# Quenched law of large numbers and quenched central limit theorem for multi-
player leagues with ergodic strengths

Jacopo Borga<EMAIL_ADDRESS>Institut für Mathematik, Universität
Zürich  Benedetta Cavalli<EMAIL_ADDRESS>Institut für
Mathematik, Universität Zürich

###### Abstract

We propose and study a new model for competitions, specifically sports multi-
player leagues where the initial strengths of the teams are independent i.i.d.
random variables that evolve during different days of the league according to
independent ergodic processes. The result of each match is random: the
probability that a team wins against another team is determined by a function
of the strengths of the two teams in the day the match is played.

Our model generalizes some previous models studied in the physical and
mathematical literature and is defined in terms of different parameters that
can be statistically calibrated. We prove a quenched – conditioning on the
initial strengths of the teams – law of large numbers and a quenched central
limit theorem for the number of victories of a team according to its initial
strength.

To obtain our results, we prove a theorem of independent interest. For a
stationary process $\bm{\xi}=(\bm{\xi}_{i})_{i\in\mathbb{N}}$ satisfying a
mixing condition and an independent sequence of i.i.d. random variables
$(\bm{s}_{i})_{i\in\mathbb{N}}$, we prove a quenched – conditioning on
$(\bm{s}_{i})_{i\in\mathbb{N}}$ – central limit theorem for sums of the form
$\sum_{i=1}^{n}g\left(\bm{\xi}_{i},\bm{s}_{i}\right)$, where $g$ is a bounded
measurable function. We highlight that the random variables
$g\left(\bm{\xi}_{i},\bm{s}_{i}\right)$ are not stationary conditioning on
$(\bm{s}_{i})_{i\in\mathbb{N}}$.

##  1 Introduction

Multi-player competitions are a recurrent theme in physics, biology,
sociology, and economics since they model several phenomena. We refer to the
introduction of [BNH07] for various examples of models of multi-player
competitions related to evolution, social stratification, business world and
other related fields.

Among all these fields, sports are indubitably one of the most famous examples
where modelling competitions is important. This is the case for two reasons:
first, there is accurate and easily accessible data; second, sports
competitions are typically head-to-head, and so they can be viewed as a nice
model of multi-player competitions with binary interactions.

Sports multi-player leagues where the outcome of each game is completely
deterministic have been studied for instance by Ben-Naim, Kahng, and Kim
[BNKK06] (see also references therein and explanations on how this model is
related to urn models). Later, Ben-Naim and Hengartner [BNH07] investigated
how randomness affects the outcome of multi-player competitions. In their
model, the considered randomness is quite simple: they start with $N$ teams
ranked from best to worst and in each game they assume that the weaker team
can defeat the stronger team with a fixed probability. They investigate the
total number of games needed for the best team to win the championship with
high probability.

A more sophisticated model was studied by Benaïm, Benjamini, Chen, and Lima
[BBCL15]: they consider a finite connected graph, and place a team at each
vertex of the graph. Two teams are called a _pair_ if they share an edge. At
discrete times, a match is played between each pair of teams. In each match,
one of the teams defeats the other (and gets a point) with probability
proportional to its current number of points raised to some fixed power
$\alpha>0$. They characterize the limiting behavior of the proportion of
points of the teams.

In this paper we propose a more realistic model for sport multi-player leagues
that can be briefly described as follows (for a precise and formal description
see Section 1.1): we start with $2n$ teams having i.i.d. initial strengths.
Then we consider a calendar of the league composed by $2n-1$ different days,
and we assume that on each day each team plays exactly one match against
another team in such a way that at the end of the calendar every team has
played exactly one match against every other team (this coincides with the
standard way how calendars are built in real football-leagues for the first
half of the season). Moreover, we assume that on each day of the league, the
initial strengths of the teams are modified by independent ergodic processes.
Finally, we assume that a team wins against another team with probability
given by a chosen function of the strengths of the two teams in the day the
match is played.

