# Generalized law of iterated logarithm for the Lorentz gas with infinite
horizon

Péter Bálint and Dalia Terhesiu  Department of Stochastics, Institute of
Mathematics, Budapest University of Technology and Economics, Műegyetem rkp.
3., H-1111, Budapest, Hungary and HUN-REN–BME Stochastics Research Group,
Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111
Budapest, Hungary<EMAIL_ADDRESS>Support of the NKFIH Research
Fund, grants 142169 and 144059, is thankfully acknowledged.Institute of
Mathematics, University of Leiden, Niels Bohrweg 1, 2333 CA Leiden, The
Netherlands<EMAIL_ADDRESS>
###### Abstract

We obtain a generalized law of iterated logarithm for a class of dependent
processes with superdiffusive behaviour. Our results apply in particular to
the Lorentz gas with infinite horizon.

##  1 Introduction and main results

The almost sure invariance principle (ASIP) with rates, also known as the
strong invariance principle (SIP) or Strassen’s invariance principle, is a
much studied problem in probability theory, see [36] for a survey. Let
$S_{n}=\sum_{i=1}^{n}Y_{i}$ denote the sum of a stationary sequence of random
variables taking values in $\mathbb{R}^{d}$, $d\geq 1$; then $S_{n}$ satisfies
an ASIP with rate $n^{\delta}$ for some $\delta\in(0,\frac{1}{2})$ if there
exists some non-degenerate covariance matrix $\sigma^{2}$ such that, possibly
on an enlarged probability space,

$|S_{n}-\sigma\cdot W(n)|=o(n^{\delta})$ (1.1)

almost surely as $n\to\infty$, where $W(n)=\sum_{i=1}^{n}Z_{i}$ denotes
standard Brownian motion at time $n$, or equivalently, the variables $Z_{i}$
are i.i.d. standard Gaussians. It is known that the ASIP implies the law of
iterated logarithms (LIL):

$\limsup_{n\to\infty}\dfrac{|S_{n}|}{\sqrt{n\cdot L(Ln)}}=a\qquad\text{ almost
surely, for some }a\in(0,\infty),$ (1.2)

where $L(t)=\max(1,\log(t))$. It can be assumed that the stationary sequence
arises as $Y_{i}=Y\circ T^{i}$ for some probability preserving transformation
(or dynamical system) $T:\Omega\to\Omega$ and $Y:\Omega\to\mathbb{R}^{d}$. The
problem of the ASIP has been studied for various classes of dynamical systems
and observables – that satisfy the classical central limit theorem (CLT) and
weak invariance principle (WIP) – by several authors, via different methods,
see [23, 24, 25, 21, 22, 10] and references therein. A common feature of these
works is that they assume (along with some decorrelation condition) that the
random variables satisfy $Y\in L^{p}$ for some $p>2$, and prove the ASIP with
some rate that depends on $p$.

We are interested in the generalization of the ASIP and the LIL when the CLT
and WIP holds with a nonstandard norming sequence, such as $\sqrt{n\log n}$.
Important dynamical examples of physical interest, for this particular norming
sequence are Sinaĭ billiards with infinite horizon with the associated flight
function (i.e., the distance in $\mathbb{R}^{d}$ between consecutive
collisions, see [33] and [8]).

A certain generalized form of the ASIP and of the LIL for i.i.d. random
variables taking values in $\mathbb{R}^{d}$ with infinite variance but still
in the domain of attraction of a normal law were obtained in [16, 17, 18, 20].
We will focus on the case of $\mathbb{R}^{d}$ regularly varying random
variables $(Y_{j})_{j\geq 0}$ of index $-2$, but remark that in the case
$d=1$, the results in [18] are not limited to regular variation. For the case
$d=1$, regular variation of index $-2$ means that, as $t\to\infty$,

$\mathbb{P}(Y>t)=(p+o(1))\ell(t)t^{-2},\quad\mathbb{P}(Y<-t)=(q+o(1))\ell(t)t^{-2}$

for some $p,q\geq 0,p+q=1$ and some slowly varying function $\ell$.

To recall the more general definition of regular variation in $\mathbb{R}^{d}$
(see [31]) let ${{\mathbb{S}}^{d-1}}=\\{t\in\mathbb{R}^{d}:|t|=1\\}$ denote
the unit sphere in $\mathbb{R}^{d}$. (Throughout, $|\;|$ denotes the Euclidean
norm.)

###### Definition 1.1

An $\mathbb{R}^{d}$-valued random variable $Y$ is _regularly varying_ with
index $-\alpha<0$ if there exists a Borel probability measure $\sigma$ on
${{\mathbb{S}}^{d-1}}$, such that

$\lim_{t\to\infty}\frac{\mathbb{P}(|Y|>\lambda t,\ Y/|Y|\in
A)}{\mathbb{P}(|Y|>t)}=\lambda^{-\alpha}\sigma(A)$

for all $\lambda>0$ and all Borel sets $A\subset{{\mathbb{S}}^{d-1}}$ with
$\sigma(\partial A)=0$. We say that $Y$ is _nondegenerate_ if
$\int_{{\mathbb{S}}^{d-1}}|u\cdot\theta|^{\alpha}\,d\sigma(\theta)>0$ for all
$u\in{{\mathbb{S}}^{d-1}}$.

Taking $A={{\mathbb{S}}^{d-1}}$, we have that $|Y|$ is a regularly varying
scalar function. Hence there exists a slowly varying function
$\ell:[0,\infty)\to(0,\infty)$ such that
$\mathbb{P}(|Y|>t)=t^{-\alpha}\ell(t)$.

Provided that $(Y_{j})_{j\geq 0}$ is an i.i.d sequence of random variables
defined on a probability space $(\Omega_{0},\mathcal{F}_{0},\mathbb{P}_{0})$
and that $Y_{0}$ is regularly varying of index $-2$ with
$\mathbb{E}(Y_{0})=0$,

$a_{n}^{-1}\sum_{j=0}^{n-1}Y_{j}\implies\mathcal{N}(0,\Sigma_{0}^{2}),\text{
as }n\to\infty,$ (1.3)

where $\implies$ denotes convergence in distribution, $\Sigma_{0}$ is a
positive definite $d\times d$ matrix and $a_{n}$ is given by
$\lim_{n\to\infty}\frac{n\tilde{\ell}(n)}{a_{n}^{2}}=1$, where
$\tilde{\ell}=1+\int_{1}^{1+x}\frac{\ell(u)}{u}du$.

###### Remark 1.2

Given several important billiard applications (infinite horizon Lorentz gas,
Bunimovich stadium, dispersing billiards with cusps), a case of special
significance is when $\ell(t)\equiv C$ for some positive constant $C$, and
thus $\tilde{\ell}(t)\sim C\log t$. We will refer to this as the Lorentz case.

We recall the result in [18, 20] for the sequence $(Y_{j})_{j\geq 0}$ of
independent random variables along with the required assumption.

  * (H0)

Let $(c_{n})_{n\geq 1}$ be a non-decreasing sequence so that

    * (i)

for any ${\varepsilon}>0$, there exists $M_{\varepsilon}\geq 1$ so that
$\frac{c_{n}}{c_{m}}\leq(1+{\varepsilon})\frac{n}{m}$ for all
$M_{\varepsilon}\leq m<n$;

    * (ii)

$\sum_{n}\mathbb{P}_{0}(|Y_{0}|>c_{n})<\infty$.

We note that given $a_{n}$ as in (1.3), $c_{n}/a_{n}\to\infty$.

###### Theorem 1.3

[17, Theorem 2], [18, Theorem 2], [19, Theorem 2.1]. Let $(Y_{j})_{j\geq 0}$
be an i.i.d sequence of $\mathbb{R}^{d}$ random variables satisfying (1.3).
Let $(c_{n})_{n\geq 1}$ be a sequence satisfying (H0).

Then, there exists a probability space
$(\Omega_{1},\mathcal{F}_{1},\mathbb{P}_{1})$ and two sequences of independent
random vectors $(Y_{j}^{*})_{j\geq 0}$, $(Z_{j})_{j\geq 0}$ so that

  * •

$Y_{j}^{*}$ is distributed as $Y_{j}$

  * •

$Z_{j}$ is distributed as $\mathcal{N}(0,I)$

and so that, almost surely, as $n\to\infty$,

$\left|\sum_{j=0}^{n-1}Y_{j}-\Gamma_{n}\,\sum_{j=0}^{n-1}Z_{j}\right|=o(c_{n}),$

where the matrix $\Gamma_{n}$ is given by
$\Gamma_{n}^{2}=cov(Y_{0}\,1_{|Y_{0}|<c_{n}})$.

When compared to the standard ASIP, a specific feature of Theorem 1.3 is the
rescaling of Brownian motion with the truncated covariance matrix
$\Gamma_{n}$, which occurs only for variables in the non-standard domain of
attraction of the normal law. This reflects the fact that in this regime the
limit phenomena are particularly sensitive to the specifics of the tail of the
distribution, which typically makes the treatment quite challenging. Indeed,
for SIP type results for i.i.d. random variables in the domain of a symmetric
stable law of index $\alpha\in(0,2)$ (see [32, Theorems 3.1 and 3.2]) there is
no need for truncation and thus, no need for an analogue of $\Gamma_{n}$, and
the main challenge is to obtain good rates (we refer to [32, Section 3]). Note
also that in this $\alpha<2$ case the approximation is, instead of sums of
independent Gaussians, by sums of independent $\alpha$-stable variables, and
thus there is no natural analogue of the sequence $\sqrt{nL(L(n))}$ of (1.2)
(for the standard CLT case), or the sequence (1.4) below (in our case), see
[32, Theorem 2.1]. Thus, although as in the case of Theorem 1.3, there is
infinite variance in the $\alpha<2$ stable law scenario too, the limit
phenomena are quite different, see also [15] for recent results in the
dynamical context.

As in [20, 18] the main purpose of Theorem 1.3 is the implication of a precise
form of a “generalized law of iterated logarithm (LIL)” for such i.i.d.
processes, which we now recall.

###### Theorem 1.4

[20, Theorem 3], [18, Theorem 1]. Assume the set up of Theorem 1.3. Set

$A(t)=\mathbb{E}(Y_{0}^{2}1_{\\{|Y_{0}|\leq
t\\}}),\,\,\,a=\sup\left\\{\alpha\geq
0:\sum_{n=1}^{\infty}\frac{1}{n}\exp\left(-\frac{\alpha^{2}c_{n}^{2}}{2nA(c_{n})}\right)=\infty\right\\}.$

Then

$\limsup_{n\to\infty}\frac{|\sum_{j=0}^{n-1}Y_{j}|}{c_{n}}=a.$

Note that in Theorem 1.4, the limit $a$ can be $0$, infinite, or a positive
finite number depending on the sequence $c_{n}$, and it is not always easy to
say for which sequences we have $a\in(0,\infty)$, a situation which can be
regarded as a “generalized law of iterated logarithm”. In special cases both
the limit $a$ and the sequence $(c_{n})_{n\geq 1}$ can be described precisely:

###### Corollary 1.5

Consider an i.i.d. sequence of random variables as in Theorems 1.3 and 1.4.
Assume, furthermore, that the density $f(x)$ of the distribution function
$\mathbb{P}(Y_{j}\leq x)$ satisfies $f(x)\sim C\,|x|^{-3}1_{|x|\geq 1}$, for
some $C>0$. Then, according to [20, p. 1622],

If $\displaystyle\ c_{n}=\sqrt{2C\cdot n\cdot L(n)\cdot
LL(n)\cdot(1+LLn\sin^{2}(LLLn))}\text{ for }L(t)=\max(1,\log t)$ (1.4) then
$\displaystyle\limsup_{n\to\infty}\frac{|\sum_{j=0}^{n-1}Y_{j}|}{c_{n}}=1.$
(1.5)

The argument in [20, p. 1622] is given for $C=1$, but it is straightforward to
get the corresponding statement for $C\neq 1$.

In this work we show that Theorem 1.3 and Theorem 1.4 essentially carry over
to dependent random variables (arising from the evolution of dynamical
systems) under some good assumptions. This includes in particular the flight
function in the infinite horizon Lorentz gas, see Remark 1.2. Here
“essentially” means that in Theorem 1.8 below (which is the generalization of
the invariance principle in Theorem 1.3) the sequence $c_{n}$ has to satisfy,
on top of (H0), an additional assumption, see (HL0) below. Yet, this
additional assumption is not much restrictive, in particular it allows the
sequence defined in (1.4).

Besides Einmahl’s results (in particular [19]) on the i.i.d case, two further
important references for our work are [21] and [8]. In [21], Gouëzel proved
the ASIP for a large class of processes with $Y\in L^{p}$ for some $p>2$. In
our argument, we adapt Gouëzel’s strategy, specifically with $p=4$, to our
situation after truncating the variables $Y_{i}$. The choice of the truncation
level is delicate and the method requires very precise bounds on the second
and the fourth moment for partial sums of the appropriately truncated random
variables. Although in [8] Chernov and Dolgopyat obtained moment bounds of
similar character in the infinite horizon Lorentz gas, we improve
significantly upon these bounds, in particular, we manage to remove some
logarithmic factors from the bound on the fourth moment, obtaining this way a
scaling essentially as good as for i.i.d. random variables, as expressed in
condition (H3)(ii) below. Another important feature of our argument is that,
on op of the truncation level $c_{n}$, we introduce an additional, slightly
lower truncation level at $d_{n}$, see Formula (1.8) and condition (HL1) for
details. This is partially inspired by [8] which also uses two truncation
levels, yet, this idea has to be implemented at the level of almost sure
convergence in our case.

Our paper is organized as follows. We state our results in sections 1.1 and
1.2 below, in particular, our abstract result is Theorem 1.8, and its main
application (to Lorentz gas with infinite horizon) is stated in Theorem 1.10.
The proof of Theorem 1.8 is broken down into several Propositions and Lemmas
in sections 3, 4 and 5 – this is essentially an adaptation of Gouëzel’s
argument from [21], which nonetheless exploits the properties of our variables
– expressed in the conditions of Theorem 1.8 – in a subtle way. These
conditions are verified for the infinite horizon Lorentz gas in sections 2 and
6. In particular, the above mentioned improvement on the bound of the fourth
moment of partial truncated sums for the infinite horizon Lorentz gas, an
interesting result on its own, is proved in Section 6, and our condition (HL1)
on the relation of the two truncation levels $c_{n}$ and $d_{n}$ is verified
in Section 6, too. In the Appendix we check that the sequence (1.4) satisfies
the required sumability condition of (HL0), completing this way the proof of
our “generalized law of iterated logarithm” ie. Theorem 1.10(ii).

