lambda
N^{3/2}}\right)\int_{Y}\psi(\mathcal{S}_{N})\mathrm{d}\nu\right]\mathrm{d}\mu={\mathbb{E}}(\psi(\mathcal{N}))\lim_{N\to\infty}\int_{X_{N}}\phi\left(\frac{V_{N}}{\Lambda
N^{3/2}}\right)\mathrm{d}\mu$
$={\mathbb{E}}(\psi(\mathcal{N}))\lim_{N\to\infty}\int_{X}\phi\left(\frac{V_{N}}{\Lambda
N^{3/2}}\right)\mathrm{d}\mu={\mathbb{E}}(\psi(\mathcal{N})){\mathbb{E}}(\phi(\mathcal{L}^{2})),$

where the first and the fourth equalities hold since $\mu(X_{N})\to 1,$ the
third inequality holds by Proposition 2.1, and the last equality holds by
Proposition 4.1. This proves (4.2).

Denote $\Sigma=\sqrt{\Lambda}$. By (4.2) and the continuous mapping theorem,

$\frac{S_{N}}{\Sigma N^{3/4}}=\frac{\sqrt{V_{N}}}{\sqrt{\Lambda
N^{3/2}}}\times\frac{\mathcal{S}_{N}}{\sqrt{V_{N}}}\Rightarrow\mathcal{L}\mathcal{N}$

completing the proof of the Theorem 1.7. $\square$

###  4.2. Proof of Proposition 4.1

We are going to use the following well known fact (see e.g. [2] Chapter 1.7,
Problem 4). If $\mathcal{X}_{n}$ is a sequence of random variables so that for
every $k\in\mathbb{Z}_{+}$

(4.3) $J_{k}=\lim_{n\to\infty}\mathbb{E}(\mathcal{X}_{n}^{k})$

exists and

(4.4) $\limsup_{k\to\infty}\left(\frac{J_{k}}{k!}\right)^{1/k}<\infty,$

then $\mathcal{X}_{n}$ converges weakly to a random variable $\mathcal{X}$.
Furthermore, $\mathbb{E}(\mathcal{X}^{k})=J_{k}$ for every
$k\in\mathbb{Z}_{+}$ and $\mathcal{X}$ is uniquely characterized by its
moments.

Now we explain our strategy of proof of Proposition 4.1 (a similar strategy
was used in [18]). We prove that there is a constant $\Lambda$ depending on
the $(T,T^{-1})$ transformation $F$ and the observable $H$ so that
$\mathcal{X}_{N}=V_{N}/(\Lambda N^{3/2})$ satisfies (4.3) (with $\mathbb{E}$
meaning integral w.r.t. $\mu$) and (4.4) with constants $J_{k}$ that do not
depend on $F$ and $H$. Consequently, there is a random variable $\mathcal{X}$
so that $\mathcal{X}_{N}=V_{N}/(\Lambda N^{3/2})$ converges weakly to
$\mathcal{X}$ for any $(T,T^{-1})$ transformation satisfying the assumptions
of Theorem 1.7. Recall from §1.2 that one such example is the one dimensional
random walk in random scenery, in which case by the result of [15],
$\mathcal{X}_{N}=V_{N}/(\Lambda N^{3/2})$ converges weakly to
$\mathcal{L}^{2}$. Thus $\mathcal{X}=\mathcal{L}^{2}$ and it has to be the
limit for all $(T,T^{-1})$ transformation satisfying the assumptions of
Theorem 1.7. It remains to check (4.3) and (4.4).

Recall that $\tau$ satisfies the MMLLT with a Gaussian density $\mathfrak{g}$.
Let $\varsigma_{1}$ be the standard deviation of this Gaussian random
variable, that is $\mathfrak{g}(z)=\varphi(z/\varsigma_{1})/\varsigma_{1}$,
where $\varphi$ is the standard Gaussian density.

Define

(4.5)
$\Lambda=\frac{\sqrt{\pi}\varsigma_{2}^{2}}{\varsigma_{1}}=\sqrt{2}\pi\varsigma_{2}^{2}\mathfrak{g}(0)$

where $\varsigma_{2}^{2}$ is given by (3.9).

######  Lemma 4.3.

For $k=1$, (4.3) holds with

(4.6)
$J_{1}=\frac{2}{\sqrt{\pi}}\int_{w\in\mathbb{R}}\int_{0<t_{1}<t_{2}<1}\frac{1}{\sqrt{t_{1}}}\frac{1}{\sqrt{t_{2}-t_{1}}}\varphi\left(\frac{w}{\sqrt{t_{1}}}\right)\varphi(0)\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}w.$

We note that the above integral can be performed explicitly. Namely
$\displaystyle J_{1}=\frac{4\sqrt{2}}{3\sqrt{\pi}}$. This shows that $\Lambda$
was chosen correctly. Indeed, in case of the simple random walk in random
scenery, formula (1.2) in [15] shows that
$J_{1}=\frac{4\sqrt{2}}{3\sqrt{\pi}}$ while by the main result of [15], the
conclusion of Proposition 4.1 holds for simple random walk in random scenery.

###### Proof of Lemma 4.3.

Fix some ${\varepsilon}>0$. We need to show that for $N$ large enough,

$\left|N^{-3/2}\Lambda^{-1}\iint\left[\sum_{n=1}^{N}{\tilde{H}}\circ
F^{n}(x,y)\right]^{2}\mathrm{d}\mu\mathrm{d}\nu-J_{1}\right|<{\varepsilon}.$

In the following proof, we will choose a small $\eta=\eta({\varepsilon})$, a
large $L=L({\varepsilon})$ a small
$\eta^{\prime}\\!\\!=\\!\\!\eta^{\prime}({\varepsilon},\eta,L)\\!\\!<\\!\\!\eta$
and finally $N=N({\varepsilon},\eta,L,\eta^{\prime})$. There will be finitely
many restrictions on these parameters, we can choose the most strict one.

Choose a partition of $X$ into sets $X_{i},i\in I$ of diameter $<\eta$ and fix
some elements $x_{i}\in X_{i}$. We have

$\iint\left[\sum_{n=1}^{N}{\tilde{H}}\circ
F^{n}(x,y)\right]^{2}\mathrm{d}\mu\mathrm{d}\nu=$ (4.7)
$\sum_{n_{1},n_{2}=1}^{N}\sum_{i_{1},i_{2}\in
I}\sum_{z_{1},z_{2}\in\mathbb{Z}}\int
p(x)\int{\tilde{H}}(f^{n_{1}}(x),G_{\tau_{n_{1}}(x)}(y)){\tilde{H}}(f^{n_{2}}(x),G_{\tau_{n_{2}}(x)}(y))\mathrm{d}\nu\mathrm{d}\mu,$

where

$p(x)=p_{\eta,n_{1},n_{2},i_{1},i_{2},z_{1},z_{2}}(x)={\mathbbm{1}}_{f^{n_{1}}x\in
X_{i_{1}}}{\mathbbm{1}}_{f^{n_{2}}x\in
X_{i_{2}}}{\mathbbm{1}}_{\tau_{n_{1}}(x)\in[z_{1},z_{1}+1]\eta}{\mathbbm{1}}_{\tau_{n_{2}}(x)\in[z_{2},z_{2}+1]\eta}.$

Split the sum in (4.7) as $T_{1}+T_{2}$, where $T_{1}$ corresponds to the
terms satisfying

  * (A1)

$\displaystyle\eta^{\prime}N<\min\\{n_{1},n_{2}\\}<\min\\{n_{1},n_{2}\\}+\eta^{\prime}N<\max\\{n_{1},n_{2}\\}$

  * (A2)

$|z_{1}|<L\sqrt{N}/\eta$

  * (A3)

$|z_{1}-z_{2}|<L/\eta$.

and $T_{2}$ stands for the terms where at least one of the conditions
(A1)–(A3) is violated. Write $T_{1}=\widetilde{\sum}(...)$.

We start by estimating $T_{1}$. Let us write $a_{N}\approx b_{N}$ if there are
constants $\eta$, $L$ and $\eta^{\prime}$ so that for $N$ large enough,
$|a_{N}-b_{N}|<{\varepsilon}\Lambda N^{3/2}/10$. We claim that

(4.8) $T_{1}\approx\widetilde{\sum}\int
p(x)\mathrm{d}\mu\int{\tilde{H}}(x_{i_{1}},G_{z_{1}\eta}(y)){\tilde{H}}(x_{i_{2}},G_{z_{2}\eta}(y))\mathrm{d}\nu.$

Indeed, by the continuity of $H$ we can choose $\eta$ so small that the
difference between the LHS and the RHS of (4.8) does not exceed

$\frac{{\varepsilon}\Lambda}{1000}\widetilde{\sum}\int
p_{\eta,n_{1},n_{2},i_{1},i_{2},z_{1},z_{2}}(x)\mathrm{d}\mu(x),$

so (4.8) follows from the anticoncentration large deviation bound.