We prove (see Section 1.4 for precise statements) a quenched law of large
numbers and a quenched central limit theorem for the number of victories of a
team according to its initial strength. Here quenched means that the results
hold a.s. conditioning on the initial strengths of the teams in the league.

###  1.1 The model

We start by fixing some simple and standard notation.

Notation. Given $n\in\mathbb{N}=\\{1,2,3,\dots\\}$, we set
$[n]=\\{1,2,\dots,n\\}$ and $[n]_{0}=\\{0,1,2,\dots,n\\}$. We also set
$\mathbb{R}_{+}=\mathbb{R}\cap[0,\infty)$.

We refer to random quantities using bold characters. Convergence in
distribution is denoted by $\xrightarrow{d}$, almost sure convergence is
denote by $\xrightarrow{a.s.}$, and convergence in probability by
$\xrightarrow{P}$.

Given a collection of sets $\mathcal{A}$, we denote by
$\sigma\left(\mathcal{A}\right)$ and $\lambda\left(\mathcal{A}\right)$ the
$\sigma$-algebra and the monotone class generated by $\mathcal{A}$,
respectively. Given a random variable $\bm{X}$, we denote by
$\sigma\left(\bm{X}\right)$ the $\sigma$-algebra generated by $\bm{X}$.

For any real-valued function $f$ we denote with $f^{2}$ the function such that
$f^{2}(\cdot)=(f(\cdot))^{2}$.

We can now proceed with the precise description of our model for multi-player
leagues.

The model. We consider a league of $2n\in 2\mathbb{N}$ teams denoted by
$\\{T_{i}\\}_{i\in[2n-1]_{0}}$ whose _initial random strengths_ are denoted by
$\\{\bm{s}_{i}\\}_{i\in[2n-1]_{0}}\in\mathbb{R}_{+}^{2n}$.

In the league every team $T_{i}$ plays $2n-1$ matches, one against each of the
remaining teams $\\{T_{j}\\}_{j\in[2n-1]_{0}\setminus\\{i\\}}$. Note that
there are in total ${2n}\choose{2}$ matches in the league. These matches are
played in $2n-1$ different days in such a way that each team plays exactly one
match every day.

For all $i\in[2n-1]_{0}$, the initial strength $\bm{s}_{i}$ of the team
$T_{i}$ is modified every day according to a discrete time
$\mathbb{R}_{+}$-valued stochastic process
$\bm{\xi}^{i}=(\bm{\xi}^{i}_{j})_{j\in\mathbb{N}}$. More precisely, the
strength of the team $T_{i}$ on the $p$-th day is equal to
$\bm{s}_{i}\cdot\bm{\xi}^{i}_{p}\in\mathbb{R}_{+}$.

We now describe the _rules_ for determining the winner of a match in the
league. We fix a function $f:\mathbb{R}_{+}^{2}\to[0,1]$ that controls the
winning probability of a match between two teams given their strengths. When a
team with strength $x$ plays a match against another team with strength $y$,
its probability of winning the match is equal to $f(x,y)$ and its probability
of loosing is equal to $1-f(x,y)$ (we are excluding the possibility of having
a draw). Therefore, if the match between the teams $T_{i}$ and $T_{j}$ is
played the $p$-th day, then, conditionally on the random variables
$\bm{s}_{i},\bm{s}_{j},\bm{\xi}^{i}_{p},\bm{\xi}^{j}_{p}$, the probability
that $T_{i}$ wins is
$f(\bm{s}_{i}\cdot\bm{\xi}^{i}_{p},\bm{s}_{j}\cdot\bm{\xi}^{j}_{p})$.
Moreover, conditionally on the strengths of the teams, the results of
different matches are independent.