###  1.1 Main results

We first present an abstract version of the result, which will make the
application to the Lorentz gas with infinite horizon easier. We phrase the
following assumptions in terms of dynamical systems and observables associated
with them. We consider an invariant probability measure preserving
transformation $(\Omega,T,\mu)$.

Let $v:\Omega\to\mathbb{R}^{d}$ and set $v_{j}=v\circ T^{j}$. Throughout, we
assume

  * (H1)

$\mathbb{E}_{\mu}(v)=0$, $v$ is regularly varying with index $-2$, let $\ell$
be the involved slowly varying function and require that

$a_{n}^{-1}\sum_{j=0}^{n-1}v_{j}\implies\mathcal{N}(0,\Sigma^{2}),\text{ as
}n\to\infty,$ (1.6)

where $\Sigma$ is a positive definite $d\times d$ matrix and where $a_{n}$ is
so that $\lim_{n\to\infty}\frac{n\ell^{*}(n)}{a_{n}^{2}}=1$, where the slowly
varying function $\ell^{*}$ satisfies $\ell^{*}(t)=C_{0}\tilde{\ell}(t)$, for
some constant $C_{0}>0$. (Recall that
$\tilde{\ell}=1+\int_{1}^{1+x}\frac{\ell(u)}{u}du$.)

###### Remark 1.6

In the Lorentz case billiard examples, the functions $\tilde{\ell}$ and
$\ell^{*}$ are as follows. For the infinite horizon Lorentz gas, we have
$\ell^{*}(t)=\tilde{\ell}(t)=C\log t$, with some $C>0$, that is, we have
$C_{0}=1$ and observe the same limit law as for iid sequences with the same
tail behaviour (see [4], [8]). For Bunimovich stadia, we have
$\ell^{*}(t)=C_{0}\tilde{\ell}(t)=C_{0}C\log t$ with $C_{0}=\frac{4+3\log
3}{4-3\log 3}$. Thus, the relevant slowly varying function differs from that
of the iid case by a nontrivial constant factor, an effect which is due to
short range correlations between consecutive large values (see [5]).

We are interested in the almost sure behaviour of $\sum_{j=0}^{n-1}v_{j}$, in
the sense of an analogue of Theorem 1.3, and with this obtain an analogue of
Theorem 1.4.

The remainder of our assumptions will be phrased in terms of the following
truncated version of $v$. For $R\in\mathbb{R}_{+}\cup\\{\infty\\}$ (and for
any $l\geq 0$, consider also $R(=R_{l})\in\mathbb{R}_{+}\cup\\{\infty\\}$)
define

$\displaystyle W^{R}=v\cdot\mathbf{1}_{|v|\leq R}\qquad\text{and
accordingly}\qquad W_{l}^{R}=W^{R}\circ
T^{l}=v_{l}\cdot\mathbf{1}_{|v_{l}|\leq R}.$ (1.7)

To ensure ‘enough independence’, we assume [21, Assumption(H)] at the level of
characteristic functions of $n+m$ blocks derived from the truncated variables
(1.7), with gaps of size $k$ in between the two blocks. More precisely:

  * (H2)

There exist ${\varepsilon}_{0}>0$, $C,c>0$ so that for any $n,m>0$,
$b_{1}<b_{2}<\ldots<b_{r+m+1}$, for any $k>0$ and for all
$t_{1},t_{2},\ldots,t_{n+m}\in\mathbb{R}^{d}$ with
$|t_{j}|\leq{\varepsilon}_{0}$, and for arbitrary truncation levels
$R_{1},R_{2},\ldots,R_{n+m}\in\mathbb{R}^{+}$

$\displaystyle\Big{|}\mathbb{E}_{\mu}$
$\displaystyle\left(e^{i\sum_{j=1}^{n}t_{j}(\sum_{l=b_{j}}^{b_{j+1}-1}W_{l}^{R_{j}})+i\sum_{j=n+1}^{n+m}t_{j}(\sum_{l=b_{j+k}}^{b_{j+1}+k-1}W_{l}^{R_{j}})}\right)$
$\displaystyle-\mathbb{E}_{\mu}\left(e^{i\sum_{j=1}^{n}t_{j}(\sum_{l=b_{j}}^{b_{j+1}-1}W_{l}^{R_{j}})}\right)\cdot\mathbb{E}_{\mu}\left(e^{i\sum_{j=r+1}^{r+m}t_{j}(\sum_{l=b_{j}+k}^{b_{j+1}+k-1}W_{l}^{R_{j}})}\right)\Big{|}$
$\displaystyle\leq C(1+\max_{0\leq j\leq
m+n}|b_{j+1}-b_{j}|)^{C(m+n)}e^{-ck}.$

In (H2) and throughout, we write $e^{itx}$ instead of $e^{i\langle
t,x\rangle}$. Note, furthermore, that the variable $t_{j}$ in the
characteristic function corresponds to the sum of the variables
$W_{l}^{R_{j}}$ for $b_{j}\leq l<b_{j+1}$, which are all truncated at the same
level $R_{j}$.

We further assume some control of moments of sums of truncated observables
along with a precise form of the covariance matrix.

  * (H3)

Let $W_{j}^{R}$ be as in (1.7) with $R_{j}=R$ for all $j$. We require that

    * (i)

$\mathbb{E}_{\mu}(W_{j}^{R})=0$, for any $R$.

For both (ii) and (iii), consider $m\in\mathbb{N}$ (the number of terms) and
$0<R(=R(m))$ (a truncation level), with $R\to\infty$ and $m\to\infty$, such
that $m=o(R^{2})$.

    * (ii)

Additionally, restricting to $m=o(R^{2-r_{1}})$ and $R=o(m^{r_{2}})$ for some
$r_{1}>0$ and $r_{2}>0$, we assume that111We will always apply (H3)(ii) in
situations when $m=2^{\alpha n}$ and $R=c_{2^{n}}$, where $n\to\infty$,
$\alpha\in(0,1)$ and the sequence $c_{k}$ (in accordance with (H0)) is equal
to $\sqrt{k}$ up to slowly varying factors. Hence $c_{2^{n}}$ is equal to
$2^{n/2}$ up to factors that grow polynomially in $n$, and the requirements on
the relation of $m$ and $R$ are satisfied.  there exists $C>0$ so that
$\int\left|\sum_{j=0}^{m-1}W_{j}^{R}\right|^{4}\,d\mu\leq C\,m\,R^{2}.$

    * (iii)

Assume there exists $C>0$ so that
$cov(\sum_{j=0}^{m-1}W_{j}^{R})=Cm\ell^{*}(R)\,\Sigma^{2}(1+o(1))$, where
$\Sigma$ and $\ell^{*}$ are as in (1.6).

Assumption (H3)(i) is for mere simplicity of the exposition. Else, inside the
proofs, we could subtract the relevant mean.

We obtain a non iid version of Theorem 1.3 for a slightly more restrictive
class of sequences $c_{n}$ than the one in (H0). For concreteness, we
specialize to the Lorentz case (see Remark 1.2), which in particular implies
$\ell^{*}(R)\asymp\tilde{\ell}(R)\asymp L(R)$ (see conditions (H1) and (H3)
above). Here and throughout, $L(t)=\max(1,\log t)$. Concerning the class of
sequences $c_{n}$, we make the following assumptions.

  * (HL0)

$c_{n}=\sqrt{n\cdot\ell^{*}(n)\cdot\ell_{1}(n)}\asymp\sqrt{n\cdot
L(n)\cdot\ell_{1}(n)}$ for some slowly varying function $\ell_{1}(n)$, such
that the following conditions hold.

    * (i)

There exists some $b>1$ such that $\ell_{1}(n)\ll(LL(n))^{b}$ (note this
assumption does not weaken our results as it puts an upper bound on $c_{n}$)

    * (ii)

$c_{n}$ satisfies (H0). As we assume Lorentz case, and $c_{n}$ is of the above
form, this is equivalent to
$\sum_{n}c_{n}^{-2}=\sum_{n}(nL(n)\ell_{1}(n))^{-1}<\infty$, in accordance
with the previous item.

    * (iii)

Our additional sumability condition is formulated in terms of $c_{2^{n}}$. It
is worth noting that $\ell^{*}(2^{n})\asymp n$ while
$L(n)\ll\ell_{1}(2^{n})\ll(L(n))^{b}$, with $b$ as in the item (i) above. We
assume $\sum_{n}\frac{2^{n}LL(n)}{(c_{2^{n}})^{2}}<\infty$, or equivalently
$\sum_{n}\frac{LL(n)}{n\ell_{1}(2^{n})}<\infty$.

###### Remark 1.7

The sequence $c_{n}$ defined in (1.4) satisfies (HL0): while (i) is immediate
and (ii) is proved in [20], item (iii) requires a justification: see Appendix
A.

We make a further assumption that allows us to simplify the exposition. To
formulate it, we introduce an additional truncation level, lower than $c_{n}$
(which is assumed to have the form specified in (HL0)). More precisely, let

$\displaystyle d_{n}=\sqrt{nL(n)\ell_{1}(n)^{-99}}.$ (1.8)

  * (HL1)

Let $n\geq 1$ and set ${\widetilde{W}}=v\cdot\mathbf{1}_{d_{2^{n}}\leq|v|\leq
c_{2^{n}}}$. Fix $r\geq 1$ and define
${\widetilde{W}}_{l}={\widetilde{W}}\circ T^{l+r}$ for $l\geq 1$. Fix
$K\in\mathbb{N}$ with $K<(d_{2^{n}})^{2}$. For $k=1,\dots,K$ and
${\varepsilon}>0$, define

$\mathcal{S}_{k}=\sum_{l=0}^{k}{\widetilde{W}}_{l};\qquad
B_{k,{\varepsilon}}=\\{x\in
M\,:\,|\mathcal{S}_{k}|\geq{\varepsilon}c_{2^{n}}\\}$

and let
$N_{k,{\varepsilon}}=B_{k,{\varepsilon}}\setminus(\cup_{k^{\prime}=0}^{k-1}B_{k^{\prime},{\varepsilon}}).$

We assume that there exists $C_{1},C_{2}>0$ so that

    * (i)

$\mathbb{E}_{\mu}\left(\mathcal{S}_{K}^{2}\right)\leq C_{1}\cdot K\cdot
L\left(\frac{c_{n}}{d_{n}}\right)\ll K\cdot LL(n)$.

    * (ii)

$\sum\limits_{k=1}^{K}\mu(N_{k,{\varepsilon}}\setminus
B_{K,{\varepsilon}/2})\leq C_{2}\frac{K\cdot LL(n)}{(c_{2^{n}})^{2}}.$

The first result below is an analogue of Theorem 1.3 in the non independent
set up of Subsection 1.1.

###### Theorem 1.8

Assume (H1)–(H3). Let $(c_{n})_{n\geq 1}$ be a sequence satisfying (HL0).
Suppose that (HL1) holds. Then, there exists a probability space
$(\Omega^{*},\mathcal{F}^{*},\mathbb{P}^{*})$ and two sequences of random
vectors $(v_{j}^{*})_{j\geq 0}$, $(Z_{j})_{j\geq 0}$ so that

  * •

$v_{j}^{*}$ is distributed as $v_{j}$;

  * •

the vectors $Z_{j},j\geq 0$ are independent and distributed as
$\mathcal{N}(0,I)$;

such that, almost surely, as $n\to\infty$,

$\left|\sum_{j=0}^{n-1}v_{j}^{*}-\Gamma_{n}\,\sum_{j=0}^{n-1}Z_{j}\right|=o(c_{n}),$

where the matrix $\Gamma_{n}$ is given by
$\Gamma_{n}^{2}=\ell^{*}(c_{n})\,\Sigma^{2}$.
Note that $\sum_{j=0}^{n-1}Z_{j}=W(n)$, where $W(n)$ is standard Brownian
motion in dimension $d$, at time $n$.

The proof of Theorem 1.8 is provided in Section 3. It relies on a combination
of arguments/results in [21] under (H2) and (H3) together with
arguments/results in [18] (some established already in previous works by the
same author).

With exactly the same proof as in [19], Theorem 1.8 gives the Generalized LIL
(the analogue of Theorem 1.4):

###### Theorem 1.9

Assume (H1)–(H3). Let $(c_{n})_{n\geq 1}$ be a sequence satisfying (HL0) and
suppose that (HL1) holds. Set

$A(t)=\ell^{*}(t)\|\Sigma\|^{2},\,\,\,a=\sup\left\\{\alpha\geq
0:\sum_{n=1}^{\infty}\frac{1}{n}\exp\left(-\frac{\alpha^{2}c_{n}^{2}}{2nA(c_{n})}\right)=\infty\right\\}.$

Then

$\limsup_{n\to\infty}\frac{\left|\sum_{j=0}^{n=1}v_{j}\right|}{c_{n}}=a,$

While the proof of Theorem 1.3 in [19] heavily relies on the i.i.d. structure,
the proof of Theorem 1.4 given Theorem 1.3 does not require independence. In
the same way, Theorem 1.8 gives Theorem 1.9. Indeed, given Theorem 1.8,
Theorem 1.9 reduces to

$\limsup_{n\to\infty}\frac{\left|\Gamma_{n}\cdot W(n)\right|}{c_{n}}=a,$

which is proved in [19, proof of Corollary 2.4] in [19, section 4.2.1] and
[19, section 4.2.2], cf. in particular [19, Formulas (4.4)-(4.5)] and [19,
Formulas (4.8)-(4.9)].

Thus, our only task is to prove Theorem 1.8. Theorem 1.8 is the main new
result of this paper and its proof is provided in Section 3.