Next, writing $m_{1}=\min\\{n_{1},n_{2}\\}$, $m_{2}=\max\\{n_{1},n_{2}\\}$ and
using the MMLLT, we find

(4.9)
$T_{1}\approx\frac{1}{\varsigma_{1}^{2}}\widetilde{\sum}\eta^{2}\frac{1}{\sqrt{m_{1}}}\varphi\left(\frac{z_{1}\eta}{\varsigma_{1}\sqrt{m_{1}}}\right)\frac{1}{\sqrt{m_{2}-m_{1}}}\varphi\left(\frac{(z_{2}-z_{1})\eta}{\varsigma_{1}\sqrt{m_{2}-m_{1}}}\right)\mu(X_{i_{1}})\mu(X_{i_{2}})\times$
$\int{\tilde{H}}(x_{i_{1}},G_{z_{1}\eta}(y)){\tilde{H}}(x_{i_{2}},G_{z_{2}\eta}(y))\mathrm{d}\nu.$

Noting that by (A1) and (A3),
$(z_{2}-z_{1})\eta/[\varsigma_{1}\sqrt{m_{2}-m_{1}}]=O(N^{-1/2})$, we see that
$\displaystyle\varphi\left(\frac{(z_{2}-z_{1})\eta}{\varsigma_{1}\sqrt{m_{2}-m_{1}}}\right)$
can be replaced by $\varphi(0)$ in (4.9) without invalidating $\approx$. Then
the only term depending on $z_{2}$ is under the integral with respect to
$\nu$. We can approximate the sum over $z_{2}$ by a Riemann integral:

(4.10)
$\eta\sum_{z_{2}=z_{1}-L/\eta}^{z_{1}+L/\eta}\int{\tilde{H}}(x_{i_{1}},G_{z_{1}\eta}(y)){\tilde{H}}(x_{i_{2}},G_{z_{2}\eta}(y))\mathrm{d}\nu=\mathcal{I}_{i_{1},i_{2}}+o_{\eta\to
0,L\to\infty,N\to\infty}(1)$

where

$\mathcal{I}_{i_{1},i_{2}}=\int_{\mathbb{R}}\int{\tilde{H}}(x_{i_{1}},y){\tilde{H}}(x_{i_{2}},G_{t}(y))\mathrm{d}\nu\mathrm{d}t.$

Combining (4.9) and (4.10), we conclude

$\displaystyle T_{1}$ $\displaystyle\approx$
$\displaystyle\frac{2}{\varsigma_{1}^{2}}\left[\sum_{i_{1},i_{2}}\mu(X_{i_{1}})\mu(X_{i_{2}})\mathcal{I}_{i_{1},i_{2}}\right]\times$
$\displaystyle\times\left[\sum_{\eta^{\prime}N<m_{1}<m_{1}+\eta^{\prime}N<m_{2}<N}\sum_{|z_{1}|<L\sqrt{N}/\eta}\eta\frac{1}{\sqrt{m_{1}}}\varphi\left(\frac{z_{1}\eta}{\varsigma_{1}\sqrt{m_{1}}}\right)\frac{1}{\sqrt{m_{2}-m_{1}}}\varphi(0)\right],$

where the factor $2$ comes from taking into account both $n_{1}<n_{2}$ and
$n_{2}<n_{1}$. Further decreasing $\eta$ and increasing $L$ if necessary and
recalling (3.9), we can replace the first sum by $\varsigma_{2}^{2}$.

Now replacing two Riemann sums with the corresponding Riemann integrals, we
obtain

$T_{1}\approx\frac{2\varsigma_{2}^{2}}{\varsigma_{1}^{2}}N^{3/2}\int_{|w|<L}\int_{\eta^{\prime}<t_{1}<t_{1}+\eta^{\prime}<t_{2}<1}\frac{1}{\sqrt{t_{1}}}\frac{1}{\sqrt{t_{2}-t_{1}}}\varphi\left(\frac{w}{\varsigma_{1}\sqrt{t_{1}}}\right)\varphi(0)\mathrm{d}t\mathrm{d}w\approx\frac{\varsigma_{2}^{2}}{\varsigma_{1}}\sqrt{\pi}N^{3/2}J_{1}.$

In order to complete the proof of (4.6), it suffices to show that for $L$
large and for $\eta,\eta^{\prime}$ small

(4.11) $|T_{2}|<{\varepsilon}N^{3/2}/10.$

We will estimate the contribution of the terms where exactly one of the
assumptions (A1)–(A3) fail (the cases when more than one fail are similar and
easier).

Suppose that (A1) fails and, moreover, $m_{1}<\eta^{\prime}N$ (the other cases
are similar). Using the anticoncentration large deviation bounds instead of
the MMLLT, we obtain that the corresponding sum is bounded by

$\sum_{m_{1}=1}^{\eta^{\prime}N}\sum_{m_{2}=0}^{N}\frac{1}{\sqrt{m_{1}}}\frac{1}{\sqrt{m_{2}+1}}\sum_{|z_{1}|<L\sqrt{N}/\eta}C(\eta,L)\leq\sqrt{\eta^{\prime}}NC^{\prime}(\eta,L)\sqrt{N}<{\varepsilon}N^{3/2}/10$

if $\eta^{\prime}$ is small enough.

Let us assume that (A2) fails. Then again using the anticoncentration large
deviation bounds instead of the MMLLT, we obtain that the corresponding sum is
bounded by

$\sum_{m_{1}=\eta^{\prime}N}^{N}\sum_{m_{2}=m_{1}+\eta^{\prime}N}^{N}\frac{1}{\sqrt{m_{1}}}\frac{1}{\sqrt{m_{2}-m_{1}}}\sum_{|z_{1}|\geq
L\sqrt{N}/\eta}C\eta\Theta(|z_{1}|\eta/\sqrt{m_{1}})$ $\leq
CN\sum_{|z_{1}|\geq L\sqrt{N}/\eta}\eta\Theta(|z_{1}|\eta/\sqrt{N})\leq
CN^{3/2}\int_{L-1}^{\infty}\Theta(r)\mathrm{d}r<{\varepsilon}N^{3/2}/10$

for $L$ large.

Finally assume that (A3) fails. Then we use the exponential mixing of $G$ to
derive

$\eta\sum_{z_{2}:|z_{2}-z_{1}|>L/\eta}\int{\tilde{H}}(x_{i_{1}},G_{z_{1}\eta}(y)){\tilde{H}}(x_{i_{2}},G_{z_{2}\eta}(y))\mathrm{d}\nu$
$\leq
2\int_{L}^{\infty}\int{\tilde{H}}(x_{i_{1}},y){\tilde{H}}(x_{i_{2}},G_{t}(y))\mathrm{d}\nu\mathrm{d}t\leq
Ce^{-cL}.$

Proceeding as in the case of $T_{1}$ we obtain that the sum corresponding to
indices when (A3) is violated is bounded by
$CN^{3/2}e^{-cL}<{\varepsilon}N^{3/2}/10$. This completes the proof of the
lemma. $\square$

######  Lemma 4.4.

For any $k\geq 2$, (4.3) holds with

(4.12)
$J_{k}=\left[\frac{2}{\sqrt{\pi}}\right]^{k}\int_{w_{1},...,w_{k}\in\mathbb{R}}\int_{0<t_{1}<...<t_{2k}<1}\prod_{j=1}^{2k}\frac{1}{\sqrt{t_{j}-t_{j-1}}}\sum_{v\in
V}\prod_{j=1}^{2k}\varphi\left(\frac{w_{v(j)}-w_{v(j-1)}}{\sqrt{t_{j}-t_{j-1}}}\right)\mathrm{d}t\mathrm{d}w,$

where $V$ is the set of all two-to-one mappings from $\\{1,2,...,2k\\}$ to
$\\{1,2,...,k\\}$ (that is, $|v^{-1}(l)|=2$ for all $l=1,...,k$).

###### Proof of Lemma 4.4.

We follow the strategy of the proof of Lemma 4.3.