###  1.2 Goal of the paper

The goal of this work is to study the model defined above when the number $2n$
of teams in the league is large. We want to look at the limiting behavior of
the number of wins of a team with initial strength $s\in\mathbb{R}_{+}$ at the
end of the league. More precisely, given $s\in\mathbb{R}_{+}$, we assume
w.l.o.g. that the team $T_{0}$ has deterministic initial strength $s$, i.e.
$\bm{s}_{0}=s$ a.s., and we set

$\bm{W}_{n}(s)\coloneqq\text{Number of wins of the team }T_{0}\text{ at the
end of a league with $2n$ players}.$ (1)

We investigate a quenched law of large numbers and a quenched central limit
theorem for $\bm{W}_{n}(s)$.

###  1.3 Our assumptions

In the following two subsections we state some assumptions on the model.

####  1.3.1 Assumptions for the law of large numbers

We make the following natural assumptions111The second hypothesis is not
needed to prove our results but it is very natural for the model. on the
function $f:\mathbb{R}_{+}^{2}\to[0,1]\>$:

  * •

$f(x,y)$ is measurable;

  * •

$f(x,y)$ is weakly-increasing in the variable $x$ and weakly-decreasing in the
variable $y$.

Recall also that it is not possible to have a draw, i.e. $f(x,y)+f(y,x)=1$,
for all $x,y\in\mathbb{R}_{+}$.

Before describing our additional assumptions on the model, we introduce some
further quantities. Fix a Borel probability measure $\nu$ on $\mathbb{R}_{+}$;
let $\bm{\xi}=(\bm{\xi}_{\ell})_{\ell\in\mathbb{N}}$ be a discrete time
$\mathbb{R}_{+}$-valued stochastic process such that

$\bm{\xi}_{\ell}\stackrel{{\scriptstyle d}}{{=}}\nu,\quad\text{for
all}\quad\ell\in\mathbb{N},$ (2)

and222This is a weak-form of the _stationarity property_ for stochastic
processes.

$\left(\bm{\xi}_{\ell},\bm{\xi}_{k}\right)\stackrel{{\scriptstyle
d}}{{=}}\left(\bm{\xi}_{\ell+\tau},\bm{\xi}_{k+\tau}\right),\quad\text{for
all}\quad\ell,k,\tau\in\mathbb{N}.$ (3)

We further assume that the process $\bm{\xi}$ is _weakly-mixing_ , that is,
for every $A\in\sigma(\bm{\xi}_{1})$ and every collection of sets
$B_{\ell}\in\sigma(\bm{\xi}_{\ell})$, it holds that

$\frac{1}{n}\sum_{\ell=1}^{n}\left|\mathbb{P}(A\cap
B_{\ell})-\mathbb{P}(A)\mathbb{P}(B_{\ell})\right|\xrightarrow{n\to\infty}0.$
(4)

The additional assumptions on our model are the following:

  * •

For all $i\in[2n-1]_{0}$, the stochastic processes $\bm{\xi}^{i}$ are
independent copies of $\bm{\xi}$.

  * •

The initial random strengths $\\{\bm{s}_{i}\\}_{i\in[2n-1]}$ of the teams
different than $T_{0}$ are i.i.d. random variables on $\mathbb{R}_{+}$ with
distribution $\mu$, for some Borel probability measure $\mu$ on
$\mathbb{R}_{+}$.

  * •

The initial random strengths $\\{\bm{s}_{i}\\}_{i\in[2n-1]}$ are independent
of the processes $\\{\bm{\xi}^{i}\\}_{i\in[2n-1]_{0}}$ and of the process
$\bm{\xi}$.

####  1.3.2 Further assumptions for the central limit theorem

In order to prove a central limit theorem, we need to make some stronger
assumptions. The first assumption concerns the mixing properties of the
process $\bm{\xi}$. For $k\in\mathbb{N}$, we introduce the two
$\sigma$-algebras
$\mathcal{A}_{1}^{k}=\sigma\left(\bm{\xi}_{1},\dots,\bm{\xi}_{k}\right)$ and
$\mathcal{A}_{k}^{\infty}=\sigma\left(\bm{\xi}_{k},\dots\right)$ and we define
for all $n\in\mathbb{N}$,

$\alpha_{n}=\sup_{\begin{subarray}{c}k\in\mathbb{N}\\\
A\in\mathcal{A}_{1}^{k},B\in\mathcal{A}_{k+n}^{\infty}\end{subarray}}\left|\mathbb{P}\left(A\cap
B\right)-\mathbb{P}(A)\mathbb{P}(B)\right|.$ (5)

We assume that

$\sum_{n=1}^{\infty}\alpha_{n}<\infty.$ (6)

Note that this condition, in particular, implies that the process $\bm{\xi}$
is _strongly mixing_ , that is, $\alpha_{n}\to 0$ as $n\to\infty$.