###  1.2 Application to the Lorentz gas map with infinite horizon

In this section, we give a brief description of the infinite horizon Lorentz
gas. For further details, see the references, in particular [8], [27], [33].
The $\mathbb{Z}^{d}$-periodic Lorentz gas describes the evolution of a point
particle moving in the $\mathbb{Z}^{d}$-periodic domain ${\widetilde{D}}$
contained either in the plane $\mathbb{R}^{2}$ (if $d=2$) or in the tube
$\mathbb{R}\times\mathbb{T}$ (if $d=1$). The collisions are elastic (equality
of pre-collision and post-collision angles). The Lorentz gas map
$\hat{T}:\hat{M}\to\hat{M}$ is the collision map on the two-dimensional phase
space (position in $\partial{\widetilde{D}}$ and unit velocity) given by
$\hat{M}=\partial{\widetilde{D}}\times(-\pi/2,\pi/2)$.

We assume that ${\widetilde{D}}$ is the lifted domain (by the canonical
projection from $\mathbb{R}^{2}$ (if $d=2$) or from
$\mathbb{R}\times\mathbb{T}$ (if $d=1$) onto $\mathbb{T}^{2}$ ) of
$D=\mathbb{T}^{2}\setminus Q$, where $Q\subset\mathbb{T}^{2}$ is a finite
union of convex obstacles (scatterers) with $C^{3}$ boundaries and
nonvanishing curvature, and pairwise disjoint boundaries. Here
$\mathbb{T}^{2}$ denotes the two dimensional torus. The Sinai (dispersing)
billiard map $T:M\to M$ corresponding to the associated collision map is
obtained from $\hat{T}:\hat{M}\rightarrow\hat{M}$ by quotienting. We denote by
$\mu$ the unique ergodic $T$-invariant smooth probability measure on $M$.

The Lorentz gas map $\hat{T}:\hat{M}\to\hat{M}$ can be viewed as a
$\mathbb{Z}^{d}$-cover, $d=1,2$, of the dispersing billiard $(T,M,\mu)$ by the
cell-change function $\kappa:M\to\mathbb{Z}^{d}$. We assume that $\kappa$ is
unbounded, so that we are in the case of _infinite_ horizon. Under certain
nondegeneracy conditions, [33] proved that $\kappa$ satisfies a central limit
theorem and local limit theorem with positive-definite covariance matrix
$\Sigma\in\mathbb{R}^{d\times d}$ and nonstandard normalisation
$a_{n}=\sqrt{n\log n}$. These nondegeneracy conditions are as follows: a) in
the case $d=2$, one requires that there exist at least two nonparallel
collisionless trajectories in the interior of ${\widetilde{D}}$; b) For $d=1$,
one requires that there exists a collisionless trajectory not orthogonal to
the direction of the $\mathbb{Z}$-cover, which is equivalent to our assumption
that $\kappa:M\to\mathbb{Z}$ is unbounded.

We recall that a different proof (via standard pairs) of the above mentioned
non standard CLT has been provided in [8] along with WIP. For other recent
results on the Lorentz gas map and flow with infinite horizon, see [3, 27, 26,
28].

The flight function $\Phi:M\to\mathbb{R}^{d}$ is given by the difference in
$\mathbb{R}^{d}$ between consecutive collision points. Moreover, it is known
that $\Phi$ can be written as $\Phi=\kappa+\chi\circ T-\chi$, where $\chi$ is
bounded.

A further property of the cell function $\kappa$, which we shall exploit is
precise information of the corresponding density function. More precisely, we
recall from [33, Proposition 6] that for some $C>0$,

$\displaystyle\mu(\kappa=N)\sim C|N|^{-3},$ (1.9)

where the event $\kappa=N$ indicates the next collision with a scatterer at
distance $|N|$ from the current position.

The following result gives the generalized LIL for $\kappa$ and $\Phi$.

###### Theorem 1.10

Consider the Sinaĭ billiard map $(M,T,\mu)$ with infinite horizon and either
the cell-change function $\kappa:M\to\mathbb{Z}^{d}$ or the flight function
$\Phi:M\to\mathbb{R}^{d}$.

Let $c_{n}$ be defined as in (1.4) (with exactly the same constant $C$ as in
(1.9)). Then

  * (i)

$\sum_{j=0}^{n-1}\kappa\circ T^{j}$ and $\sum_{j=0}^{n-1}\Phi\circ T^{j}$
satisfy the conclusion of Theorem 1.8.

  * (ii)

$\limsup_{n\to\infty}\frac{\left|\sum_{j=0}^{n=1}\kappa\circ
T^{j}\right|}{c_{n}}=1,$ and a similar statement holds for $\Phi$.

#### Notations.

Throughout the proofs, we use “big O” and $\ll$ notation interchangeably,
writing $b_{n}=O(c_{n})$ or $b_{n}\ll c_{n}$ if there are constants $C>0$,
$n_{0}\geq 1$ such that $b_{n}\leq Cc_{n}$ for all $n\geq n_{0}$. Furthermore,
$b_{n}\asymp c_{n}$ means that $b_{n}\ll c_{n}$ and $c_{n}\ll b_{n}$ hold
simultaneously. This is not to be confused with the stronger property
$b_{n}\sim c_{n}$ which means that $b_{n}=c_{n}\cdot(1+o(1))$.

##  2 Proof of Theorem 1.10

In this section we derive Theorem 1.10 from Theorems 1.8 and 1.9 with $c_{n}$
be defined as in (1.4).

We already know that such $c_{n}$ satisfies (HL0): see Remark 1.7. Thus, to
obtain (i) of Theorem 1.10, we need to verify that assumptions (H1)–(H3) hold.
In this scenario, the probability measure preserving map $(\Omega,T,\mu)$ is
replaced by $(M,T,\mu)$ and the truncated observable $W_{l}^{R_{l}}$ as in
(1.7) is defined with $\kappa$ (respectively, $\Phi$) instead of $v$. It is
known that $\kappa$ and $\Phi$ satisfy (H1), that is they are regularly
varying observables with index $-2$ and as shown in [33] (via the Young tower
constructed in [7, 34]), $\kappa$ and $\Phi$ satisfy (1.6) with
$a_{n}=\sqrt{n\log n}$. It is known that $\mu(\kappa)=0$ and (H3)(i) holds by
symmetry. (H3)(iii) is known to hold with $C=2$ and $\ell^{*}=\log$ (see [8]).
(H3)(ii) and (HL1) are verified in Section 6. (H2) is verified below.

Item (ii) of Theorem 1.10 follows from Theorem 1.9. The precise description of
$c_{n}$ and limit $a$ is possible because using (1.9) we obtain that
$A(t)=CL(t)\|\Sigma\|^{2}$; here $C$ is the same as in (1.9). Given $c_{n}$ as
in (1.4) (with exactly this positive $C$), it is easy to see that $a=1$.

Verifying (H2). We first recall a fact established in [21], which shows that
(H2) holds under very mild assumptions on the transfer operator.

Let $T:\Omega\to\Omega$ be a non-singular transformation w.r.t. $m=Leb$ and
let $L:L^{1}(m)\to L^{1}(m)$ be the transfer operator given by
$\int_{\Omega}Lf\,g\,dm=\int_{\Omega}f\,g\circ T\,dm$, for $f\in L^{1}(m)$ and
$g$ bounded and measurable.

Now we define perturbations of the transfer operator $L$ associated to our
observable $v:\Omega\to\mathbb{R}^{d}$ (satisfying (H1)) and its truncations
$W^{R}$ (as defined in Formula (1.7)). Given a perturbation parameter
$t\in\mathbb{R}^{d}$ and a truncation level
$R\in\mathbb{R}^{+}\cup\\{\infty\\}$, let

$L_{t,R}f=L(e^{itW^{R}}f).$

For $t=0$ and arbitrary $R$, we apparently have $L_{0,R}=L_{0}=L$.

The main difference between our (H2) and [21, Condition (H)] is that (H2)
involves several truncation levels, but this does not create a problem in
checking that (H2) holds under mild conditions on the perturbed operators
$L_{t,R}$. Indeed, we can proceed essentially as in [21, Section 2].

To start, let us slightly generalize the following notion from [21, Formula
(2.2)].

###### Definition 2.1

Recall the setting and the notation of section 1.1, in particular, let $\mu$
be an invariant measure for $T$. Suppose that there exists a Banach space
$\mathcal{B}$ (of functions or distributions), and moreover, some
$u_{0}\in\mathcal{B}$ and $\xi_{0}\in\mathcal{B}^{\prime}$ (the dual of
$\mathcal{B}$) so that for any $t_{l}\in\mathbb{R}^{d}$,
$|t_{l}|\leq{\varepsilon}_{0}$ with $0\leq l\leq n-1$, and any
$R_{l}\in\mathbb{R}^{+}\cup\\{\infty\\}$, again with $0\leq l\leq n-1$, we
have

$\displaystyle\mathbb{E}_{\mu}(e^{i\sum_{l=0}^{n-1}t_{l}W_{l}^{R_{l}}})=\langle\xi_{0},L_{t_{n-1},R_{n-1}}L_{t_{n-2},R_{n-2}}\ldots
L_{t_{1},R_{1}}L_{t_{0},R_{0}}u_{0}\rangle.$ (2.1)

In this case we say that the random variables $v\circ T^{l}$ along with their
truncations $W_{l}^{R_{l}}$ are coded by
$(\mathcal{B},L_{t,R},u_{0},\xi_{0})$.

###### Lemma 2.2

[21, Section 2] Suppose that the random variables $v\circ T^{l}$ are coded by
some $(\mathcal{B},L_{t,R},u_{0},\xi_{0})$ in the sense above. Assume,
furthermore, that the family of operators $L$ and $L_{t,R}$ satisfy the
properties (I1) and (I2) below.

  1. (I1)

There exists $C>0$ and $\delta_{0}<1$ such that $L(=L_{0})$ as an operator
acting on $\mathcal{B}$ satisfies $L=\Pi+Q$ where $\Pi$ is a one dimensional
projection, while $\Pi Q=Q\Pi$ and for any $n\geq 1$
$\|Q^{n}\|_{\mathcal{B}\to\mathcal{B}}\leq C\delta_{0}^{n}$.

  2. (I2)

There exists $C^{\prime}>0$ and ${\varepsilon}_{0}>0$ such that for all
$|t|\leq{\varepsilon}_{0}$, all $R\in\mathbb{R}^{+}\cup\\{\infty\\}$, and all
$n\in\mathbb{N}$ we have $\|L_{t,R}^{n}\|_{\mathcal{B}\to\mathcal{B}}\leq
C^{\prime}$.

Then (H2) holds.
Moreover, instead (I2) it suffices to check the following (strong) continuity
condition.

  1. (I2c)

The family $L_{t,R}$ depends, in the operator norm of the space $\mathcal{B}$,
continuously on the parameter $t\in\mathbb{R}^{d}$ at $t=0$, uniformly in the
truncation level $R$. That is $\|L_{t,R}-L\|_{\mathcal{B}\to\mathcal{B}}\to 0$
as $t\to 0$, uniformly in the truncation level $R$.

###### Proof.

Let us note first that conditions (I1) and (I2c) together imply condition
(I2). This is proved for one parameter families $L_{t}$ in [21, Proposition
2.3]. To see that this extends to two parameter families $L_{t,R}$ if the
continuity is uniform in $R$, recall that the essence of the argument is that
under (I1) and (I2c), there exists some decomposition
$L_{t_{R}}=\lambda_{t,R}\Pi_{t,R}+Q_{t,R}$ such that $\|Q_{t,R}^{n}\|\leq
C\delta_{0}^{n}$ for small enough $t$, uniformly in $n$ and $R$, $\Pi_{t,R}$
is a one dimensional projection that limits to $\Pi$ as $t\to 0$, and thus
$|\lambda_{t,R}|\leq 1$ for small $t$, again uniformly in $R$.

Now let us proceed to the verification of (H2) under (I1) and (I2). To
simplify the exposition, we introduce the following shorthand notations.
Keeping in mind that for any fixed $t_{j}$, ($j=1,\dots,n+m$), in (H2) we
consider the sum of the random variables $W_{l}^{R_{j}}$, with $b_{j}\leq
l<b_{j+1}$, all of which are truncated at the same level $R_{j}$, let

$L_{j}=L_{t_{j},R_{j}};\qquad\text{i.e.}\qquad
L_{j}f=L(e^{it_{j}W^{R_{j}}}f)\qquad\text{and}\quad e_{j}=b_{j+1}-b_{j}.$

We record the following immediate consequence of (I2):

$\|L_{j}^{e_{j}}\|_{\mathcal{B}\to\mathcal{B}}\leq
C^{\prime},\quad\text{uniformly in }e_{j}.$ (2.2)

As the characteristic functions are coded by our perturbed transfer operators
in the sense of Definition 2.1, using the above notations, the left hand side
of (H2) can be written as

$\displaystyle\mathbb{E}_{\mu}\left(e^{i\sum_{j=1}^{n}t_{j}\sum_{l=b_{j}}^{b_{j+1}-1}W_{l}^{R_{j}}+i\sum_{j=n+1}^{n+m}t_{j}\sum_{l=b_{j}+k}^{b_{j+1}+k-1}W_{l}^{R_{j}}}\right)$
$\displaystyle=\left\langle\xi_{0}\ ,\ L_{n+m}^{e_{n+m}}\cdots
L_{n+1}^{e_{n+1}}L^{k}L_{n}^{e_{n}}\cdots L_{1}^{e_{1}}u_{0}\right\rangle$
$\displaystyle=\left\langle\xi_{0}\ ,\ L_{n+m}^{e_{n+m}}\cdots
L_{n+1}^{e_{n+1}}(L^{k}-\Pi)L_{n}^{e_{n}}\cdots
L_{1}^{e_{1}}u_{0}\right\rangle$ $\displaystyle+\left\langle\xi_{0}\ ,\
L_{n+m}^{e_{n+m}}\cdots L_{n+1}^{e_{n+1}}(\Pi)L_{n}^{e_{n}}\cdots
L_{1}^{e_{1}}u_{0}\right\rangle.$

This together with the bounds in (I1) (in particular
$\|Q^{n}\|_{\mathcal{B}\to\mathcal{B}}\leq C\delta_{0}^{n}$) and (2.2), allows
us to proceed exactly the same way as in [21, Step 2, (H) holds, Subsection
2.5] to conclude that (H2) holds. ∎

Given Lemma 2.2, we just need to recall previous results on the Sinaĭ billiard
map $(M,T,\mu)$ with infinite horizon and the cell-change function
$\kappa:M\to\mathbb{Z}^{d}$, where (I1) and (I2c) were shown to hold in the
anisotropic Banach spaces constructed for the billiard map in [12, 13, 14],
and further studied in [3]. We recall that the probability measure preserving
map $(\Omega,T,\mu)$ is replaced by $(M,T,\mu)$ and the truncated observable
$W_{l}^{R_{l}}$ is defined with $\kappa$ instead of $v$.