Fix some ${\varepsilon}>0$. We need to show that for $N$ large enough,

$\left|N^{-3k/2}\Lambda^{-k}\int\left(\int\left[\sum_{n=1}^{N}{\tilde{H}}\circ
F^{n}(x,y)\right]^{2}\mathrm{d}\nu\right)^{k}\mathrm{d}\mu-
J_{k}\right|<{\varepsilon}.$

Now we have

$\displaystyle\int\left(\int\left[\sum_{n=1}^{N}{\tilde{H}}\circ
F^{n}(x,y)\right]^{2}\mathrm{d}\nu\right)^{k}\mathrm{d}\mu$ $\displaystyle=$
$\displaystyle\sum_{n_{1},n_{2},...,n_{2k}=1}^{N}\sum_{i_{1},i_{2}...,i_{2k}\in
I}\sum_{z_{1},z_{2},...,z_{2k}\in\mathbb{Z}}$ $\displaystyle\int
p(x)\left[\prod_{j=1}^{k}\int{\tilde{H}}(f^{n_{2j-1}}(x),G_{\tau_{n_{2j-1}}(x)}(y)){\tilde{H}}(f^{n_{2j}}(x),G_{\tau_{n_{2j}}(x)}(y))\mathrm{d}\nu\right]\mathrm{d}\mu$

where

$p(x)=p_{\eta,\underline{n},\underline{i},\underline{z}}(x)=\left[\prod_{j=1}^{2k}{\mathbbm{1}}_{f^{n_{j}}x\in
X_{i_{j}}}\right]\left[\prod_{j=1}^{2k}{\mathbbm{1}}_{\tau_{n_{j}}(x)\in[z_{j},z_{j}+1]\eta}\right].$

Let us order the numbers $n_{1},...,n_{2k}$ as $m_{1}\leq m_{2}\leq...\leq
m_{2k}$ and denote $m_{0}=0$. As before, we write the sum in (4.2) as
$T_{1}+T_{2}$, where $T_{1}$ corresponds to the terms satisfying

  * (A1’)

for every $j=1,...,2k$, $m_{j}-m_{j-1}>\eta^{\prime}N$

  * (A2’)

$|z_{2j-1}|<L\sqrt{N}/\eta$ for all $j=1,...,k$

  * (A3’)

$|z_{2j-1}-z_{2j}|<L/\eta$ for all $j=1,...,k$

and write $T_{1}=\widetilde{\sum}(...)$. As in (4.8),

(4.14) $T_{1}\approx\widetilde{\sum}\int
p(x)\mathrm{d}\mu\prod_{j=1}^{k}\int{\tilde{H}}(x_{i_{2j-1}},G_{z_{2j-1}\eta}(y)){\tilde{H}}(x_{i_{2j}},G_{z_{2j}\eta}(y))\mathrm{d}\nu,$

where $\approx$ now means that the difference between the LHS and the RHS an
be made smaller than ${{\varepsilon}\Lambda^{k}N^{3k/2}}/{10}$ provided that
$\eta$ and $\eta^{\prime}$ are small enough and $L$ and $N$ are large enough.
To compute $\int p(x)d\mu$, we use the MMLLT for times
$m_{1}<m_{2}<...<m_{2k}$. Note however that the range of summation for
different $z_{j}$’s are different and so it is important to keep track of the
index $i$ so that $n_{i}=m_{j}$. To this end, define to permutation $\rho$
(uniquely defined by the tuple $(n_{1},...,n_{2k})$ if (A1’) holds) so that
$m_{j}=n_{\rho(j)}$. Writing $\rho(0)=0$ and $z_{0}=0$ and using the MMLLT we
obtain

(4.16) $\displaystyle T_{1}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\varsigma_{1}^{2k}}\widetilde{\sum}\eta^{2k}\left[\prod_{j=1}^{2k}\frac{1}{\sqrt{m_{j}-m_{j-1}}}\varphi\left(\frac{(z_{\rho(j)}-z_{\rho(j-1)})\eta}{\varsigma_{1}\sqrt{m_{j}-m_{j-1}}}\right)\right]\times$
$\displaystyle\left[\prod_{j=1}^{2k}\mu(X_{i_{j}})\right]\left[\prod_{j=1}^{k}\int{\tilde{H}}(x_{i_{2j-1}},G_{z_{{2j-1}}}(y)){\tilde{H}}(x_{i_{2j}},G_{z_{{2j}}}(y))\mathrm{d}\nu\right].$

By (A1’) and (A3’), $\rho(l)$ can be replaced by $2\lceil\rho(l)/2\rceil-1$
(the biggest odd integer not bigger than $\rho(j)$) for $l=j-1,j$ in the
subscripts of $z$ in (4.16). Consequently, $z_{2j}$ only appears in (4.16). As
before, we compute

$\sum_{i_{1},...,i_{2k}}\left[\prod_{j=1}^{2k}\mu(X_{i_{j}})\right]\eta^{k}\sum_{\begin{subarray}{c}z_{2},z_{4},...,z_{2k}:\\\
|z_{2j}-z_{2j-1}|<L/\eta\end{subarray}}\left[\prod_{j=1}^{k}\int{\tilde{H}}(x_{i_{2j-1}},G_{z_{{2j-1}}}(y)){\tilde{H}}(x_{i_{2j}},G_{z_{{2j}}}(y))\mathrm{d}\nu\right]$
$=\varsigma_{2}^{2k}+o(1).$

Thus we arrive at

$T_{1}\approx\frac{\varsigma_{2}^{2k}}{\varsigma_{1}^{2k}}\widehat{\sum}\eta^{k}\left[\prod_{j=1}^{2k}\frac{1}{\sqrt{m_{j}-m_{j-1}}}\varphi\left(\frac{(z_{2\lceil\rho(j)/2\rceil-1}-z_{2\lceil\rho(j-1)/2\rceil-1})\eta}{\varsigma_{1}\sqrt{m_{j}-m_{j-1}}}\right)\right],$

where $\widehat{\Sigma}$ refers to the sum for $n_{1},...,n_{2k}$,
$z_{1},z_{3},...,z_{2k-1}$, satisfying (A1’), (A3’). Next, observe that
$u:\\{1,...,2k\\}\to\\{1,3...,2k-1\\}$ defined by
$u(j)=2\lceil\rho(j)/2\rceil-1$ is a two-to-one mapping. Let $U$ be the set of
all such mappings. The summand in the last displayed formula only depends on
$(n_{1},...,n_{2k})$ through $(m_{1},...,m_{2k})$ and $u$. Furthermore, for
any given $(m_{1},...,m_{2k})$ and $u$, there are exactly $2^{k}$
corresponding tuples $(n_{1},...,n_{2k})$. Thus

$T_{1}\approx 2^{k}\frac{\varsigma_{2}^{2k}}{\varsigma_{1}^{2k}}\sum_{u\in
U}\sum_{\begin{subarray}{c}m_{1},...m_{2k}\\\ \text{satisfying
(A1')}\end{subarray}}\sum_{\begin{subarray}{c}z_{1},..,z_{2k-1}\\\
\text{satisfying
(A3')}\end{subarray}}\eta^{k}\left[\prod_{j=1}^{2k}\frac{1}{\sqrt{m_{j}-m_{j-1}}}\varphi\left(\frac{(z_{u(j)}-z_{u(j-1)})\eta}{\varsigma_{1}\sqrt{m_{j}-m_{j-1}}}\right)\right].$

Now we are ready to replace the last two sums (Riemann sums) by the
corresponding Riemann integral with $t_{j}\sim m_{j}/N$ and $w_{l}\sim
z_{2l-1}\eta/\sqrt{N}$. To simplify the notation a little, we denote
$v(j)=(u(j)+1)/2$, thus $v$ is a 2-to-1 mapping from $\\{1,2,...,2k\\}$ to
$\\{1,2,...,k\\}$ (the set of all such mapping is denoted by $V$). We obtain

$\displaystyle T_{1}$ $\displaystyle\approx$ $\displaystyle
2^{k}\frac{\varsigma_{2}^{2k}}{\varsigma_{1}^{2k}}N^{3k/2}\sum_{v\in V}$
$\displaystyle\int_{|w_{1}|,...,|w_{k}|<L}\int_{\begin{subarray}{c}0=t_{0}<t_{1}<...<t_{2k}<1:\\\
t_{j}-t_{j-1}<\eta^{\prime}\end{subarray}}\prod_{j=1}^{2k}\frac{1}{\sqrt{t_{j}-t_{j-1}}}\prod_{j=1}^{2k}\varphi\left(\frac{w_{v(j)}-w_{v(j-1)}}{\varsigma_{1}\sqrt{t_{j}-t_{j-1}}}\right)\mathrm{d}t\mathrm{d}w.$

Substituting $w$ to $w/\varsigma_{1}$ in the above integrals, we obtain

$T_{1}\approx\frac{\varsigma_{2}^{2k}}{\varsigma_{1}^{k}}\pi^{k/2}N^{3k/2}J_{k}.$

As in Lemma 4.3, $T_{2}$ is negligible. This completes the proof of Lemma 4.4.
$\square$

To finish the proof of Proposition 4.1, it remains to verify (4.4). To this
end, first note that since $\varphi$ decays quickly, there is a constant $M$
so that for every $v\in V$ and every $t_{1},...,t_{2k}\in(0,1]$,

$\int_{\mathbb{R}^{k}}\prod_{j=1}^{2k}\varphi\left(\frac{w_{v(j)}-w_{v(j-1)}}{\varsigma_{1}\sqrt{t_{j}-t_{j-1}}}\right)\mathrm{d}w<M^{k}.$

Noting that $|V|=(2k)!/2^{k}$, we have

$|J_{k}|\leq
C^{k}(2k)!\int_{0<t_{1}<...<t_{2k}<1}\prod_{j=1}^{2k}\frac{1}{\sqrt{t_{j}-t_{j-1}}}\mathrm{d}t.$

The above integral can be computed explicitly and is equal to
$\displaystyle\frac{\Gamma(1/2)^{2k}}{\Gamma(k+1)}=\frac{\pi^{k}}{k!}.$ Noting
that $(2k)!/(k!)<C^{k}k!$, we find that $\displaystyle|J_{k}|\leq C^{k}k!$
whence (4.4) follows. This completes the proof of Proposition 4.1.