Finally, we assume that there exist two sequences $p=p(n)$ and $q=q(n)$ such
that:

  * •

$p\xrightarrow{n\to\infty}+\infty$ and $q\xrightarrow{n\to\infty}+\infty$,

  * •

$q=o(p)$ and $p=o(n)$ as $n\to\infty$,

  * •

$np^{-1}\alpha_{q}=o(1)$,

  * •

$\frac{p}{n}\cdot\sum_{j=1}^{p}j\alpha_{j}=o(1)$.

######  Remark 1.1.

The latter assumption concerning the existence of the sequences $p$ and $q$ is
not very restrictive. For example, simply assuming that
$\alpha_{n}=O(\frac{1}{n\log(n)})$ ensures that the four conditions are
satisfied for $p=\frac{\sqrt{n}}{\log\log n}$ and $q=\frac{\sqrt{n}}{(\log\log
n)^{2}}$. Indeed, in this case, the first two conditions are trivial, the
fourth one follows by noting that $\sum_{j=1}^{p}j\alpha_{j}=O(p)$ thanks to
the assumption in Eq. 6, and finally the third condition follows by standard
computations.

######  Remark 1.2.

Note also that as soon as $p=O(\sqrt{n})$ then the fourth condition is
immediately verified. Indeed,
$\sum_{j=1}^{p}j\alpha_{j}\leq\sqrt{p}\sum_{j=1}^{\sqrt{p}}\alpha_{j}+p\sum_{j=\sqrt{p}}^{p}\alpha_{j}=o(p)$.

###  1.4 Main results

####  1.4.1 Results for our model of multi-player leagues

Let $\bm{V},\bm{V^{\prime}},\bm{U},\bm{U}^{\prime}$ be four independent random
variables such that $\bm{V}\stackrel{{\scriptstyle
d}}{{=}}\bm{V^{\prime}}\stackrel{{\scriptstyle d}}{{=}}\nu$ and
$\bm{U}\stackrel{{\scriptstyle d}}{{=}}\bm{U}^{\prime}\stackrel{{\scriptstyle
d}}{{=}}\mu$. Given a deterministic sequence
$\vec{s}=(s_{i})_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$, denote by
$\mathbb{P}_{\vec{s}}$ the law of the random variable
$\frac{\bm{W}_{n}(s)}{2n}$ when the initial strengths of the teams
$(T_{i})_{i\in[2n-1]}$ are equal to $\vec{s}=(s_{i})_{{i\in[2n-1]}}$, i.e. we
study $\frac{\bm{W}_{n}(s)}{2n}$ on the event

$\bm{s}_{0}=s\quad\text{ and
}\quad(\bm{s}_{i})_{{i\in[2n-1]}}=(s_{i})_{{i\in[2n-1]}}.$

######  Theorem 1.3.

(Quenched law of large numbers) Suppose that the assumptions in Section 1.3.1
hold. Fix any $s\in\mathbb{R}_{+}$. For $\mu^{\mathbb{N}}$-almost every
sequence $\vec{s}=(s_{i})_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$,
under $\mathbb{P}_{\vec{s}}$ the following convergence holds

$\frac{\bm{W}_{n}(s)}{2n}\xrightarrow[n\to\infty]{P}\ell(s),$ (7)

where

$\ell(s)=\mathbb{E}\left[f\left(s\cdot\bm{V},\bm{U}\cdot\bm{V^{\prime}}\right)\right]=\int_{\mathbb{R}^{3}_{+}}f\left(s\cdot
v,u\cdot v^{\prime}\right)d\nu(v)d\nu(v^{\prime})d\mu(u).$ (8)

We now state our second result.