Condition (I1) in Lemma 2.2 is stated for example in [12, Theorem 2.5], [13,
Theorem 2.2], or [14, Theorem 2.4]. It is recalled also in [3, Section 5.2].

Condition (I2c) in Lemma 2.2 is verified in a stronger form in [3, Section
5.1]. Literally, [3] considers only the untruncated case $R=\infty$, in
particular, [3, Formula (29)] implies that there exists some exponent $\nu>0$
and a constant $C>0$ such that

$\|L_{t,\infty}-L\|_{\mathcal{B}\to\mathcal{B}}\leq C|t|^{\nu}.$

In fact, [3, Formula (29)] also investigates how the modulus of continuity
depends on the size of the scatterers, but this can be taken into account as a
uniform constant in the present case. To see that the estimate extends to
finite truncation levels $R<\infty$ with the same $\nu$ and $C$, note that the
argument in [3, Section 5.1] relies on the growth lemma type estimates [3,
Formula (19)] and [3, Formula (25)], along with the bound
$\left|(e^{it\kappa}-1)|_{W_{i}}\right|\ll|t|^{\nu}|\kappa|^{\nu}|_{W_{i}}$,
where $W_{i}$ is some unstable curve. Now the effect of the truncation is that
for unstable curves $W_{i}$ on which $|\kappa|$ exceeds $R$ the difference
$e^{it\kappa^{R}}-1$ simply vanishes, thus these curves make no contribution,
and the total sum is definitely less than in the untruncated case $R=\infty$.

##  3 Proof of Theorem 1.8

The crucial ingredient for the proof of Theorem 1.8 is Proposition 3.1. To
state it, let us introduce the following notation:

$\text{Given }j\geq 1,\text{ let
}{\underline{j}}=2^{\lfloor\log_{2}j\rfloor},$ (3.1)

that is, ${\underline{j}}$ denotes the smallest power of $2$ that does not
exceed $j$.

###### Proposition 3.1

Assume (H1)–(H3). Let $(c_{n})_{n\geq 1}$ be defined as in (HL0). Given $j\geq
1$, let ${\underline{j}}$ be as in (3.1), and let
$W_{j}^{c_{{\underline{j}}}}$ be as in (1.7) with $R=c_{{\underline{j}}}$.

There exists a sequence of independent Gaussian random vectors $Z_{j}$
distributed as $\mathcal{N}(0,I)$ so that almost surely, as $n\to\infty$,

$\left|\sum_{j=1}^{n}(W_{j}^{c_{{\underline{j}}}}-\sigma_{j}Z_{j})\right|=o(c_{n}),$

where $\sigma_{j}$ is a positive definite symmetric matrix so that
$\sigma_{j}^{2}=\ell^{*}(c_{{\underline{j}}})\,\Sigma^{2}(1+o(1))$(with
$\Sigma$ and $\ell^{*}$ as in (H3)(iii)).

The proof of Proposition 3.1 is provided in Section 4. The remainder of the
proof of Theorem 1.8 comes down to recalling a few results from [18], which do
not require any form of independence.

###### Lemma 3.2

[18, Proof of Theorem 1, Step 2] [19, Item (iii) of Section 3.2] Let
$\Gamma_{n}=\ell^{*}(c_{n})\Sigma^{2}$. Let $Z_{j}$ the independent Gaussians
defined in Proposition 3.1. Then, almost surely, as $n\to\infty$,

$\sum_{j=1}^{n}(\Gamma_{j}-\sigma_{j})Z_{j}=o(c_{n}).$

We can now complete

Proof of Theorem 1.8 By Proposition 3.1 and Lemma 3.2,
$\sum_{j=1}^{n}W_{j}^{c_{{\underline{j}}}}-\sum_{j=1}^{n}\Gamma_{j}Z_{j}=o(c_{n}).$
By (HL0)(iii),

$\sum_{j=1}^{\infty}\mathbb{P}(v_{j}\neq
W_{j}^{c_{{\underline{j}}}})\ll\sum_{j=1}^{\infty}c_{{\underline{j}}}^{2}=\sum_{n=1}^{\infty}\frac{2^{n}}{c_{2^{n}}^{2}}<\infty,$

and thus, by the Borel-Cantelli lemma,
$\sum_{j=1}^{n}v_{j}-\sum_{j=1}^{n}W_{j}^{c_{{\underline{j}}}}=o(c_{n})$.
Hence,

$\displaystyle\sum_{j=1}^{n}v_{j}-\sum_{j=1}^{n}\Gamma_{j}Z_{j}=o(c_{n}).$
(3.2)

By [19, Equation (3.16)] (see also [18, Proof of Theorem 1, Steps 3 and 4]),
$\max_{1\leq k\leq
n}\left|\sum_{j=1}^{k}(\Gamma_{n}-\Gamma_{j})Z_{j}\right|=o(c_{n}).$ This
together with (3.2) implies that

$\max_{1\leq k\leq
n}\left|\sum_{j=1}^{k}v_{j}-\Gamma_{n}\sum_{j=1}^{k}Z_{j}\right|=o(c_{n}).$

∎

##  4 Proof of Proposition 3.1

Proceeding as in [21, Section 5], we divide $\mathbb{N}$ into intervals
$[2^{n},2^{n+1})$ and further divide each such interval into intervals of
‘good’ length with gaps between them.

Fix $\beta\in(0,1)$ and ${\varepsilon}_{1}\in(0,1-\beta)$. Decompose each
interval $[2^{n},2^{(n+1)})$ into a union of $F(n)=2^{\lfloor\beta n\rfloor}$
intervals $I_{n,j},0\leq j<F(n)$ of equal length and place $F(n)$ gaps
$J_{n,j},0\leq j<F(n)$ between them. We choose the lengths of the gaps in the
same way as in [21, Section 5]. In particular, for $1\leq j\leq F(n)$, write
$j=\sum_{k=0}^{2^{\lfloor n\beta\rfloor}}\alpha_{k}(j)2^{k}$,
$\alpha_{k}(j)\in\\{0,1\\}$ and let $r$ be the smallest number so that
$\alpha_{r}(j)\neq 0$. In this notation,

$\displaystyle|J_{n,j}|=2^{\lfloor{\varepsilon}_{1}n\rfloor}2^{r}\text{ for
}j\neq 0\text{ and
}|J_{n,0}|=2^{\lfloor{\varepsilon}_{1}n\rfloor}2^{\lfloor\beta n\rfloor}.$
(4.1)

As such, the lengths of the gaps add up to
$2^{\lfloor{\varepsilon}_{1}n\rfloor}2^{\lfloor\beta n\rfloor-1}(\lfloor\beta
n\rfloor+2)$ and one can take

$\displaystyle|I_{n,j}|=2^{n-[n\beta]}-(\lfloor\beta
n\rfloor+2)2^{\lfloor{\varepsilon}_{1}n\rfloor-1}.$ (4.2)

The first proposition below uses the gaps (via the fast decorrelation
expressed in (H2)) to show the sums $\sum_{l\in I_{n,j}}W_{l}^{c_{2^{n}}}$ can
be regarded independent for distinct intervals $I_{(n,j)}$ (at the price of an
almost surely $o(c_{n})$ error). It is analogous to [21, Proposition 5.1]
under assumption (H2). For its statement we need the lexicographical order
$\prec$ (as used in [21, Section 5]): we write
$(n^{\prime},j^{\prime})\prec(n,j)$ if the interval
$I_{n^{\prime},j^{\prime}}$ is to the left of $I_{n,j}$ for $0\leq
j^{\prime}<F(n^{\prime})$ and $0\leq j<F(n)$.

###### Lemma 4.1

Assume (H1)–(H3). Let $X_{n,j}=\sum_{l\in I_{n,j}}W_{l}^{c_{2^{n}}}$.

There exists a coupling between $(X_{n,j})_{n\in\mathbb{N},\,0\leq j<F(n)}$
and a sequence of independent random variables
$(\hat{X}_{n,j})_{n\in\mathbb{N},\,0\leq j<F(n)}$ with $\hat{X}_{n,j}$
distributed like $X_{n,j}$, such that, almost surely as $(n,j)\to\infty$,

$\left|\sum_{(n^{\prime},j^{\prime})\prec(n,j)}(X_{n^{\prime},j^{\prime}}-\hat{X}_{n^{\prime},j^{\prime}})\right|=o\left(c_{2^{n}}\right).$

The proof of this lemma is provided in Subsection 5.1.

The next lemma is an analogue of [21, Section 5.2] (coupling with Gaussians)
under the current assumptions.

###### Lemma 4.2

Assume (H1)–(H3), (HL0) and (HL1). Let
$(\hat{X}_{n,j})_{n\in\mathbb{N},\,0\leq j<F(n)}$ be the independent random
variables obtained in Lemma 4.1.

There exits a coupling between $\hat{X}_{n,j}$ and a sequence of independent
Gaussian random variables ${\widetilde{Z}}_{n,j}$ with covariance matrices

$cov({\widetilde{Z}}_{n,j})=cov(\hat{X}_{n,j})=\ell^{*}(c_{2^{n}})\cdot|I_{n,j}|\cdot\Sigma^{2}(1+o(1))$

so that almost surely as $(n,j)\to\infty$,

$\left|\sum_{(n^{\prime},j^{\prime})\prec(n,j)}(\hat{X}_{n^{\prime},j^{\prime}}-{\widetilde{Z}}_{n^{\prime},j^{\prime}})\right|=o(c_{2^{n}}).$

The proof of this lemma is provided in Subsection 5.2.

As a consequence of the two previous lemmas, we obtain

###### Corollary 4.3

Let $X_{n,j}$ and ${\widetilde{Z}}_{n,j}$ as defined in Lemmas 4.1and in 4.2,
respectively. Then

$\left|\sum_{(n^{\prime},j^{\prime})\prec(n,j)}(X_{n^{\prime},j^{\prime}}-{\widetilde{Z}}_{n^{\prime},j^{\prime}})\right|=o\left(c_{2^{n}}\right).$

We turn to maxima of sums and gaps.

###### Lemma 4.4

Assume (H3), (HL0) and (HL1). Let $i_{n,j}$ be the smallest element of
$I_{n,j}$. Then the following holds almost surely as $(n,j)\to\infty$

$\max_{r<|I_{n,j}|}\left|\sum_{l=i_{n,j}}^{i_{n,j}+r}W_{l}^{c_{2^{n}}}\right|=o\left(c_{2^{n}}\right)$

and a similar statement holds for ${\widetilde{Z}}_{n^{\prime},j^{\prime}}$.

The proof of Lemma 4.4 is given in Subsection 5.3.

The next lemma shows that the gaps $J_{n,j}$ (see (4.1)) can be neglected.

###### Lemma 4.5

Let $c_{n}$ as in (HL0) and assume (H2) and (H3). Let
$\mathcal{J}=\cup_{n,j}J_{n,j}$. Then, for any ${\varepsilon}>0$, almost
surely, as $k\to\infty$,

$\left|\sum_{l<k,l\in\mathcal{J}}W_{l}^{c_{k}}\right|=o\left(k^{\beta/2+2{\varepsilon}+{\varepsilon}_{1}}\right).$

A similar statement holds for ${\widetilde{Z}}_{n^{\prime},j^{\prime}}$.

Lemma 4.5 is proved in Subsection 5.4 by controlling the $L^{2}$ norms of the
truncated variables in the same manner as in [21, Proof of Lemma 5.9].

We can complete

Proof of Proposition 3.1 First note that, as for any $(n,j)$ the Gaussian
variable ${\widetilde{Z}}_{n,j}$ has covariance
$|I_{n,j}|\ell^{*}(c_{2^{n}})\Sigma^{2}(1+o(1))$222Note that for $l\in
I_{n,j}$ we have ${\underline{l}}=2^{n}$., it can be regarded as a sum of
independent Gaussians, ${\widetilde{Z}}_{n,j}=\sum_{l\in
I_{n,j}}{\widetilde{Z}}_{l}$, where ${\widetilde{Z}}_{l}$ is distributed as
$\mathcal{N}(0,\sigma_{l}^{2})$, with
$\sigma_{l}^{2}=\ell^{*}(c_{{\underline{l}}})\Sigma^{2}(1+o(1))$ as in the
statement of the Proposition.

Now let $\mathcal{I}=\cup_{n,j}I_{n,j}$ be the union of the main blocks. By
Corollary 4.3 and Lemma 4.4 ,
$\left|\sum\limits_{l\in\mathcal{I},l<k}\left(W_{l}^{c_{{\underline{l}}}}-{\widetilde{Z}}_{l}\right)\right|=o(c_{{\underline{k}}})=o(c_{k})$,
almost surely, as $k\to\infty$. By Lemma 4.5,
$\left|\sum\limits_{l\in\mathcal{J},l<k}W_{l}^{c_{{\underline{l}}}}\right|=o\left(k^{\beta/2+2{\varepsilon}+{\varepsilon}_{1}}\right)=o(c_{k})$;
in the last equalities we have used that $\beta<1$ and that
${\varepsilon},{\varepsilon}_{1}$ can be arbitrarily small and also that
$k\in\mathcal{J}$. Thus,
$\left|\sum_{j=1}^{k}W_{j}^{c_{{\underline{j}}}}-{\widetilde{Z}}_{j}\right|=o(c_{k})$,
almost surely, as $k\to\infty$.

By definition, ${\widetilde{Z}}_{j}=\sigma_{j}Z_{j}$, where as in the
statement of Proposition 3.1,
$\sigma_{j}^{2}=\ell^{*}(c_{{\underline{j}}})\Sigma^{2}(1+o(1))$, and $Z_{j}$
is a sequence of Gaussian random vectors distributed as $\mathcal{N}(0,I)$.
Thus, almost surely, as $k\to\infty$,

$\displaystyle\left|\sum_{j=1}^{k}W_{j}^{c_{{\underline{j}}}}-{\widetilde{Z}}_{j}\right|=\left|\sum_{j=1}^{k}(W_{j}^{c_{{\underline{j}}}}-\sigma_{j}\,Z_{j})\right|=o(c_{k}).$

∎

##  5 Proofs of the lemmas used in the proof of Theorem 1.8

###  5.1 Proof of Lemma 4.1

The proof of this lemma is analogous to [21, Proof of Proposition 5.1]. We
recall the main elements partly for completeness, partly for use in some
proofs below.