##  5\. Polynomially mixing flows in dimension one.

In this section, we extend the result of Theorem 1.7 to some flows with
polynomial mixing rates. We use the setting of [10]. Recall that a flow
$G_{t}$ is called partially hyperbolic if there is a $G_{t}$ invariant
splitting $TY=E_{u}\oplus E_{cs}$ and positive constants
$C_{1},C_{2},\lambda_{1},\lambda_{2}$ such that for each $t\geq 0$

(i) $\|dG_{-t}|E_{u}\|\leq C_{1}e^{-\lambda_{1}t};$

(ii) For any unit vectors $v_{u}\in E_{u},$ $v_{cs}\in E_{cs}$ we have
$\|dG_{t}|v_{cs}\|\leq C_{2}e^{-\lambda_{2}t}\|dG_{t}v_{u}\|.$

For partially hyperbolic flows the leaves of $E_{u}$ are tangent to the leaves
of an absolutely continuous foliation $W^{u}.$

Fix constants $R,{\bar{v}},{\bar{C}}_{1},\alpha_{1}$. Let
$\mathfrak{A}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1})$ denote the collection of
sets $D$ which belong to one leaf of $W^{u}$ and satisfy

${\rm diam}(D)\leq R,\quad{\rm mes}(D)\geq{\bar{v}},\quad{\rm
mes}(\partial_{\varepsilon}D)\leq{\bar{C}}_{1}{\varepsilon}^{\alpha_{1}}$

for all ${\varepsilon}>0$, where $\partial_{\varepsilon}D$ is the
${\varepsilon}$ neighborhood of the boundary of $D.$ We say that the sets from
$\mathfrak{A}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1})$ have bounded geometry.

Fix ${\bar{C}}_{2},\alpha_{2}>0.$ Let
$\mathfrak{M}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1},{\bar{C}}_{2},\alpha_{2})$
denote the set of linear functionals of the form

${\mathbb{E}}_{\ell_{D,\rho}}(A)=\int_{D}A(y)\rho(y)\mathrm{d}y,$

where $D\in\mathfrak{A}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1}),$ $\rho$ is a
probability density on $D$, $\ln\rho\in C^{\alpha_{2}}(D),$
$\|\ln\rho\|_{C^{r}}\leq{\bar{C}}_{2}.$ We shall often use the existence of
almost Markov decomposition established in [10, Section 2]: if $R$,
${\bar{C}}_{1},$ and ${\bar{C}}_{2}$ are large enough and
$\alpha_{1},\alpha_{2}$ and ${\bar{v}}$ are small enough, then given
$\ell\in\mathfrak{M}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1},{\bar{C}}_{2},\alpha_{2})$
and $t>0$ we can decompose

(5.1) ${\mathbb{E}}_{\ell}(A\circ
G_{t})=\sum_{s}c_{s}{\mathbb{E}}_{\ell_{s}}(A)+O\left(\|A\|_{C^{0}}\theta^{t}\right)$

with
$\ell_{s}\in\mathfrak{M}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1},{\bar{C}}_{2},\alpha_{2})$
and $\displaystyle\sum_{s}c_{s}=1+O(\theta^{t})$ for some $\theta<1.$

We say that a measure is u-Gibbs if its conditional measures on unstable
leaves have smooth densities. The existence of almost Markov decomposition
implies that u-Gibbs measures belong to the convex hull of
$\mathfrak{M}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1},{\bar{C}}_{2},\alpha_{2})$
for appropriate choice of
$R,{\bar{v}},{\bar{C}}_{1},\alpha_{1},{\bar{C}}_{2},\alpha_{2}$ (see [5,
§11.2]).

From now on we shall fix constants involved in the definition of
$\mathfrak{M}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1},{\bar{C}}_{2},\alpha_{2})$
which satisfies (5.1) and whose convex hull contains u-Gibbs measures. For the
sake of simplicity, we will write $\mathfrak{M}$ instead of
$\mathfrak{M}(R,{\bar{v}},{\bar{C}}_{1},\alpha_{1},{\bar{C}}_{2},\alpha_{2}).$

We say that the unstable leaves become equidistributed at rate $a(t)$ if there
is $r>0$ such that for all $\ell\in\mathfrak{M}$ we have

(5.2) $\left|{\mathbb{E}}_{\ell}(A\circ G_{t})-\nu(A)\right|\leq
a(t)\|A\|_{C^{r}}$

where $\nu$ is the reference invariant measure for $G_{t}.$ We note that if
(5.2) holds with $a(t)\to 0$, then $\nu$ belongs to the convex hull of
$\mathfrak{M}$, whence it is a u-Gibbs measure (in fact, it is the unique SRB
measure for $G_{t}$, see [10, Corollary 2]).

######  Theorem 5.1.

Theorem 1.7 remains valid if the assumption that $G$ enjoys exponential mixing
of all orders is replaced by the assumption that $G$ is partially hyperbolic
and unstable leaves become equidistributed with rate $Kt^{-\gamma}$ for some
$\gamma>1$ and $K>0$.

######  Corollary 5.2.

Suppose that $\tau$ satisfies the assumption of Theorem 1.7, $G_{t}$ is a
topologically transitive Anosov flow whose stable and unstable foliations are
not jointly integrable, and $\nu$ is the SRB measure. Then (1.4) holds for
sufficiently smooth functions.

Indeed, according to [9, Theorem 3], for Anosov flows, unstable leaves are
equidistributed on $C^{r}$ at rate $Kt^{-\gamma(r)}$ where
$\gamma(r)\to\infty$ as $r\to\infty.$

The proof of Theorem 5.1 requires a modification of Proposition 2.1. We shall
use the same notation as in that proposition.

######  Proposition 5.3.

Suppose that $G_{t}$ is partially hyperbolic with unstable leaves
equidistributed at rate $Kt^{-\gamma}$ for some $\gamma>1.$ Suppose that for
some $0<\delta<1/20$ such that

(5.3) $\gamma-1>\frac{16\delta}{1-20\delta}$

and $\kappa>0$ we have

(a) $\mathfrak{m}_{N}(t\in\mathbb{R}:|t|\leq N^{(1/2)+\delta})\leq
N^{1/4+\delta}$ and $\mathfrak{m}_{N}(t:|t|>N^{(1/2)+\delta})=0$;

(b) for every $t\in\mathbb{R}$, $\mathfrak{m}_{N}([t,t+1])\leq
N^{\delta-(1/4)};$

(c) $\displaystyle\iint_{|t_{1}-t_{2}|>\sqrt{\ln
N}}\frac{\mathrm{d}\mathfrak{m}_{N}(t_{1})\mathrm{d}\mathfrak{m}_{N}(t_{2})}{|t_{1}-t_{2}|^{\gamma}}\leq\ln^{-\kappa}N;$

(d) $\displaystyle\lim_{N\to\infty}\nu(\mathcal{S}_{N}^{2})=1.$

Then $\mathcal{S}_{N}\Rightarrow\mathcal{N}(0,1)$ as $N\to\infty.$

######  Lemma 5.4.

Under the assumptions of Theorems 5.1, there are subsets $X_{N}\subset X$ such
that $\displaystyle\lim_{N\to\infty}\mu(X_{N})=1$ and for any sequence
$x_{N}\in X_{N}$ the measures $\\{\mathfrak{m}_{N}(x_{N})\\}$, defined by
(4.1), satisfy the conditions of Proposition 5.3.