######  Theorem 1.4.

(Quenched central limit theorem) Suppose that the assumptions in Section 1.3.1
and Section 1.3.2 hold. Fix any $s\in\mathbb{R}_{+}$. For
$\mu^{\mathbb{N}}$-almost every sequence
$\vec{s}=(s_{i})_{i\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}}_{+}$, under
$\mathbb{P}_{\vec{s}}$ the following convergence holds

$\frac{\bm{W}_{n}(s)-\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]}{\sqrt{2n}}\xrightarrow{d}\bm{\mathcal{N}}\left(0,\sigma(s)^{2}+\rho(s)^{2}\right),$
(9)

where, for $F_{s}(x,y)\coloneqq\mathbb{E}\left[f\left(s\cdot
x,y\cdot\bm{V}^{\prime}\right)\right]$ and
$\tilde{F}_{s}\left(x,y\right)\coloneqq
F_{s}\left(x,y\right)-\mathbb{E}[F_{s}\left(\bm{V},y\right)]$,

$\sigma(s)^{2}=\mathbb{E}\left[F_{s}(\bm{V},\bm{U})-\left(F_{s}(\bm{V},\bm{U})\right)^{2}\right]=\ell(s)-\mathbb{E}\left[\left(F_{s}(\bm{V},\bm{U})\right)^{2}\right]$
(10)

and

$\rho(s)^{2}=\mathbb{E}\left[\tilde{F}_{s}(\bm{V},\bm{U})^{2}\right]+2\cdot\sum_{k=1}^{\infty}\mathbb{E}\left[\tilde{F}_{s}(\bm{\xi}_{1},\bm{U})\tilde{F}_{s}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right],$
(11)

the last series being convergent.

######  Remark 1.5.

The assumption that the initial strengths of the teams
$(\bm{s}_{i})_{i\in\mathbb{N}}$ are independent could be relaxed from a
theoretical point of view, but we believe that this is a very natural
assumption for our model.

####  1.4.2 Originality of our results and analysis of the limiting variance:
a quenched CLT for functionals of ergodic processes and i.i.d. random
variables

We now comment on the originality of our results and contextualize them in
relation to the established literature and prior studies on sums of dependent
and not equi-distributed random variables. Additionally, we give some informal
explanations on the two components $\sigma(s)^{2}$ and $\rho(s)^{2}$ of the
variance of the limiting Gaussian random variable in Eq. 9.

We start by noticing that, without loss of generality, we can assume that for
every $j\in[2n-1]$, the team $T_{0}$ plays against the team $T_{j}$ the $j$-th
day of the league. Denoting by $W_{j}=W_{j}(s)$ the event that the team
$T_{0}$ wins against the team $T_{j}$, then $\bm{W}_{n}(s)$ rewrites as

$\bm{W}_{n}(s)=\sum_{j=1}^{2n-1}\mathds{1}_{W_{j}}.$ (12)

Note that the Bernoulli random variables $(\mathds{1}_{W_{j}})_{j\in[2n-1]}$
are only independent conditionally on the process
$(\bm{\xi}^{0}_{j})_{j\in[2n-1]}$. In addition, under our assumptions, we have
that the conditional parameters of the Bernoulli random variables are given by
$\mathbb{P}_{\vec{s}}\left(W_{j}\middle|\bm{\xi}^{0}_{j}\right)=F_{s}(\bm{\xi}^{0}_{j},s_{j})$.
Therefore, under $\mathbb{P}_{\vec{s}}$, the random variable $\bm{W}_{n}(s)$
is a sum of Bernoulli random variables that are _neither independent nor
identically distributed_. As a consequence, the proofs of the quenched law of
large numbers (see Section 2) and of the quenched central limit theorem (see
Section 3) do not follow from a simple application of classical results in the
literature.