The proof exploits assumption (H2) and the Strassen Dudley theorem (see, for
instance, [6, Theorem 6.9]), which we now recall. The Prokhorov metric
$\pi(Y,Z)$ between two random variables $Y$ and $Z$ taking values in some
metric space $M$ is defined as

$\inf\\{{\varepsilon}>0\,|\,\mathbb{P}(Y\in A)<\mathbb{P}(Z\in
A^{{\varepsilon}})+{\varepsilon}\text{ and }\mathbb{P}(Z\in A)<\mathbb{P}(Y\in
A^{{\varepsilon}})+{\varepsilon},\forall A\in\mathcal{B}(M)\\},$

where $\mathcal{B}(M)$ denotes the collection of Borel sets of $M$, and
$A^{{\varepsilon}}$ denotes the ${\varepsilon}$-neighborhood of $A$.
Specifically, for real-valued random variables $Y$ and $Z$ with distribution
functions $P_{Y}$ and $P_{Z}$ this reduces to

$\inf\\{{\varepsilon}>0\,|\,P_{Y}(x-{\varepsilon})-{\varepsilon}\leq
P_{Z}(x)\leq P_{Y}(x+{\varepsilon})+{\varepsilon},\forall x\in\mathbb{R}\\}.$

In particular, if $|P_{Y}(x)-P_{Z}(x)|\leq{\varepsilon}$ for every
$x\in\mathbb{R}$, then $\pi(Y,Z)\leq{\varepsilon}$.

###### Theorem 5.1

[21, Theorem 3.4, Lemma 3.5] Let $Y,Z$ be two random variables taking values
in $\mathbb{R}^{dM}$ with probability distributions $P_{Y},P_{Z}$ and
characteristic functions $\Psi_{Y},\Psi_{Z}$. Let $\pi(P_{Y},P_{Z})$ denote
the Prokhorov distance between these distributions.

(a) If $\pi(P_{Y},P_{Z})<c$ then there exists a coupling between $Y$ and $Z$
such that $\mathbb{P}(|Y-Z|>c)<c$.

(b) There exists $C_{d}>0$ so that for any $T^{*}>0$,

$\displaystyle\pi(P_{Y},P_{Z})$
$\displaystyle\leq\sum_{j=1}^{M}\left|P_{Y}(|y_{j}|>T^{*})-P_{Z}(|z_{j}|>T^{*})\right|+\sum_{j=1}^{M}P_{Y}(|y_{j}|>T^{*})$
$\displaystyle+(C_{d}T^{*})^{(dM)/2}\left(\int_{\mathbb{R}^{dM}}|\Psi_{Y}(t)-\Psi_{Z}(t)|^{2}\,dt\right)^{1/2}.$

We can now recall the main steps of [21, Proof of Proposition 5.1]. We
emphasise that these steps require (H2), (H3), Theorem 5.1 and that the
observable of interest (in our set up $W_{l}^{c_{2^{n}}}$) is in $L^{1}(\mu)$.
Proof of Lemma 4.1. Recall $X_{n,j}=\sum_{l\in I_{n,j}}W_{l}^{c_{2^{n}}}$ Let
$X_{n}=(X_{n,j})_{0\leq j<F(n)}$ be vectors in $\mathbb{R}^{dF(n)}$. To
exploit (H2), we shall proceed as in [21] working with variants of $X_{n,j}$
whose characteristic functions will be supported on a neighborhood of $0$.
More precisely, given ${\varepsilon}_{0}$ as in (H2), [21, Proposition 3.8]
ensures that there exists a symmetric random variable $V$ in $\mathbb{R}^{d}$,
$V\in L^{q}$, for every $q\geq 2$, whose characteristic function is supported
in $\\{|t|\leq{\varepsilon}_{0}\\}$. Let $V_{n,j}$ be independent copies of
$V$, independent of everything else. Set $X_{n,j}^{*}=X_{n,j}+V_{n,j}$ and let
$X_{n}^{*}=(X_{n,0}^{*},X_{n,1}^{*},\ldots,X_{n,F(n)-1}^{*})$ be vectors in
$\mathbb{R}^{dF(n)}$.

Let $Q_{n}$ be a random variable distributed like $X_{n}$ and consider the
$\mathbb{R}^{dD(n)}$ random vectors $(X_{1},X_{2},\ldots,X_{n-1},X_{n})$ and
$(X_{1},X_{2},\ldots,X_{n-1},Q_{n})$, where $D(n)=\sum_{j=1}^{n}F(j)\ll
2^{\beta n}$. Note that $J_{n,0}$ is a gap of length $k\approx
2^{{\varepsilon}_{1}n+\beta n}$ between between $(X_{j})_{j<n}$ and $X_{n}$.
Let $\phi$ and $\gamma$ be the characteristic functions of
$(X_{1},X_{2},\ldots,X_{n-1},X_{n})$ and $(X_{1},X_{2},\ldots,X_{n-1},Q_{n})$.
Reasoning as in the first part of the proof of [21, Lemma 5.2], we have under
the current (H2) that for all $|t_{m,j}|\leq{\varepsilon}_{0}$ and for $n$
large enough,

$|\phi-\gamma|\ll e^{-c2^{{\varepsilon}_{1}n+\beta n}}.$

Set $Q_{n}^{*}=Q_{n}+V_{n}$, where $V_{n}$ is an independent copy of $V$,
independent of everything else. Consider the $\mathbb{R}^{dD(n)}$ random
vectors $(X_{1}^{*},X_{2}^{*},\ldots,X_{n-1}^{*},X_{n}^{*})$ and
$(X_{1}^{*},X_{2}^{*},\ldots,X_{n-1}^{*},Q_{n}^{*})$ with characteristic
functions $\phi^{*}$ and $\gamma^{*}$, which are obtained by multiplying
$\phi,\gamma$ by the characteristic function of $V$. Thus,
$|\phi^{*}-\gamma^{*}|\ll e^{-c2^{{\varepsilon}_{1}n+\beta n}}$.

As in the last paragraph of [21, Proof of Lemma 5.2], the distance between the
distributions requires only that for all $n$,
$\|W_{l}^{c_{2^{n}}}\|_{L^{1}(\mu)}\leq C$, for some $C>0$ independent of $n$
and $l$; this is true because $v\in L^{1}(\mu)$. We apply Theorem 5.1(b) with
$T^{*}=e^{2^{{\varepsilon}_{1}n/2}}$ and obtain, by the Markov inequality,
that the terms associated to the tails of the distributions are bounded by
$\ll 2^{n}e^{-2^{{\varepsilon}_{1}n/2}}$.

Putting the previous two facts together and using Theorem 5.1,

$\displaystyle\pi\left((X_{1}^{*},X_{2}^{*},\ldots,X_{n-1}^{*},X_{n}^{*}),(X_{1}^{*},X_{2}^{*},\ldots,X_{n-1}^{*},Q_{n}^{*})\right)\ll
4^{-n}.$ (5.1)

Using (5.1) and proceeding as in [21, Proof of Corollary 5.3], we obtain that
there exists a coupling between $(X_{1}^{*},X_{2}^{*},\ldots)$ and
$(R_{1}^{*},R_{2}^{*},\ldots)$, where the variables
$R_{n}^{*}=(R_{n,j}^{*})_{j<F(n)}$ are distributed as the $X_{n}^{*}$, but
independent of each other, so that

$\displaystyle\mu\left(|X_{n,j}^{*}-R_{n,j}^{*}|\geq C4^{-n}\right)\leq
C4^{-n}.$ (5.2)

The next step is then to show – based on (H2), with an argument analogous to
[21, Proof of Lemma 5.4] – that

$\displaystyle\pi\left((R_{n,j}^{*})_{0\leq j<F(n)},(\hat{X}_{n,j}^{*})_{0\leq
j<F(n)}\right)\leq C4^{-n},$ (5.3)

where $\hat{X}_{n,j}^{*}=\hat{X}_{n,j}+V_{n,j}$ with
$(\hat{X}_{n,j})_{n\in\mathbb{N},0\leq j<F(n)}$ a sequence of independent
random variables distributed like $X_{n,j}$. We recall the main ingredients of
the argument in [21, Proof of Lemma 5.4] needed to obtain (5.3), partly for
completeness, partly for use in the proof of Lemma 4.5 below. The main idea of
[21, Proof of Lemma 5.4] is to make the variables $(R_{n,j}^{*})_{0\leq
j<F(n)/2}$ independent of $(R_{n,j}^{*})_{0\leq j<F(n)/2}$ by using a large
gap $J_{n,F(n)/2}$ and proceed in each remaining half using the gap in the
middle of this half and repeating the procedure. For $0\leq i<\lfloor
n\beta\rfloor$ define $\tilde{Y}_{n,j}^{i}$ so that for $0\leq k<2^{\lfloor
n\beta\rfloor-1}$, the random variable
$\mathcal{\tilde{Y}}_{n,k}^{i}=(\tilde{Y}_{n,j}^{i})_{k2^{i}\leq
j<(k+1)2^{i}}$ is distributed like $(X_{n,j}^{*})_{k2^{i}\leq j<(k+1)2^{i}}$
and so that $\mathcal{\tilde{Y}}_{n,k}^{i}$ is independent of
$\mathcal{\tilde{Y}}_{n,k^{\prime}}^{i}$ if $k\neq k^{\prime}$. The argument
of [21, Proof of Lemma 5.4] simply goes through provided that the distance
between the characteristic function of the random variables
$\mathcal{\tilde{Y}}_{n,k}^{i}$ and
$(\mathcal{\tilde{Y}}_{n,2k}^{i-1},\mathcal{\tilde{Y}}_{n,2k+1}^{i-1})$ is
$O(e^{-c2^{n{\varepsilon}_{1}+i}})$. Since $\mathcal{\tilde{Y}}_{n,k}^{i}$
consists of $2^{i}$ sums of variables $W_{l}^{c_{2^{n}}}$ along blocks of
length $2^{n-\lfloor n\beta\rfloor}$ with a gap of size $\approx
2^{n{\varepsilon}_{1}+i}$ in the middle, (H2) applies and, the required result
$O(e^{-c2^{n{\varepsilon}_{1}+i}})$ follows. Finally, let
$\hat{X}_{n,j}^{*}=\tilde{Y}_{n,j}^{0}$.

By (5.1) and (5.3) and Borel-Cantelli lemma (after summing over $j$ and $n$),

$\left|\sum_{(n^{\prime},j^{\prime})\prec(n,j)}(X_{n^{\prime},j^{\prime}}^{*}-\hat{X}_{n^{\prime},j^{\prime}}^{*})\right|=o(2^{n/2})=o(c_{2^{n}}).$

To obtain the result for $X_{n,j}$, one needs to argue that the contribution
of $V_{n,j}$ is negligible. As in [21, Proof of Proposition 5.1], this can be
done using the law of iterated logarithms for the centered, independent random
variables $V_{n,j}\in L^{2}(\mu)$:

$\left|\sum_{(n^{\prime},j^{\prime})\prec(n,j)}V_{n,j}\right|\leq
CN_{n,j}^{1/2}\cdot\sqrt{\log\log(N_{n,j})}.$

where $N_{n,j}$ is the cardinality of the set
$\\{(n^{\prime},j^{\prime}):(n^{\prime},j^{\prime})\prec(n,j)\\}$. Since
$N_{n,j}\leq\sum_{n^{\prime}=1}^{n}\sum_{j^{\prime}=0}^{F(n^{\prime})-1}1\leq
C2^{\beta n}$,

$\left|\sum_{(n^{\prime},j^{\prime})\prec(n,j)}V_{n,j}\right|=o(2^{(\beta
n+{\varepsilon}n)/2}),$

for any ${\varepsilon}>0$. Since $\beta<1$ and ${\varepsilon}$ is arbitrary,
$o(2^{(\beta n+{\varepsilon}n)/2})=o(2^{n/2})$, we obtain

$\displaystyle\left|\sum_{(n^{\prime},j^{\prime})\prec(n,j)}(X_{n^{\prime},j^{\prime}}-\hat{X}_{n^{\prime},j^{\prime}})\right|=o(2^{n/2})=o(c_{2^{n}}),$
(5.4)

as required. $\Box$

###  5.2 Proof of Lemma 4.2

Let $(\hat{X}_{n,j})_{n\in\mathbb{N},\,0\leq j<F(n)}$ be the independent
random variables obtained in Lemma 4.1.

In this section we exploit assumption (HL1)(i) with
$d_{n}=\sqrt{nL(n)\ell_{1}(n)^{-99}}$ as in (1.8) and $r=i_{n,j}$. More
precisely, fix $(n,j)$ and for $l=0,\dots,|I_{n,j}|$ let

${\widetilde{W}}=v\cdot\mathbf{1}_{d_{2^{n}}\leq|v|\leq
c_{2^{n}}},\qquad{\widetilde{W}}_{l}=W_{l+i_{n,j}}^{c_{2^{n}}}-W_{l+i_{n,j}}^{d_{2^{n}}}={\widetilde{W}}\circ
T^{l+i_{n,j}}.$

Note that

$\ell^{*}(c_{2^{n}})\asymp\ell^{*}(d_{2^{n}})\asymp
n\quad\text{and}\quad\frac{c_{n}}{d_{n}}\asymp(\ell_{1}(n))^{50}$ (5.5)

Let $Y_{n,j}=\sum_{l\in I_{n,j}}W_{l}^{d_{2^{n}}}$, and let $\hat{Y}_{n,j}$ be
distributed as $Y_{n,j}$, but independent for
$(n^{\prime},j^{\prime})\neq(n,j)$.

The first result below says that the additional truncation in terms of
$d_{2^{n}}$ causes an error which is almost surely $o(c_{2^{n}})$.