###### Proof.

The fact that conditions (a) and (b) hold for arbitrary $\delta>0$ are
verified as before. Condition (d) is immediate from the definition of
$\mathfrak{m}_{N}.$ In order to verify condition (c) we note that by
Proposition 4.1, for any $\kappa>0$, $\mu(x:V_{N}(x)<N^{3/2}\ln^{-\kappa}N)\to
0$ as $N\to\infty.$ Therefore it is enough to check that if $\delta$ and
$\kappa$ are sufficiently small, then

$\mu(x:\mathcal{R}_{N}(x)<N^{3/2}\ln^{-2\kappa}N)$

is close to 1, where

$\mathcal{R}_{N}(x):=\sum_{n_{1},n_{2}\in\mathbb{Z}:|n_{1}-n_{2}|>\sqrt{\ln
N}}\frac{\ell(x,n_{1},N)\ell(x,n_{2},N)}{|n_{1}-n_{2}|^{\gamma}}$

and $\ell(x,t,N)$ is the local time defined by (3.6). Note that

$\mu\left(\ell(n_{1},N)\ell(n_{2},N)\right)=\sum_{1\leq j_{1},j_{2}\leq
N}\mu(x:|\tau_{j_{1}}(x)-n_{1}|\leq 1,|\tau_{j_{2}}(x)-n_{2}|\leq 1).$

Therefore, using the anticoncentration large deviation bound, we conclude that

$\mu(\mathcal{R}_{N})\leq C\sum_{1\leq j_{1}\leq j_{2}\leq
N}\sum_{|n_{1}-n_{2}|\geq\sqrt{\ln
N}}\frac{1}{\sqrt{j_{1}}\sqrt{j_{2}-j_{1}+1}}\Theta\left(\frac{n_{1}}{\sqrt{j_{1}}}\right)\frac{1}{|n_{2}-n_{1}|^{\gamma}}$
$\leq
C\frac{\sqrt{N}}{\ln^{(\gamma-1)/2}N}\sum_{j_{1}}\sum_{n_{1}}\frac{1}{\sqrt{j_{1}}}\;\Theta\left(\frac{n_{1}}{\sqrt{j_{1}}}\right)\leq
C\frac{\sqrt{N}}{\ln^{(\gamma-1)/2}N}\sum_{j_{1}}\int_{0}^{\infty}\Theta\left(r\right)\mathrm{d}r\leq
C\frac{N^{3/2}}{\ln^{(\gamma-1)/2}N}.$

Now the result follows by the Markov inequality provided that
$\gamma-1>4\kappa.$ $\square$

Thus in order to prove Theorem 5.1 it suffices to establish Proposition 5.3.

###### Proof of Proposition 5.3..

We divide the interval $[-N^{1/2+\delta},N^{1/2+\delta}]$ into big blocks of
size $N^{\beta_{1}}>0$ separated by small blocks of size $N^{\beta_{2}}$ where
the parameters $\beta_{2}<\beta_{1}$ will be chosen later. Let $J_{j}$ denote
the $j$-th big block and $L$ be the union of the small blocks. Let
$\mathcal{S}_{N}^{\prime}=\int_{t\in
L}A_{t}(G_{t}y)\mathrm{d}\mathfrak{m}_{N}(y).$ Note that
$\nu(\mathcal{S}^{\prime}_{N})=0.$ We claim that
$\nu((\mathcal{S}_{N}^{\prime})^{2})\to 0$ as $N\to\infty$ provided that

(5.4) $\beta_{1}>\beta_{2}+3\delta.$

Indeed

(5.5)
$\nu\left(\left(\mathcal{S}_{N}^{\prime}\right)^{2}\right)\leq\iint_{L\times
L}\nu(A_{t_{1}}(A_{t_{2}}\circ
G_{t_{2}-t_{1}}))\mathrm{d}\mathfrak{m}_{N}(t_{1})\mathrm{d}\mathfrak{m}_{N}(t_{2}).$

According to [10, Theorem 2], $\displaystyle\nu(A_{t_{1}}(A_{t_{2}}\circ
G_{t_{2}-t_{1}}))=O\left(|t_{2}-t_{1}|^{-\gamma}\right).$ Therefore property
(b) shows that the integral of the RHS of (5.5) with respect to $t_{2}$ is
$O\left(N^{-(1/4)+\delta}\right)$. To integrate with respect to $t_{1}$ we
divide $L$ into unit intervals. Noting that the mass of each interval is
$O\left(N^{\delta-(1/4)}\right)$ and the number of intervals is
$O\left(N^{(1/2)+\delta-\beta_{1}+\beta_{2}}\right)$, we conclude that
$\displaystyle\nu\left(\left(\mathcal{S}_{N}^{\prime}\right)^{2}\right)=O\left(N^{\beta_{2}+3\delta-\beta_{1}}\right)$
which tends to zero under (5.4).

Thus the main contribution to the variance comes from the big blocks. Let

$\tau=N^{1/2+\delta},\quad
T_{j}=T_{j}(y)=\int_{J_{j}}A_{t}(G_{\tau+t}y)\mathrm{d}\mathfrak{m}_{N}(t),\quad
S_{N,j}=\sum_{k=1}^{j}T_{k}.$

Fix any $\xi\in\mathbb{R}$. We will show that there is a sequence
${\varepsilon}_{N}\to 0$, depending only $|\xi|$ so that for each
$\ell\in\mathfrak{M}$ we have

$|{\mathbb{E}}_{\ell}\left(e^{i\xi
S_{N,{\bar{N}}}}\right)-e^{-\xi^{2}/2}|\leq{\varepsilon}_{N}$

where ${\bar{N}}$ is the number of big blocks. Let
$\displaystyle\Phi_{j}(\xi)={\mathbb{E}}_{\ell}\left(e^{i\xi S_{N,j}}\right)$
with $\Phi_{0}(\xi)=1$.

######  Lemma 5.5.

If

(5.6) $\beta_{1}<1/4-2\delta,$

then

$\Phi_{j}(\xi)=\Phi_{j-1}(\xi)\left[1-\frac{\xi^{2}}{2}v_{j}\right]+O\left(v_{j}N^{-\delta}+w_{j}+{\hat{{\varepsilon}}}_{N}\right),$

where

$v_{j}\\!=\\!\nu(T_{j}^{2}),\quad w_{j}\\!=\\!\iint_{(t_{1},t_{2})\in
J_{j}\atop|t_{1}-t_{2}|\geq\sqrt{\ln
N}}\frac{\mathrm{d}\mathfrak{m}_{N}(t_{1})\mathrm{d}\mathfrak{m}_{N}(t_{2})}{|t_{1}-t_{2}|^{\gamma}},\quad{\hat{{\varepsilon}}}_{N}\\!=\\!N^{\delta-1/4-\beta_{2}(\gamma-1)}.$

Note that by property (b) and (5.6), we have
$T_{j}\\!\\!=\\!\\!O(N^{-\delta}).$ Hence $v_{j}\\!\\!=\\!\\!O(N^{-2\delta})$
and so the estimate of Lemma 5.5 can be rewritten as

$\Phi_{j}(\xi)=\Phi_{j-1}(\xi)e^{-\xi^{2}v_{j}/2}+O\left(v_{j}N^{-\delta}+w_{j}+{\hat{{\varepsilon}}}_{N}\right).$

Repeating this process, we obtain

(5.7)
${\mathbb{E}}_{\ell}\left(e^{i\xi\mathcal{S}_{N,{\bar{N}}}}\right)=\Phi_{\bar{N}}(\xi)=\exp\left[-\frac{\xi^{2}}{2}\sum_{j=1}^{N}v_{j}\right]+O\left(\sum_{j}\left[v_{j}N^{-\delta}+w_{j}\right]+{\bar{N}}{\hat{{\varepsilon}}}_{N}\right).$

Next we claim that

(5.8) $\sum_{j=1}^{\bar{N}}v_{j}=1+o_{N\to\infty}(1)$

provided that

(5.9) $\beta_{2}(\gamma-1)>2\delta.$

Indeed, $\displaystyle
1+o(1)\\!\\!=\\!\\!\nu(S_{N,{\bar{N}}}^{2})\\!\\!=\\!\\!\sum_{j=1}^{\bar{N}}v_{j}\\!+\\!\sum_{j_{1}\neq
j_{2}}\nu\left(T_{j_{1}}T_{j_{2}}\right).$ The second term here is at most

$C\sum_{n_{1},n_{2}:|n_{1}-n_{2}|\geq
N^{\beta_{2}}}\frac{\mathfrak{m}_{N}([n_{1},n_{1}+1])\mathfrak{m}_{N}([n_{2},n_{2}+1])}{|n_{1}-n_{2}|^{\gamma}}.$

Summing over $n_{2}$ using assumption (b) we are left with

$C\sum_{n_{1}}\mathfrak{m}_{N}([n_{1},n_{1}+1])N^{\delta-(1/4)-\beta_{2}(\gamma-1)}$
$=C\mathfrak{m}_{N}([-N^{1/2+\delta},N^{1/2+\delta}])N^{\delta-(1/4)-\beta_{2}(\gamma-1)}\leq{\bar{C}}N^{2\delta-\beta_{2}(\gamma-1)}=o_{N\to\infty}(1),$

where in the last inequality we also used assumption (a).

This proves that (5.9) implies (5.8). (5.8) shows in particular that
$\displaystyle\sum_{j}v_{j}N^{-\delta}\\!\\!=\\!\\!O(N^{-\delta}).$ Also
$\displaystyle\sum_{j}w_{j}\\!=\\!o(1)$ due to assumption (c) of Proposition
5.3, while

${\bar{N}}{\hat{{\varepsilon}}}\\!\leq\\!N^{1/4+2\delta-\beta_{1}-\beta_{2}(\gamma-1)}\\!=\\!o(1)$

provided that

(5.10) $\beta_{1}+\beta_{2}(\gamma-1)>\frac{1}{4}+2\delta.$

Plugging these estimates into (5.7) we conclude that for all
$\ell\in\mathfrak{M}$ we have

(5.11) ${\mathbb{E}}_{\ell}\left(e^{i\xi
S_{N,{\bar{N}}}}\right)=e^{-\xi^{2}/2}+o_{N\to\infty}(1)$

if $\beta_{1}$ and $\beta_{2}$ satisfy (5.4), (5.6), (5.9), and (5.10). Thus
we need $\beta_{1}$ and $\beta_{2}$ to satisfy

$\beta_{1}<\frac{1}{4}-2\delta,\quad\beta_{2}<\beta_{1}-3\delta,\quad\beta_{1}+\beta_{2}(\gamma-1)>\frac{1}{4}+2\delta,\quad\beta_{2}(\gamma-1)>2\delta.$

Since $\beta_{1}$ can be chosen arbitrary close to $\frac{1}{4}-2\delta$ and
$\beta_{2}$ can be chosen arbitrarily close to $\beta_{1}-3\delta$, the above
inequalities are compatible if (5.3) holds. It then follows that (5.11) holds
on the convex hull of $\mathfrak{M}$ which includes $\nu.$ This completes the
proof of Proposition 5.3 modulo Lemma 5.5. $\square$

###### Proof of Lemma 5.5.

Let $J_{j}=[n_{j}^{-},n_{j}^{+}].$ Denote
$m_{j}=\tau+\frac{n_{j-1}^{+}+n_{j}^{-}}{2}.$ We use the almost Markov
decomposition (5.1):

${\mathbb{E}}_{\ell}(A\circ
G_{m_{j}})=\sum_{s}c_{s}{\mathbb{E}}_{\ell_{s}}(A)+O({\tilde{\varepsilon}}_{N})$

where ${\tilde{\varepsilon}}_{N}=\theta^{N^{\beta_{2}}}$,
$\ell_{s}=(D_{s},\rho_{s})\in\mathfrak{M}$ and
$\displaystyle\sum_{s}c_{s}=1-O({\tilde{\varepsilon}}_{N}).$

Fix arbitrary $y_{s}\in G_{-m_{j}}D_{s}.$ Then

${\mathbb{E}}_{\ell}\left(e^{i\xi
S_{N,j}}\right)=\sum_{s}c_{s}{\mathbb{E}}_{\ell_{s}}\left(e^{i\xi[S_{N,j-1}(y_{s})+{\tilde{T}}_{j}(y)]}\right)+O({\tilde{\varepsilon}}_{N})$

where

${\tilde{T}}_{j}(y)=T_{j}(G_{-m_{j}}y)=\int_{J_{j}}A_{t}(G_{t-m_{j}}y)\mathrm{d}\mathfrak{m}_{N}(t).$

Next

${\mathbb{E}}_{\ell_{s}}\left(e^{i\xi{\tilde{T}}_{j}}\right)=1+i\xi{\mathbb{E}}({\tilde{T}}_{j})-\frac{\xi^{2}}{2}{\mathbb{E}}_{\ell_{s}}({\tilde{T}}_{j}^{2})+O\left({\mathbb{E}}_{\ell_{s}}\left(\left|{\tilde{T}}_{j}\right|^{3}\right)\right).$

Note that
$\displaystyle{\mathbb{E}}({\tilde{T}}_{j})=O\left(N^{\beta_{2}(\gamma-1)-(1/4)+\delta}\right)$
due to the equidistribution of unstable leaves.

To estimate ${\mathbb{E}}_{\ell_{s}}({\tilde{T}}_{j}^{2})$ we split

${\mathbb{E}}_{\ell_{s}}({\tilde{T}}_{j}^{2})=\iint_{J_{j}\times
J_{j}}{\mathbb{E}}_{\ell_{s}}[A_{t_{1}}(G_{t_{1}-m_{j}}y)A_{t_{2}}(G_{t_{2}-m_{j}}y)]\mathrm{d}\mathfrak{m}_{N}(t_{1})\mathrm{d}\mathfrak{m}_{N}(t_{2})=I+{I\\!\\!I}$

where $I$ includes the terms where $|t_{1}-t_{2}|\leq\sqrt{\ln N}$ and
${I\\!\\!I}$ includes the other terms.

According to [10, Theorem 2],

${\mathbb{E}}_{\ell_{s}}\left((A_{t_{1}}\circ G_{t_{1}-m_{j}})(A_{t_{2}}\circ
G_{t_{2}-m_{j}})\right)=O\left(|t_{2}-t_{1}|^{-\gamma}\right).$

Therefore ${I\\!\\!I}=O(w_{j}).$

To estimate $I$, we note that

$\displaystyle{\mathbb{E}}_{\ell_{s}}\left((A_{t_{1}}\circ
G_{t_{1}-m_{j}})(A_{t_{2}}\circ
G_{t_{2}-m_{j}})\right)={\mathbb{E}}_{\ell_{s}}\left((D_{t_{1},t_{2}}\circ
G_{t_{1}-m_{j}})\right)$

where $D_{t_{1},t_{2}}=A_{t_{1}}(A_{t_{2}}\circ G_{t_{2}-t_{1}})$ with
$\displaystyle\|D_{t_{1},t_{2}}\|_{C^{r}}\leq K^{t_{2}-t_{1}}\leq
N^{\delta/2}.$

Hence using the equidistribution of unstable leaves, we obtain

${\mathbb{E}}_{\ell_{s}}\left((A_{t_{1}}\circ G_{t_{1}-m_{j}})(A_{t_{2}}\circ
G_{t_{2}-m_{j}})\right)=\nu(D_{t_{1},t_{2}})+O\left(N^{\delta/2}(t_{1}-m_{j})^{-\gamma}\right)$
$=\nu(A_{t_{1}}(A_{t_{2}}\circ
G_{t_{2}-t_{1}}))+O\left(N^{\delta/2}(t_{1}-m_{j})^{-\gamma}\right).$

It follows that

$I=\iint_{t_{1},t_{2}\in J_{j}\atop|t_{1}-t_{2}|\leq\sqrt{\ln
N}}\nu(A_{t_{1}}(A_{t_{2}}G_{t_{2}-t_{1}}))\mathrm{d}\mathfrak{m}_{N}(t_{1})\mathrm{d}\mathfrak{m}_{N}(t_{2})+O\left(N^{(5\delta/2)-(1/2)-\beta_{2}(\gamma-1)}\sqrt{\ln
N}\right)$
$=v_{j}+O(w_{j})+O\left(N^{3\delta-(1/2)-\beta_{2}(\gamma-1)}\right).$

Finally

${\mathbb{E}}_{\ell_{s}}(|{\tilde{T}}_{j}|^{3})\leq
N^{-\delta}{\mathbb{E}}_{\ell_{s}}(|{\tilde{T}}_{j}|^{2})=O\left(\left[v_{j}+w_{j}\right]N^{-\delta}+N^{2\delta-(1/2)-\beta_{2}(\gamma-1)}\right).$

Summing over $s$ and using the fact that
$\displaystyle\sum_{s}c_{s}{\mathbb{E}}_{\ell_{s}}\left(e^{i\xi
S_{N,{j-1}}(y_{s})}\right)={\mathbb{E}}_{\ell}\left(e^{i\xi
S_{N,j-1}}\right)+O({\tilde{\varepsilon}}_{N})$, we obtain the result.
$\square$

## Acknowledgement

C.D. was partially supported by Nankai Zhide Foundation, and AMS-Simons travel
grant. D.D. was partially supported by NSF DMS 1956049. A.K. was partially
supported by NSF DMS 1956310. P.N. was partially supported by NSF DMS 2154725.