###### Lemma 5.2

For any $\varepsilon>0$ we have

$\sum_{n=1}^{\infty}\sum_{j=0}^{F(n)-1}\mu\left(\left|\hat{X}_{n,j}-\hat{Y}_{n,j}\right|>\varepsilon
c_{2^{n}}\right)<\infty.$

###### Proof.

We will use Chebyshev’s inequality, hence need a bound on the second moment of

$\hat{X}_{n,j}-\hat{Y}_{n,j}=\sum_{l\in
I_{n,j}}(W_{l}^{c_{2^{n}}}-W_{l}^{d_{2^{n}}}).$

Using (HL1)(i) and (5.5),

$\displaystyle\mathbb{E}_{\mu}(|\hat{X}_{n,j}-\hat{Y}_{n,j}|^{2})$
$\displaystyle\ll\sum_{l\in
I_{n,j}}\mathbb{E}_{\mu}\left((W_{l}^{c_{2^{n}}}-W_{l}^{d_{2^{n}}})^{2}\right)=\sum_{l\in
I_{n,j}}\mathbb{E}_{\mu}\left(({\widetilde{W}}_{l})^{2}\right)$
$\displaystyle\ll|I_{n,j}|L\left(\frac{c_{2^{n}}}{d_{2^{n}}}\right)\ll|I_{n,j}|LL(n).$
(5.6)

Hence, the Chebyshev inequality implies

$\displaystyle\sum_{j=0}^{F(n)-1}\mu\left(\left|\hat{X}_{n,j}-\hat{Y}_{n,j}\right|>\varepsilon
c_{2^{n}}\right)\ll$ $\displaystyle\frac{F(n)\cdot|I_{n,j}|\cdot
LL(n)}{(c_{2^{n}})^{2}}\ll\frac{LL(n)}{n\ell_{1}(2^{n})}$

which is sumable in $n$, by assumption (HL0)(iii). ∎

It remains to couple the independent variables $\hat{Y}_{n,j}$ truncated at
the lower level $d_{2^{n}}$ to appropriate Gaussians, as stated in the next
Lemma, the analogue of [21, Lemma 5.6] in the present set up.

###### Lemma 5.3

Let $n\in\mathbb{N}$. There exists a coupling between
$(\hat{Y}_{n,0},\hat{Y}_{n,1},\ldots,\hat{Y}_{n,F(n)-1})$ and
$(S_{n,0},S_{n,1},\ldots,S_{n,F(n)-1})$, where $S_{n,j}$ are independent,
centered Gaussian vectors $covS_{n,j}=cov\hat{Y}_{n,j}$ so that for any
${\varepsilon}>0$,

$\sum_{n}\mu\left(\max_{1\leq i\leq
F(n)}\left|\sum_{j=0}^{i-1}Y_{n,j}-S_{n,j}\right|\geq\frac{c_{2^{n}}}{L(n)}\right)<\infty.$

###### Proof.

We apply [35, Corollary 3] as recalled in [21, Proposition 5.5] to
$\hat{Y}_{n,j}$. We briefly recall the statement. Let $Y_{0},\ldots,Y_{b-1}$
be independent centered $\mathbb{R}^{d}$-valued random variables. Let $p\geq
2$ and $M=\left(\sum_{j=0}^{b-1}\mathbb{E}|Y_{j}|^{p}\right)^{1/p}$. Assume
that there exists a sequence $0=m_{0}<m_{1}<\ldots m_{s}=b$ so that given
$\Xi_{k}=Y_{m_{k}}+\ldots+Y_{m_{k+1}-1}$, $B_{k}=cov\Xi_{k}$, the following
holds for all $w\in\mathbb{R}^{d}$, for all $0\leq k\leq s$ and for some
$C>1$:

$\displaystyle 100M^{2}|w|^{2}\leq\langle B_{k}w,w\rangle\leq
100CM^{2}|w|^{2}.$ (5.7)

Then there exits a coupling between $(Y_{0},\ldots Y_{b-1})$ and a sequence of
independent Gaussian random vectors $(S_{0},\ldots S_{b-1})$ with
$covS_{j}=covY_{j}$ and so that

$\displaystyle\mu\left(\max_{1\leq i\leq
b}\left|\sum_{j=0}^{i-1}Y_{j}-S_{j}\right|\geq
Mz\right)<c_{0}z^{-p}+e^{-C_{0}z},$ (5.8)

for all $z\geq C_{0}\log s$ with $C_{0}$ a constant that depends only on $C$,
dimension $d$ and exponent $p$.

We will apply this statement with $b=F(n)$, $Y_{j}=\hat{Y}_{n,j}$, $p=4$ and
$z=n^{1/4}\cdot L(n)$. Let us check that the conditions hold. By (H3)(ii) we
have (recall $|I_{n,j}|=2^{(1-\beta)n}$ with $\beta<1$ while
$2^{n/2}=o(d_{2^{n}})$, so the required relation of number of terms and
truncation level, $|I_{n,j}|=o((d_{2^{n}})^{2})$, holds)

$\mathbb{E}(|\hat{Y}_{n,j}|^{4})\ll|I_{n,j}|(d_{2^{n}})^{2},$

and thus

$\displaystyle
M=\left(\sum_{j=0}^{F(n)-1}\mathbb{E}(|\hat{Y}_{n,j}|^{4})\right)^{1/4}\ll$
$\displaystyle\left(F(n)2^{(1-\beta)n}d_{2^{n}}^{2}\right)^{1/4}\ll
2^{n/4}d_{2^{n}}^{1/2}$ $\displaystyle\ll$ $\displaystyle
2^{n/2}\left(L(2^{n})\ell_{1}(2^{n})^{-80}\right)^{1/4}\ll
2^{n/2}n^{1/4}(L(n))^{-20}.$ (5.9)

This implies

$M^{2}\ll 2^{n}\sqrt{n},$

while, by (H3) (ii)

$\sum_{j=0}^{F(n)-1}cov\hat{Y}_{n,j}\asymp\sum_{j=0}^{F(n)-1}|I_{n,j}|\ell^{*}(d_{2^{n}})\gg
2^{n}n$

which is larger in magnitude than $M^{2}$. On the other hand, recalling that
$\Sigma$ is non degenerate (by (H1)), there must exits some constant $C_{1}$
so that for any large $n$, for any $0\leq m<m^{\prime}<F(n)$ for any vector
$w$ in $\mathbb{R}^{d}$,

$C_{1}^{-1}(m^{\prime}-m)2^{(1-\beta)n}|w|^{2}\leq\Big{\langle}\sum_{j=m}^{m^{\prime}-1}covY_{n,j}w,w\Big{\rangle}\leq
C_{1}(m^{\prime}-m)2^{(1-\beta)n}|w|^{2}$

and each individual term is bounded by

$|covY_{n,j}|\,|w|^{2}\leq\mathbb{E}_{\mu}(|Y_{n,j}|^{2})|w|^{2}\leq(\mathbb{E}_{\mu}(|Y_{n,j}|^{4}))^{1/2}|w|^{2}\leq
M^{2}|w|^{2}.$

So, one can group $Y_{n,j}$ into consecutive blocks so that (5.7) holds with
some constant $C$. Note also that, for the number of blocks,
$s=\sqrt{n}\ell_{2}(n)$ with some slowly varying function $\ell_{2}$, hence
the condition $z\gg\log s$ holds with the above specified $z=n^{1/4}L(n)$.

Now, using (5.9), we have

$z\cdot
M=n^{1/4}L(n)2^{n/2}n^{1/4}(\ell_{1}(2^{n}))^{-20}\ll\sqrt{2^{n}n\ell_{1}(2^{n})}(\ell_{1}(2^{n}))^{-1}\ll\frac{c_{2^{n}}}{L(n)}$

where we have used that $L(n)\ll\ell_{1}(2^{n})$ by (HL0)(ii) (see also
(HL0)(iii)). The conclusion follows by (5.7) as

$\displaystyle\mu\left(\max_{1\leq i\leq
F(n)}\left|\sum_{j=0}^{i-1}Y_{n,j}-S_{n,j}\right|\geq\frac{c_{2^{n}}}{L(n)}\right)\,<\,$
$\displaystyle\mu\left(\max_{1\leq i\leq
b}\left|\sum_{j=0}^{i-1}Y_{j}-S_{j}\right|\geq Mz\right)$ $\displaystyle<\,$
$\displaystyle
c_{0}z^{-4}+e^{-C_{0}z}\,<\,c_{0}(nL(n)^{4})^{-1}+e^{-C_{0}n^{1/4}}$

which is sumable in $n$. ∎

Lemmas 5.2 and 5.3 imply that there exist a coupling such that

$\max_{1\leq i\leq
F(n)}\left|\sum_{j=0}^{i-1}\hat{X}_{n,j}-S_{n,j}\right|=o(c_{2^{n}})$

almost surely, as $n\to\infty$. As

$\sum_{m=0}^{n}c_{2^{m}}\leq\left(\sum_{m=0}^{n}2^{m/2}\right)\sqrt{\ell^{*}(2^{n})\ell_{1}(2^{n})}\ll
c_{2^{n}},$

it follows that

$\sum_{(n^{\prime},j^{\prime})\prec(n,j)}\hat{X}_{n^{\prime},j^{\prime}}-S_{n^{\prime},j^{\prime}}=o(c_{2^{n}}),$

almost surely, as $(n,j)\to\infty$.

We can now complete

Proof of Lemma 4.2 Let $S_{n,j}$ be independent, centered Gaussian vectors
obtained in Lemma 5.3. We still need to construct a sequence of independent
Gaussian random vectors ${\widetilde{Z}}_{n,j}$ with
$cov({\widetilde{Z}}_{n,j})=\ell^{*}(c_{2^{n}})\cdot|I_{n,j}|\cdot\Sigma^{2}$
and a coupling between $S_{n,j}$ and ${\widetilde{Z}}_{n,j}$. We may rely on
[21, Lemma 5.7], which provides such a construction, such that

$\sum_{(n^{\prime},j^{\prime})\prec(n,j)}\left(\frac{S_{n^{\prime},j^{\prime}}}{\sqrt{\ell^{*}(c_{2^{n}})}}-\frac{{\widetilde{Z}}_{n^{\prime},j^{\prime}}}{\sqrt{\ell^{*}(c_{2^{n}})}}\right)=o(2^{(\beta+{\varepsilon}_{1})n/2}).$

The conclusion follows – together with Lemma 5.3– as
$c_{n}=o(\sqrt{\ell^{*}(c_{2^{n}})}\cdot 2^{(\beta+{\varepsilon}_{1})n/2})$,
and $\beta+{\varepsilon}_{1}<1$. ∎

###  5.3 Proof of Lemma 4.4

Throughout this section we use the notation introduced in Section 5.2. First,
we need a strengthening of Lemma 5.2, which will be derived from (HL1)(ii).

###### Lemma 5.4

For any ${\varepsilon}>0$

$\sum_{n=1}^{\infty}\sum_{j=0}^{F(n)-1}\mu\left(\max_{r<|I_{n,j}|}\left|\sum_{l=0}^{r}{\widetilde{W}}_{l}\right|\geq{\varepsilon}c_{2^{n}}\right)<\infty.$
(5.10)

###### Proof.

Note that Lemma 5.2 can be formulated as

$\sum_{n=1}^{\infty}\sum_{j=0}^{F(n)-1}\mu\left(\left|\sum_{l=0}^{|I_{n,j}|}{\widetilde{W}}_{l}\right|\geq{\varepsilon}c_{2^{n}}\right)<\infty.$
(5.11)

Next, we exploit (HL1)(ii) with $r=i_{n,j}$ and $K=|I_{n,j}|$. Note that $r$
and $K$ are fixed since $n$ and $j$ are fixed. In concrete terms, for
$k=1,\dots,K$, we apply (HL1)(ii) to

$\mathcal{S}_{k}=\sum_{l=0}^{k}{\widetilde{W}}_{l};\qquad
B_{k,{\varepsilon}}=\\{x\in
M\,|\,|\mathcal{S}_{k}|\geq{\varepsilon}c_{2^{n}}\\}$

and
$N_{k,{\varepsilon}}=B_{k,{\varepsilon}}\setminus(\cup_{k^{\prime}=0}^{k-1}B_{k^{\prime},{\varepsilon}}).$
By (HL1)(ii) $\sum_{k=1}^{K}\mu(N_{k,{\varepsilon}}\setminus
B_{K,{\varepsilon}/2})\ll K\frac{LL(n)}{(c_{2^{n}})^{2}}.$

By the argument used in (5.2), $\mu(B_{K,{\varepsilon}/2})\ll\frac{K\cdot
LL(n)}{(c_{2^{n}})^{2}}.$ Thus,

$\mu\left(\cup_{k=0}^{K}B_{k,{\varepsilon}}\right)\ll\frac{K\cdot
LL(n)}{(c_{2^{n}})^{2}}.$

The required sumability for (5.10) follows by the argument used in the proof
of Lemma 5.2. ∎

Proof of Lemma 4.4. By Lemma 5.4, it remains to prove

$\sum_{n=1}^{\infty}\sum_{j=0}^{F(n)-1}\mu\left(\max_{r<|I_{n,j}|}\left|\sum_{l=i_{n,j}}^{i_{n,j}+r}W_{l}^{d_{2^{n}}}\right|\geq{\varepsilon}c_{2^{n}}\right)<\infty.$

By the Markov inequality

$\displaystyle\mu\left(\max_{r<|I_{n,j}|}\left|\sum_{l=i_{n,j}}^{i_{n,j}+r}W_{l}^{c_{2^{n}}}\right|\geq{\varepsilon}c_{2^{n}}\right)\leq\left({\varepsilon}c_{2^{n}}\right)^{-4}\int\max_{r<|I_{n,j}|}\left|\sum_{l=i_{n,j}}^{i_{n,j}+r}W_{l}^{d_{2^{n}}}\right|^{4}\,d\mu.$
(5.12)

By [30, Theorem B], there exists $C>0$ such that

$\displaystyle\int\max_{r<|I_{n,j}|}\left|\sum_{l=i_{n,j}}^{i_{n,j}+r}W_{l}^{d_{n}}\right|^{4}\,d\mu\leq
C\cdot\int\left|\sum_{l\in I_{n,j}}W_{l}^{d_{2^{n}}}\right|^{4}\,d\mu$

By (H3)(ii) with $R=d_{2^{n}}$ and $n$ replaced by $|I_{n,j}|$ (note that
$|I_{n,j}|\asymp 2^{(1-\beta)n}=o((d_{2^{n}})^{2-r_{1}})$ for some $r_{1}>0$),

$\int\left|\sum_{l=0}^{|I_{n,j}|}W_{l}^{d_{2^{n}}}\right|^{4}\,d\mu\,\ll\,|I_{n,j}|(d_{2^{n}})^{2}.$

Now recall that $d_{n}^{2}=n\cdot L(n)\cdot(\ell_{1}(n))^{-99}$ while
$c_{n}^{2}=n\cdot L(n)\cdot\ell_{1}(n)$

$\displaystyle\mu\left(\max_{r<|I_{n,j}|}\left|\sum_{l=i_{n,j}}^{i_{n,j}+r}W_{l}^{d_{2^{n}}}\right|\geq{\varepsilon}c_{2^{n}}\right)$
$\displaystyle\ll\frac{|I_{n,j}|d_{2^{n}}^{2}}{c_{2^{n}}^{4}}$
$\displaystyle\ll\frac{|I_{n,j}|}{c_{2^{n}}^{2}(\ell_{1}(2^{n}))^{101}}\ll\frac{|I_{n,j}|}{2^{n}\cdot
n\cdot(\ell_{1}(2^{n}))^{100}}$

Now summing on $j=0,\dots,F(n)-1$, we may use
$\sum_{j=0}^{F(n)-1}|I_{n,j}|\asymp 2^{n}$, and get

$\sum_{j=0}^{F(n)-1}\mu\left(\max_{r<|I_{n,j}|}\left|\sum_{l=i_{n,j}}^{i_{n,j}+r}W_{l}^{d_{2^{n}}}\right|\geq{\varepsilon}c_{2^{n}}\right)\ll(n\cdot(\ell_{1}(2^{n}))^{100})^{-1}.$

This bound is sumable in $n$ due to assumption (HL0)(iii). $\Box$

###  5.4 Proof of Lemma 4.5

Since we are looking at the truncated process for which the second moment is
finite, it makes sense to apply the Gal-Koksma strong law of large numbers
[21, Proposition 3.7]. As such, we can proceed as [21, Section 5.4]. We recall
the main steps in our set up, which are slightly different because the effects
of the truncation have to be dealt with.