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#  $\mathcal{G}$lidar3DJ: A View-Invariant gait identification via flash lidar
data correction

###### Abstract

Gait recognition is a leading remote-based identification method, suitable for
real-world surveillance and medical applications. Model-based gait recognition
methods have been particularly recognized due to their scale and view-
invariant properties. We present the first model-based gait recognition
methodology, $\mathcal{G}$lidar3DJ using a skeleton model extracted from
sequences generated by a single flash lidar camera. Existing successful model-
based approaches take advantage of high quality skeleton data collected by
Kinect and Mocap, for example, are not practicable for application outside the
laboratory. The low resolution and noisy imaging process of lidar negatively
affects the performance of state-of-the-art skeleton-based systems, generating
a significant number of outlier skeletons. We propose a rule-based filtering
mechanism that adopts robust statistics to correct for skeleton joint
measurements. Quantitative measurements validate the efficacy of the proposed
method in improving gait recognition.

††This paper is accepted to be published in: 2019 IEEE International
Conference on Image Processing, Sept 22-25, 2019, Taipei, Taiwan. IEEE
Copyright Notice: ⓒIEEE 2019 Personal use of this material is permitted.
Permission from IEEE must be obtained for all other uses, in any current or
future media, including reprinting/republishing this material for advertising
or promotional purposes, creating new collective works, for resale or
redistribution to servers or lists, or reuse of any copyrighted component of
this work in other works.

Index Terms—  gait recognition, lidar, feature correction

##  1 Introduction

Gait recognition has been an active area of research in the last decade due to
the widespread application in forensic cases, surveillance, and medical
studies of patients affected by motion-related diseases like Parkinson’s
disease [1]. Gait recognition uses the features from both structure and motion
for person identification. Previous studies have shown the relative uniqueness
of gait for individuals [2]. Unlike other biometric features such as the iris,
face, and fingerprint, gait recognition does not require a subject’s
cooperation, nor does it require high quality data. Under uncontrolled real-
world conditions, there are many scenarios in which direct contact between
subjects and sensors is not possible, or there is a considerable distance
between cameras and subjects that makes reliable data acquisition difficult or
impossible. Under such conditions, many biometric methods fail; whereas,
several studies have shown promising results for person identification with
the gait-based biometric features [3, 4]. Furthermore, unlike color and
texture, which are among the prevalent features in many identification
studies, features extracted from gait are resilient to changes in clothing and
lighting conditions.

Fig. 1: Sample frames of lidar data. From top, first row: intensity data,
second row: range data, third row: sample frames with correctly detected
skeleton, last row: frames with faulty skeletons

In recent years, depth-sensing cameras such as Kinect and lidar have become
popular for gait analysis due to their ability to provide range (depth) and
intensity data [5, 6, 7, 8, 9]. Each pixel in the range data provides distance
information, so three-dimensional information (with the additional range
dimension) can be recorded in time. Unlike ordinary cameras, the performance
of depth cameras is not affected by changes in lighting conditions. In this
work, we use data that was collected by a single flash lidar camera. A lidar
sensor is a time-of-flight camera that uses laser beam to measure the distance
of targets from the camera. A laser beam can be focused into small spots to
suit the objects of interest and does not expand considerably by the object’s
surface, which gives a lidar sensor the capability of providing detailed
images of a scene. As a result of such properties, lidar sensors have found
application in areas such as archaeology, forestry, geology, geography, space
missions, transportation and autonomous vehicles.

Fig. 2: Pipeline for person identification using joint location correction

Gait recognition methods using video data are generally divided into two main
categories, model-free methods and model-based methods. Model-free approaches
require features from clean silhouettes [10, 11]. Model-based methods fit a
model, like a skeleton, to human body and exploit features from the fitted
model for recognition. Unlike model-free methods, model-based approaches are
view and scale-invariant suited for real-world scenarios. On the other hand,
model-based methods are computationally expensive. But, with depth-sensing
modalities like Kinect that provide a direct estimation of joint positions,
this expense is not an issue. However, Kinect sensors have the issues of
limited range and unreliability of range information in outdoor environments,
in particular under direct sunlight, where the high intensity infrared of
environment cannot be easily differentiated from the infrared light of the
sensor [12]. Compared with Kinect, a flash lidar camera has a drastically
extended range ($>1000$ meters) and its performance is not affected in outdoor
environments due to the high irradiance power of pulsed laser compared with
the background [13].

Fig. 3: Left: Skeleton model, Right: features. Each green arrow shows one
3-dimensional vector in the feature vector.

Existing state-of-the-art model-based methods avoid the challenge of erroneous
features by adopting high-quality skeleton data provided by Kinect or Mocap.
In contrast, the data collected by a flash lidar camera is noisy and has low
resolution that negatively affects the skeleton extraction performance. Faulty
skeleton models result in features that are plagued with missing and erroneous
measurements that in turn present a major challenge for a successful gait
identification. This work takes on the challenging task of gait identification
using flash lidar data. Our main contributions can be described as follows.
First, we present the first model-based approach for gait recognition using
flash lidar data. The only existing lidar-based person identification methods
are model-free, and rely on silhouette extraction from a point cloud [14, 15].
Second, we propose a rule-based filtering mechanism to correct for erroneous
skeleton joint coordinates by modeling each joint location as a time sequence
and adopting robust statistics measures of the nearest neighbors. Third,
experiments are performed using the flash lidar data to evaluate the
performance of the proposed methodology.

##  2 Proposed methodology

Figure 2 demonstrates the workflow of the proposed methodology. For a video
sequence like $V$ with $f$ frames, the input into the proposed gait
recognition system are the intensity $I=[I_{1},I_{2},...,I_{f}]$ and range
data $R=[R_{1},R_{2},...,R_{f}]$, recorded by a single flash lidar camera,
where images are preprocessed to reduce the noise. Figure 1 shows sample
intensity and range data in the first and second row. OpenPose, a state-of-
the-art real-time pose detector [16], is leveraged to extract a skeleton model
from the intensity information of lidar. The employed skeleton model is
illustrated in the left side of Figure 3. For each frame, we present the
2-dimensional coordinates of skeleton joints in the vectorized form

$\vspace{-.3cm}J_{i}=[x_{k},y_{k}]^{M_{j}}_{k=1}\in\Re^{2N}$ (1)

where $M_{j}$ is the number of joints and $(x_{k},y_{k})$ are the coordinates
of the $kth$ joint in the image frame of reference. The sample frames with the
correct detected skeleton models can be seen in the third row of Figure 1.
Next, the range data is used to project the 2D locations of joints, provided
by OpenPose, into a real-world coordinate system:

$\vspace{-.3cm}L^{i}_{j}=\frac{2}{N_{pixels}}\times\tan(\frac{\theta_{aov}}{2})\times
Lp^{i}_{j}\times D^{i}_{camera}$ (2)

where $L^{i}_{j}$ is the real-world location of joint $i$ in the direction $j$
and $Lp^{i}_{j}$ is the corresponding location in the image coordinate system.
$N_{pixels}$ is the number of pixels in the $j$ direction, $\theta_{aov}$ is
the angle of view, and $D^{i}_{camera}$ is the range value of joint $i$.
Several factors in the data negatively affect the quality of features that are
computed from the resulting joints. As the subjects loom closer to the camera,
range data are affected by noise. The intensity data lack color, and there is
similarity between human clothing, skin and the background. The last row in
Figure 1 shows a few examples of the detected faulty skeletons, which are the
result of noisy nature of lidar data and erroneous joint localization of
OpenPose. Features that are computed from such faulty skeletons contain
missing and erroneous values that presents a big challenge for a successful
gait recognition. To resolve this problem, we present a filtering mechanism to
correct joint location values and extract features after joint correction.
Furthermore, to incorporate the dynamic of the motion, we perform feature
concatenation with a new criterion.

Fig. 4: Sample joint location time sequence in one direction before (top),
after joint correction (middle), and after smoothing (bottom).

##  3 Joint location correction

The top row of Figure 4 shows a typical joint location time sequence in one
direction. The significant number of missing values and sudden jumps in the
joint location sequence results into features that are plagued with erroneous
and outlier measurements. The outcome will be gait identification with lower
accuracy. To resolve this problem, we propose a rule-based short-memory median
filter to correct joint location time sequence. $L_{x}$, the joint location
time sequence in the x direction, extended over $F_{n}$ frames, is defined as
follows

$L_{x}=[L_{x}(t)]^{F_{n}}_{t=1}~{}~{}~{}~{}~{}~{}L_{x}(t)\in\Re\vspace{-.2cm}$
(3)

where $L_{y}$ and $L_{z}$ can be defined in the same way. We correct $L_{x}$
at time $t$, if there is a missing value ($L_{x}(t)=0$), or a sudden jump. We
define the sudden jump using the relative distance of the current and previous
values of the joint location

$\frac{\left|L_{x}(t)-L_{x}(t-1)\right|}{L_{x}(t-1)}>median(\frac{\partial
L_{x}}{\partial t})~{}~{}~{}~{},\frac{\partial L_{x}}{\partial t}\neq
0\vspace{-.25cm}$ (4)

Each instant corresponds with one frame, and $(\frac{\partial L_{x}}{\partial
t})$ is the derivative of joint location in the x direction with respect to
time. If any of the above conditions are satisfied, the corrected joint
location value at frame $t$ will be calculated according to the following
equation:

$\begin{cases}\text{$L_{x}(t)=median(W(t))$}\\\ \text{$W(t)=[L_{x}(i)\mid
L_{x}(i)\neq 0]^{t-1}_{i=t-F}$}\\\
\text{$\mathbf{card}(W(t))=P_{nz},~{}~{}~{}~{}F\geq P_{nz}$}\\\
\end{cases}\vspace{-.25cm}$ (5)

where $P_{nz}$ is the number of previous closest non-zero neighbors of joint
location values at time $t$. $L_{x}(i)$ is the joint location value at
previous closest non-zero neighbors of frame $t$. We define $W(t)$ as the
array of closest nonzero neighbors at time $t$. $W(t)$ is updated
continuously, containing at each instant the previous nonzero values of joint
location sequence that are closest in time to the current instant. This
includes the corrected joint location values.

The length of $W(t)$ is selected to follow the local pattern of $L_{x}(t)$. We
used $\mathbf{card}(W(t))=3$ as the smallest possible number of previous
nearest neighbors. $\mathbf{card}(W(t))=1$ and $\mathbf{card}(W(t))=2$ are not
proper choices for this problem. Choosing $\mathbf{card}(W(t))=1$ corresponds
with the previous nearest non-zero neighbor. However, if the nearest non-zero
neighbor is noisy, the error will propagate as a result of correction. On the
other hand, considering $\mathbf{card}(W(t))=2$ for the length of $W$,
transforms median into mean. Median is selected as a robust statistic that is
less affected by outliers. In this problem, by choosing median over mean, the
effect of erroneous joint location values can be ameliorated. Furthermore,
median filtering is effective in removing the impulse-like signal features,
which occur in this application due to be the pattern of missing joints and
jumps in the joint location sequence. The middle row of Figure 4 shows a
sample joint location sequence along one direction after applying joint
correction.

Figure 5 shows a comparison between the proposed rule-based median filter with
the moving median. Our rule-based filter is in particular successful in
correcting missing values when they occur over consecutive frames. This is
mainly the result of the way each of the above two filters work. In
particular, the moving median uses the neighborhood information irrespective
of their values. In contrast, our rule-based median filter uses the previous
neighbors’ values only if they are nonzero and there is no sudden jump between
consecutive values. However, as can be seen in Figure 4 and 5, this can also
cause the flattening of the signal in some regions.

Table 1: List of three-dimensional vectors in the feature vector (L refers to
the left joints and R refers to the right joints)

Neck to R Shoulder | Neck to L Shoulder | Neck to R Hip | Neck to L Hip
---|---|---|---
R Shoulder to R Elbow | L Shoulder to L Elbow | R Hip to R Knee | L Hip to L Knee
R Elbow to R Wrist | L Elbow to L Wrist | R Knee to R Ankle | L Knee to L Ankle

Finally, in order to alleviate the effects of signal flattening and lower
amplitude impulses, both the result of joint correction in regions with
consecutive missing values or sudden jumps, we employ RLowess (locally
weighted scattered plot smoothing) [17], that locally fits first order
polynomial using weighted linear regression, where regression weights are
estimated through a robust procedure. The robustness of the employed method is
essential due to the existence of low-amplitude impulses that act as outliers.
In Figure 4 the last row shows the smoothed joint location time sequence.

Fig. 5: From top to bottom: original joint location in one direction, after
joint correction with the proposed rule-based median, after applying moving
median filter of the same window length

##  4 feature vectors

Like [18], our features is comprised by a set of 3-dimensional vectors
measured between selected joints of skeleton model. Compared to features that
describe the distance between joints [19], or to features that only consider
angles between selected joints [20], we can implicitly encode both the
distance, and the angles of selected joints in the skeleton in different
postures. Each three-dimensional vector $V_{ij}$ in our feature vector is
defined according to equation 6:

$\vspace{-.2cm}V_{ij}=(X_{i}-X_{j},Y_{i}-Y_{j},Z_{i}-Z_{j}),~{}~{}where~{}~{}i\neq
j$ (6)

where $i$ and $j$ are the indices of selected joints. Unlike features in [18]
that were defined with respect to a reference joint, our features are
formulated between different joints. Table 1 describes the name of the joints
that form each of the three-dimensional vectors. Figure 3, right illustrates
the described vectors.

Table 2: Correct identification scores (accuracy and F-score) for the proposed
features (**), the methods in [4] and [19]. Features are computed without
joint correction.

| [4] | [19] | **
---|---|---|---
Accuracy | 43.07% | 45.67% |  56.26%
F-Score | 42.41% | 43.72% |  57.24%

Table 3: Correct identification scores for $\mathcal{G}$lidar3DJ,
$\mathcal{G}$lidar3DJ with feature concatenation (F-C), and the methods in [4]
and [19]. Features are computed from corrected joints.

| [4] | [19] |  $\mathcal{G}$lidar3DJ |  $\mathcal{G}$lidar3DJ(F-C)
---|---|---|---|---
Accuracy | 61.20% | 70.59% | 81.24% |  85.11%
F-Score | 57.41% | 65.15% | 80.30% |  84.33%

##  5 feature concatenation

When two individuals have similar body measurements in multiple postures,
motion dynamics can play a crucial role in gait identification. A common
practice to include motion dynamics in many model-based methods is to compute
features like speed and step length using ankle-to-ankle distance sequence
[21, 22], or to calculate moments like variance, maximum and average of
selected features in each gait cycle [4, 19, 23]. The gait cycle is estimated
by looking at the distance between the two ankle joints time sequence, which
is straightforward with clean joint position data. Figure 6 shows a comparison
between ankle-to-ankle distance sequences, one from Kinect, and one from
joint-corrected samples of our lidar data set. As can be seen in this figure,
while a clear cyclic pattern can be observed for the Kinect sample, it is
difficult to determine a gait cycle in the lidar time sequence. To compensate
for such shortcomings, instead of calculating feature moments, we concatenate
feature vectors in the consecutive frames to encode the dynamics of the
motion.

To find a proper window length for feature concatenation, we use the idea of
gait cycle; however, the gait cycle can vary from subject to subject and even
during a course of walking sequence. To resolve this issue, we first remove
the small and large values of gait cycle from each of the ankle-to-ankle
distance curve. These small and large values occur in the beginning and end of
the motion, as well as when the subjects change their motion direction. Once
such outlier gait cycles are removed, majority voting is performed on the
remaining gait cycles. The gait cycle that appears the most is selected as the
length of window for feature concatenation.

Fig. 6: Ankle-to-ankle distance for Kinect (left) vs. lidar(right).

##  6 Results

The data set recorded by a single flash lidar camera includes 34 sequences of
walking from 10 subjects in which the camera is fixed during all the actions.
The walking action is performed in three different ways, capturing multiple
views of the subjects: walking toward and away from the camera, walking on a
diamond shape, and walking on a diamond shape while holding a yard stick in
one hand. We used 70% of the sequences for training and the rest for testing,
where the classifier is tested on a type of walking that it was not trained
on. K-nearest neighbors of $N=7$ and Manhattan distance is adopted as our
classifier. The performance of the proposed approach is compared with the
works in [4] and [19], which are among state-of-the-art model-based methods.
Table 2 shows the correct identification scores with the proposed features,
the features in [4] and [19] without joint correction. Results in Table 3
report identification scores with the proposed joint correction. It also shows
the scores after applying feature concatenation on $\mathcal{G}$lidar3DJ. By
comparing the results, it is clear that joint correction can drastically
improve gait identification accuracy in all of the cases. The results also
demonstrate the advantage of feature concatenation over the model-based
approaches that rely on a combination of static anthropometric-based
attributes, and statistical moments that describe dynamic features [4, 19].

##  7 Conclusion

In this work, we introduce $\mathcal{G}$lidar3DJ, the first model-based
approach for gait recognition using flash lidar data. Our model-based approach
is scale and view invariant, which is essential for real-world application. We
address a major limitation of the current state-of-the-art model-based methods
that require high quality Mocap or Kinect data. The noisy, low resolution
flash lidar data used in this study present challenges to the performance of
state-of-the-art skeleton detectors, degrading joint localization and gait
identification accuracy. A rule-based short-memory median filter is presented
that improves the quality of features and gait recognition accuracy. The
proposed median filter has the potential application for predicting missing
values in time series. Furthermore, we introduce a new feature concatenation
criterion to incorporate the dynamic of motion and improve gait recognition
methodology. Experimental results support the effectiveness of
$\mathcal{G}$lidar3DJ for gait recognition despite noisy data with faulty
missing features.

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