Our key Lemma, analogous to [21, Proof of Lemma 5.9], is

###### Lemma 5.5

For any ${\varepsilon}>0$, there exists $C>0$ so that for any interval
$J\subset\mathbb{N}$,

$\mathbb{E}_{\mu}\left(\sum_{l\in
J\cap\mathcal{J}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\leq
C\,|J\cap\mathcal{J}|^{1+{\varepsilon}}.$

This allows to complete in the same manner as in [21, Section 5.4],

Proof of Lemma 4.5 The Gal-Koksma strong law of large numbers [21, Propostion
3.7] says that if $(X_{l})_{l\geq 0}$ are centered random variables and if
$\mathbb{E}\left|\sum_{l=m}^{m+n-1}X_{l}\right|^{2}\leq Cn^{r}$ for some
$r\geq 1$ and some $C>0$, then for any ${\varepsilon}_{2}>0$,
$\frac{\sum_{l=1}^{N}X_{l}}{N^{r/2+{\varepsilon}_{2}}}\to 0$, almost surely,
as $N\to\infty$.

We can apply Gal-Koksma to $W_{l}^{c_{{\underline{l}}}}$; this is possible
because by assumption (H3)(i), $W_{l}^{c_{l}}$ are centered.

By Lemma 5.5 for any ${\varepsilon}>0$, $\mathbb{E}_{\mu}\left(\sum_{l\in
J\cap\mathcal{J}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\leq
C|J\cap\mathcal{J}|^{1+{\varepsilon}}.$ Taking $k\in[2^{n},2^{n+1})$ and
recalling (4.1),

$|J\cap\mathcal{J}|\leq\sum_{m=1}^{n}\sum_{j=0}^{F(m)-1}|J_{m,j}|\ll\sum_{m=1}^{n}m2^{{\varepsilon}_{1}m+\beta
m}\ll n2^{{\varepsilon}_{1}n+\beta n}\ll k^{\beta+3{\varepsilon}_{1}/2}.$

It follows from Gal-Koksma that
$\sum_{l<k,l\in\mathcal{J}}W_{l}^{c_{l}}=o\left(k^{\beta/2+2{\varepsilon}+{\varepsilon}_{1}}\right),$
as desired. ∎

It remains to provide

Proof of Lemma 5.5 As already mentioned, this proof is analogous to [21, Proof
of Lemma 5.9]. The only aspect which is different in our set up is that for
estimating second moments of (truncated) sums we need to use (H3)(iii) instead
of [21, Proposition 4.1] and the current (H2) (instead of [21, Assumption
(H)]).

To start, let us recall that $L(\cdot)=\log(\max(\cdot,e))$ and introduce the
notation

$N=\max_{l\in J}L(l).$ (5.13)

Via (H3), second moment moment bounds of sums of truncated variables will
include factors of the type $\ell^{*}(c_{l})$ for various $l\in J$. all these
contributions can be estimated using

$\ell^{*}(c_{2^{n}})\asymp L(2^{n})\ll N.$

Now use, as in the second displayed equation inside [21, Proof of Lemma 5.9],
the convexity inequality

$\displaystyle\left(\sum_{l\in
J\cap\mathcal{J}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\leq 3\left(\sum_{l\in
J_{0}}W_{l}^{c_{{\underline{l}}}}\right)^{2}+3\left(\sum_{l\in
J_{1}}W_{l}^{c_{{\underline{l}}}}\right)^{2}+3\left(\sum_{l\in
J_{0}}W_{l}^{c_{{\underline{l}}}}\right)^{2},$

where $J_{0},J_{2}$ are the first and last intervals in $J\cap\mathcal{J}$ and
where $J_{1}$ is the remaining part.

Note that $\mathbb{E}_{\mu}\left(\sum_{l\in
J_{0}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll\mathbb{E}_{\mu}\left(\sum_{l\in
J_{0}}W_{l}^{\max_{l\in J_{0}}c_{l}}\right)^{2}$ and
$\mathbb{E}_{\mu}\left(\sum_{l\in
J_{2}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll\mathbb{E}_{\mu}\left(\sum_{l\in
J_{2}}W_{l}^{\max_{l\in J_{2}}c_{l}}\right)^{2}$. This together with (H3)
gives

$\displaystyle\mathbb{E}_{\mu}\left(\sum_{l\in
J_{0}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll|J_{0}|\cdot\ell^{*}(\max_{l\in
J_{0}}c_{l}),\qquad\mathbb{E}_{\mu}\left(\sum_{l\in
J_{2}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll|J_{2}|\cdot\ell^{*}(\max_{l\in
J_{2}}c_{l}).$

Under (HL0), for all $m$ large enough, $\ell^{*}(m)\approx L(m)$. Recalling
the assumption on $c_{n}$ in (HL0),

$\displaystyle\mathbb{E}_{\mu}\left(\sum_{l\in
J_{0}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll|J_{0}|\cdot
N,\qquad\mathbb{E}_{\mu}\left(\sum_{l\in
J_{2}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll|J_{2}|\cdot N.$ (5.14)

We claim that for any ${\varepsilon}>0$,

$\displaystyle\mathbb{E}_{\mu}\left(\sum_{l\in
J_{1}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll|J_{1}|^{1+{\varepsilon}}.$
(5.15)

This together with (5.14) gives the conclusion.

If $J_{1}$ is empty, there is nothing to do. If $J_{1}$ is not empty, the
proof of the claim (5.15) follows the argument of [21, Proof of Equation
(5.18)] with again, the only difference that we have to use (H3) instead of
[21, Proposition 4.1] and we need the current (H2).

Recall that as before (4.1), an interval has rank $r$ if its length is
$2^{\lfloor{\varepsilon}_{1}n\rfloor+r}$. Let $\mathcal{J}^{(n,r)}$ denote the
union of intervals $J_{n,j}$ of rank $0\leq r\leq\rfloor n\beta\rfloor$.
Recall, furthermore, (5.13). The number of sets $\mathcal{J}^{(n,r)}$
intersecting $J_{1}$ is at most $\sum_{n=0}^{N}(\rfloor n\beta\rfloor+1)\ll
N^{2}$. Proceeding as in the argument used in obtaining [21, Equation (5.19)],
we can rely on the convexity inequality and get

$\displaystyle\left(\sum_{l\in J_{1}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll
N^{2}\sum_{n,r}\left(\sum_{l\in
J_{1}\cap\mathcal{J}^{(n,r)}}W_{l}^{c_{{\underline{l}}}}\right)^{2}.$ (5.16)

Let us fix some relevant $(n,r)$ and enumerate the intervals in
$\mathcal{J}^{(n,r)}$ as $K_{1},\ldots,K_{t}$ for $t=2^{\rfloor
n\beta\rfloor-1-r}$, $0\leq r\leq\rfloor n\beta\rfloor$. Let $T_{s}=\sum_{l\in
K_{s}}W_{l}^{c_{{\underline{l}}}}$. To complete the proof of Lemma 5.5, the
key bound, analogous to [21, Equation (5.20)], is that for any subset
$S\subset\\{1,\dots,t\\}$

$\displaystyle\mathbb{E}_{\mu}\left(\sum_{s\in
S}T_{s}\right)^{2}\ll\left(\sum_{s\in
S}\mathbb{E}_{\mu}(T_{s}^{2})+|S|\right)\cdot N.$ (5.17)

(5.17) can be proved using (H2) and (H3), see below, but let us first see how
to obtain (5.15) from (5.17). By (H3),
$\mathbb{E}_{\mu}(T_{s}^{2})\ll\left(\sum_{l\in K_{s}}W_{l}^{\max_{l\in
K_{s}}c_{l}}\right)^{2}\ll|K_{s}|\cdot N$. This type of bound holds for any
set $K$ which is a union of intervals in $\mathcal{J}^{(n,r)}$. Taking
$K=J_{1}\cap\mathcal{J}^{(n,r)}$ and combining with (5.16),

$\displaystyle\left(\sum_{l\in J_{1}}W_{l}^{c_{{\underline{l}}}}\right)^{2}\ll
N^{2}\sum_{n,r}|J_{1}\cap\mathcal{J}^{(n,r)}|\cdot N$

and the claim (5.15) follows as $N\ll|J_{1}]^{{\varepsilon}}$ for any
${\varepsilon}>0$, whenever $J_{1}$ is nonempty (note that if $J_{1}$ is
nonempty, it has to contain at least one interval of length $\asymp 2^{\beta
N}$).

It remains to justify (5.17) proceeding as in the proof of [21, Equation
(5.20)]. Let $(U_{1},\ldots,U_{t})$ be independent random variables such that
$U_{s}$ is distributed like $T_{s}$. We recall that given ${\varepsilon}_{0}$
as in (H2), [21, Proposition 3.8] ensures that there exists a symmetric random
variable $V$ in $\mathbb{R}^{d}$, $V\in L^{q}$, $q\geq 2$, whose
characteristic function is supported in $\\{|t|\leq{\varepsilon}_{0}\\}$. Let
$V_{1},\ldots,V_{t},V_{1}^{\prime},\ldots,V_{t}^{\prime}$ be independent
random variables distributed like $V$ and define $\tilde{T}_{s}=T_{s}+V_{s}$,
$\tilde{U}_{s}=U_{s}+V_{s}^{\prime}$. Using intervals of rank greater than $k$
as gaps and using the procedure recalled in justifying (5.3) above we obtain
(in the same manner as in the proof of [21, Equation (5.21)]),

$\displaystyle\pi\left((\tilde{T}_{1},\ldots,\tilde{T}_{t}),(\tilde{U}_{1},\ldots,\tilde{U}_{t})\right)\ll
e^{2^{\delta n}},$

for some $\delta>0$. Now we may proceed as in [21]: by Theorem 5.1 there
exists a coupling between the $(\tilde{T}_{j})$ and the $(\tilde{U}_{j})$
$(j=1,\dots,t)$ such that, outside a set $O$ of measure $\ll e^{-2^{\delta
n}}$, we have $|\tilde{T}_{j}-\tilde{U}_{j}|\ll e^{-2^{\delta n}}$ for any
$j=1,\dots,t$. Then, for any subset $S\subset\\{1,\dots,t\\}$ we have

$\left\|\sum_{s\in
S}\tilde{T}_{s}\right\|_{L^{2}}\leq\left\|\mathbf{1}_{O}\cdot\sum_{s\in
S}\tilde{T}_{s}\right\|_{L^{2}}+\left\|\mathbf{1}_{O^{c}}\cdot\sum_{s\in
S}(\tilde{T}_{s}-\tilde{U}_{s})\right\|_{L^{2}}+\left\|\sum_{s\in
S}\tilde{U}_{s}\right\|_{L^{2}}.$ (5.18)

To estimate the first term in (5.18), [21] uses an $L^{p}$ bound for a
relevant $p>2$, here we proceed slightly differently; we rely on (H3i) and get

$\displaystyle\left\|\mathbf{1}_{O}\cdot\sum_{s\in
S}\tilde{T}_{s}\right\|_{L^{2}}$
$\displaystyle\leq\left\|\mathbf{1}_{O}\right\|_{L^{4}}\cdot\left\|\sum_{s\in
S}\tilde{T}_{s}\right\|_{L^{4}}$ $\displaystyle\ll e^{-2^{\delta
n}/4}\cdot\sum_{s\in S}\left\|\tilde{T}_{s}\right\|_{L^{4}}\ll e^{-2^{\delta
n}/4}\cdot|S|\cdot 2^{n/4}\cdot c_{2^{n}}^{1/2}$ $\displaystyle\ll
e^{-2^{\delta n}/4}\cdot 2^{2n}\leq C$

for some $C>0$. From this point on, the argument can be completed literally as
in [21]; the second term in (5.18) can be bounded by some $C>0$, and thus

$\displaystyle\left\|\sum_{s\in S}T_{s}\right\|_{L^{2}}$
$\displaystyle\leq\left\|\sum_{s\in
S}\tilde{T}_{s}\right\|_{L^{2}}+\left\|\sum_{s\in S}V_{s}\right\|_{L^{2}}$
$\displaystyle\leq C+\left\|\sum_{s\in
S}U_{s}\right\|_{L^{2}}+\left\|\sum_{s\in
S}V_{s}\right\|_{L^{2}}+\left\|\sum_{s\in S}V^{\prime}_{s}\right\|_{L^{2}}$
$\displaystyle\leq C+(\sum_{s\in
S}\mathbb{E}_{\mu}(T_{s}^{2}))^{1/2}+C|S|^{1/2}.$

∎

##  6 Verification of (H3)(ii) and (HL1) for the Lorentz gas

###  6.1 Verification of (H3)(ii): bound on the fourth moment

Let $\mathcal{S}=\sum_{i=0}^{n-1}W^{R}\circ T^{i}$, that is, the number of
terms is $n$ and the truncation level is $R$. Throughout, we assume
$n=o(R^{2-r_{1}})$ and $R=O(n^{r_{2}})$, where $r_{1}>0$ and $r_{2}>0$ are
some fixed exponents - this implies also $\log R\asymp\log n$. Our aim is to
bound $\mathbb{E}(\mathcal{S}^{4})$. For brevity let $Y_{i}=W^{R}\circ T^{i}$
and $Y=Y_{0}$. We also assume $\mathbb{E}Y=0$ which holds in the Lorentz gas
by symmetry. We have

$\mathbb{E}\mathcal{S}^{4}\ll\sum_{0\leq i\leq j\leq k\leq l\leq
n-1}\mathbb{E}(Y_{i}Y_{j}Y_{k}Y_{l})$

We distinguish several cases depending on the indices $i,j,k,l$. For some
large constant $C$, let

Case 1. Either $|i-j|\geq C\log n$ or $|k-l|\geq C\log n$. By decay of
correlations (see eg. [4, Lemma 4.1]) the contribution of such terms is

$\ll n^{4}\cdot R^{4}\theta^{C\log n}=O(1)$

here $n^{4}$ is the number of terms and for $C$ sufficiently large the decay
of correlations can be made $\theta^{C\log n}\ll n^{-10r_{2}+10}$ so that it
kills all polynomially growing factors.

Case 2. Both $|i-j|\leq C\log n$ and $|k-l|\leq C\log n$ but $|j-k|\geq C\log
n$, we may again apply decay of correlations but now the expectations
$\mathbb{E}(Y_{i}Y_{j})$ (and $\mathbb{E}(Y_{k}Y_{l})$) are nonzero. Using the
Cauchy-Schwartz inequality, any such term contributes by
$(\mathbb{E}(Y^{2}))^{2}\ll(\log R)^{2}$ (correlations give an $O(1)$
correction) hence the total contribution is

$n^{2}(\log R)^{2}(\log n)^{2}\ll n^{2}(\log R)^{4}\ll\ll nR^{2-r_{1}}(\log
R)^{4}\ll nR^{2}.$

Case 3. $|i-j|\leq C\log n$, $|k-l|\leq C\log n$ and $|j-k|\leq C\log n$. As
further bounds will rely on it, first let us estimate
$\mathbb{E}(Y_{i}^{2}Y_{j}^{2})=\mathbb{E}(Y^{2}Y_{j-i}^{2})$ . We again
distinguish two cases.

Case a, $i=j$. We only have $\mathbb{E}Y^{4}\ll\sum_{m=1}^{R}m^{4}m^{-3}\ll
R^{2}$.

Case b, $j>i$. Let

$M_{m}=\\{\omega\in M|\lfloor|v(\omega)|\rfloor=m\\}$ (6.1)

for some $m\geq 1$. Then $\mu(M_{m})\ll m^{-3}$ and (see [8, Formula (9.6)])

$\mu(M_{m}\cap T^{-(j-i)}M_{m^{\prime}})\ll m^{-9/4}(m^{\prime})^{-2}.$

This gives

$\mathbb{E}(Y^{2}Y_{j-i}^{2})\ll\sum_{m=1}^{R}\sum_{m^{\prime}=1}^{R}m^{2}(m^{\prime})^{2}m^{-9/4}(m^{\prime})^{-2}\ll
R^{3/4}R=R^{7/4}.$

Now let us get back to the investigation of Case 3 above. Let us consider
first the terms when $i=j$ and $k=l$. If, furthermore, $j=k$ then we get a
single term when all indices coincide contributing with

$\mathbb{E}(Y^{4})\ll R^{2},$

see above. Otherwise, if $i=j<k=l$, (yet $k-j<C\log n$) we may apply Case b
above to obtain

$\mathbb{E}(Y_{j}^{2}Y_{k}^{2})\ll R^{7/4}$

and as there are $\ll n\log n$ such terms (first fix $(i=)j$, then can pick
$k(=l)$ in $\ll\log n$ ways) their total contribution is

$\ll R^{7/4}n\log n\ll nR^{2}.$

Now if either $j>i$ or $l>k$ (yet the conditions of Case 3 remain valid),
first we apply a Cauchy-Schwartz to such terms and get

$\ll(\mathbb{E}(Y_{i}^{2}Y_{j}^{2}))^{1/2}\cdot(\mathbb{E}(Y_{k}^{2}Y_{l}^{2}))^{1/2}.$

As either $j>i$ or $l>k$, to at least one of the factors Case b applies and we
get $\ll R^{7/8}R=R^{15/8}$. the total contribution of such terms is

$nR^{15/8}(\log R)^{3}\ll nR^{2}.$

###  6.2 Verification of (HL1)

Recall that ${\widetilde{W}}=v\cdot\mathbf{1}_{d_{2^{n}}\leq|v|\leq
c_{2^{n}}}$ and that ${\widetilde{W}}_{l}={\widetilde{W}}\circ T^{l+r}$ for
$l\geq 1$ and $r$ fixed.

###  6.3 Verification of (HL1)(i)

By exponential decay of correlations, see [8, Proposition 9.1], the
contribution of cross terms is of $O(K)$, and thus (see also [8, Proof of
Lemma 11.1])

$\displaystyle\mathbb{E}_{\mu}\left((\mathcal{S}_{K})^{2}\right)\ll K\cdot
L\left(\frac{c_{2^{n}}}{d_{2^{n}}}\right)\ll K\cdot LL(n),$ (6.2)

as required.

###  6.4 Verification of (HL1)(ii)

Recall that $K$ is fixed so that $K<(d_{2^{n}})^{2}$ and that for
$k=1,\dots,K$ and ${\varepsilon}>0$,

$\mathcal{S}_{k}=\sum_{l=0}^{k}{\widetilde{W}}_{l};\qquad
B_{k,{\varepsilon}}=\\{x\in
M\,:\,|\mathcal{S}_{k}|\geq{\varepsilon}c_{2^{n}}\\}$

and
$N_{k,{\varepsilon}}=B_{k,{\varepsilon}}\setminus(\cup_{k^{\prime}=0}^{k-1}B_{k^{\prime},{\varepsilon}}).$

We need to show that $\sum_{k=1}^{K}\mu(N_{k,{\varepsilon}}\setminus
B_{K,{\varepsilon}/2})\leq C_{2}\frac{K\cdot LL(n)}{(c_{2^{n}})^{2}}.$ This
type of bound is essentially given in [8, pp.696–697] and [4, pp. 506–508],
nevertheless, we summarize the main ingredients here.

Note that the points $x\in N_{k,{\varepsilon}}\setminus B_{K,{\varepsilon}/2}$
simultaneously satisfy three conditions:

$|\mathcal{S}_{k}|\geq{\varepsilon}\cdot
c_{2^{n}};\qquad|\mathcal{S}_{K}-\mathcal{S}_{k}|\geq\frac{{\varepsilon}}{2}\cdot
c_{2^{n}};\qquad{\widetilde{W}}_{k}\neq 0.$ (6.3)

Let us consider the third condition: ${\widetilde{W}}(x)\neq 0$ implies that
$x\in M_{m}$ for some $m$ with $d_{2^{n}}\leq m\leq c_{2^{n}}$ (recall (6.1)
for the definition of $M_{m}$). Now we may use [9, Lemma 5.1] (see also [33,
Lemma 16]): there exist constants $p,q>0$ such that for any $C>0$ there exists
a subset ${\widetilde{M}}_{m}\subset M_{m}$ such that

$\displaystyle\mu(M_{m}\setminus{\widetilde{M}}_{m})\ll
m^{-p}\mu(M_{m});\quad\text{and}$ $\displaystyle\quad x\in{\widetilde{M}}_{m}\
\Rightarrow\ T^{i}x\not\in M_{r}\ \text{for}\ r>m^{1-q}\ \text{whenever}\
i=1,\dots C\cdot L(m).$

Since

$\mu\left(\cup_{m\geq
d_{2^{n}}}(M_{m}\setminus{\widetilde{M}}_{m})\right)\ll\sum_{m\geq
d_{2^{n}}}m^{-(3+p)}\ll(d_{2^{n}})^{-(2+p)}=o(c_{2^{n}}^{-2}),$

we may assume that whenever ${\widetilde{W}}_{k}\neq 0$, we have
$T^{(k+i_{n,j})}x\in{\widetilde{M}}_{m}$ for some $m$ with $d_{2^{n}}\leq
m\leq c_{2^{n}}$. However, this implies ${\widetilde{W}}_{k^{\prime}}=0$ for
$k^{\prime}=(k+1),\dots,k+Cn$ (recall $L(d_{2^{n}})\asymp n$). In terms of the
three conditions of (6.3) this implies that the contribution of
$\mu(N_{k,{\varepsilon}}\setminus B_{K,{\varepsilon}/2})$ can be disregarded
unless $K-k>Cn$. Furthermore, the values ${\widetilde{W}}_{k}\neq 0$ and
${\widetilde{W}}_{k^{\prime}}\neq 0$ for $k^{\prime}>k+Cn$ are strongly
decorrelated (by exponential decay of correlations, see eg. [4, Lemma 4.1])
and are of order $\gamma^{n}$ for some $\gamma<1$ that can be made arbitrary
small by choosing $C$ large. To conclude, let

$F_{k}=|\mathcal{S}_{k}|\cdot\mathbf{1}_{N_{k,{\varepsilon}}};\qquad
G_{k}=|\mathcal{S}_{K}-\mathcal{S}_{k}|.$

Then, by the Chebyshev inequality,

$\displaystyle\mu(N_{k,{\varepsilon}}\setminus B_{K,{\varepsilon}/2})\leq$
$\displaystyle\mu\left(F_{k}\geq{\varepsilon}c_{2^{n}}\ \text{and}\
G_{k}\geq\frac{{\varepsilon}}{2}c_{2^{n}}\right)$ $\displaystyle\leq$
$\displaystyle\,\mu\left(F_{k}\cdot
G_{k}\geq\frac{{\varepsilon}^{2}}{2}(c_{2^{n}})^{2}\right)$ $\displaystyle\ll$
$\displaystyle\,\frac{\mu(F_{k}^{2}G_{k}^{2})}{c_{2^{n}}^{4}}\ll\left[\mu(F_{k}^{2})\mu(G_{k}^{2})+\gamma^{-n}\right]/c_{2^{n}}^{4}$

and the contribution from $\gamma^{-n}$ is under control. Reasoning as in
(6.2), the second moment of each term in the sum defining $G_{k}$ is of
$O(LL(n))$. Thus,

$\mu(G_{k}^{2})\ll(K-k)\cdot LL(n)\ll KLL(n).$

To bound $\mu(F_{k}^{2})$, note that for $x\in N_{k,{\varepsilon}}$

$|\mathcal{S}_{k}(x)|\leq|\mathcal{S}_{k-1}(x)|+|{\widetilde{W}}_{k}(x)|\ll{\varepsilon}c_{2^{n}}+c_{2^{n}}\ll
c_{2^{n}}$

thus

$\mu(F_{k}^{2})\ll(c_{2^{n}})^{2}\cdot\mu(N_{k,{\varepsilon}})\ll(c_{2^{n}})^{2}\cdot(d_{2^{n}})^{-2}.$

Putting all these estimates together and recalling that $K\leq d_{2^{n}}^{2}$
by assumption,

$\sum_{k=1}^{K}\mu(N_{k,{\varepsilon}}\setminus B_{K,{\varepsilon}/2})\ll
K\cdot\frac{K\cdot LL(n)}{(c_{2^{n}})^{2}(d_{2^{n}})^{2}}\ll\frac{K\cdot
LL(n)}{(c_{2^{n}})^{2}},$

as required.

##  Appendix A The sequence $c_{n}$ defined in (1.4) satisfies (HL0)(iii)

The argument in [20, p. 1622] shows that given $c_{n}$ as in (1.4),
$\sum\limits_{k=1}^{\infty}(c_{k})^{-2}<\infty$. We need to show that
$\sum\limits_{n=1}^{\infty}\frac{2^{n}LL(n)}{(c_{2^{n}})^{2}}<\infty$.

Proceeding as in the argument in [20, p. 1622], a change of variables shows
that

$\displaystyle\sum_{n}\frac{2^{n}LL(n)}{(c_{2^{n}})^{2}}\ll\int_{1}^{\infty}\frac{y}{1+e^{y}\sin^{2}y}\,dy.$

We just need to argue that the above integral is finite, which can be seen as
follows. Let $U_{k}=B_{\frac{1}{(k\pi)^{3}}}(k\pi)$, $k\geq 1$ and set
$V_{k}=[(k-\frac{1}{2})\pi,(k+\frac{1}{2})\pi]\setminus U_{k}$. Note that

$\displaystyle\int_{\frac{\pi}{2}}^{\infty}\frac{y}{1+e^{y}\sin^{2}y}\,dy=\sum_{k}\left(\int_{U_{k}}+\int_{V_{k}}\right)\frac{y}{1+e^{y}\sin^{2}y}\,dy.$

On $V_{k}$ we have $\sin^{2}y\geq\left(\frac{1}{2k\pi}\right)^{6}$. Note that
$|V_{k}|=\pi$. Thus,
$\int_{V_{k}}\frac{y}{1+e^{y}\sin^{2}y}\,dy\leq\pi\frac{(2k\pi)^{7}}{e^{k\pi}}$,
which is sumable in $k$.

On $U_{k}$ we have $\sin^{2}y\geq 0$. Note that
$|U_{k}|=\frac{2}{(k\pi)^{3}}$. Thus,
$\int_{U_{k}}\frac{y}{1+e^{y}\sin^{2}y}\,dy\leq\frac{4k\pi}{(k\pi)^{3}}$,
which is also sumable in $k$.

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