We recall that it is quite simple to relax the identically distributed
assumption in order to obtain a central limit theorem: the Lindeberg
criterion, see for instance [Bil95, Theorem 27.2], gives a sufficient (and
almost necessary) criterion for a sum of independent random variables to
converge towards a Gaussian random variable (after suitable renormalization).
Relaxing independence is more delicate and there is no universal theory to do
it. In the present paper, we combine two well-known theories to obtain our
results: the theory for stationary ergodic processes (see for instance [Ros56,
Ibr62, IL71, Pel92, Bra07]) and the theory for $m$-dependent random variables
(see for instance [HR48, Ber73, RW00]) and dependency graphs (see for instance
[PL82, Jan88, BR89, Jan04, FMN20]).

To explain the presence of the two terms in the variance, we start by noticing
that using the law of total conditional variance, the conditional variance of
$\mathds{1}_{W_{j}}$ is given by
$\operatorname{Var}_{\vec{s}}\left(\mathds{1}_{W_{j}}\middle|\bm{\xi}^{0}_{j}\right)=F_{s}(\bm{\xi}^{0}_{j},s_{j})-(F_{s}(\bm{\xi}^{0}_{j},s_{j}))^{2}$.
The term $\sigma(s)^{2}$ arises as the limit of

$\frac{1}{2n}\sum_{j=1}^{2n-1}\operatorname{Var}_{\vec{s}}\left(\mathds{1}_{W_{j}}\middle|\bm{\xi}^{0}_{j}\right),$

and this is in analogy with the case of sums of independent but not
identically distributed Bernoulli random variables. The additional term
$\rho(s)^{2}$, on the contrary, arises from the fluctuations of the
conditional parameters of the Bernoulli variables, i.e. $\rho(s)^{2}$ comes
from the limit of

$\frac{1}{2n}\sum_{j=1}^{2n-1}\operatorname{Var}\left(\mathbb{P}_{\vec{s}}\left(W_{j}\middle|\bm{\xi}^{0}_{j}\right)\right)=\frac{1}{2n}\sum_{j=1}^{2n-1}\operatorname{Var}\left(F_{s}(\bm{\xi}^{0}_{j},s_{j})\right).$
(13)

Note that an additional difficulty arises from the fact that the sums in the
last two equations are not independent (but we will show that they are
asymptotically independent).

To study the limit in Eq. 13 we prove in Section 3 the following general
result that we believe to be of independent interest.

######  Theorem 1.6.

Suppose that the assumptions in Eqs. 2 and 3, and in Section 1.3.2 hold. Let
$g:\mathbb{R}^{2}_{+}\to\mathbb{R}_{+}$ be a bounded, measurable function, and
define $\tilde{g}:\mathbb{R}^{2}_{+}\to\mathbb{R}$ by $\tilde{g}(x,y)\coloneqq
g(x,y)-\mathbb{E}\left[g\left(\bm{V},y\right)\right]$. Then, the quantity

$\rho_{g}^{2}\coloneqq\mathbb{E}\left[\tilde{g}(\bm{V},\bm{U})^{2}\right]+2\cdot\sum_{k=1}^{\infty}\mathbb{E}\left[\tilde{g}(\bm{\xi}_{1},\bm{U})\tilde{g}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right]$
(14)

is finite. Moreover, for $\mu^{\mathbb{N}}$-almost every sequence
$(s_{i})_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$, the following
convergence holds

$\frac{\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)}{\sqrt{2n}}\xrightarrow{d}\bm{\mathcal{N}}(0,\rho_{g}^{2}).$
(15)

The main difficulty in studying the sum
$\sum_{j=1}^{2n-1}g\left(\bm{\xi}_{j},s_{j}\right)$ is that fixing a
deterministic realization
$(s_{i})_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$ imposes that the
random variables $g\left(\bm{\xi}_{j},s_{j}\right)$ are not stationary
anymore.

In many of the classical results available in the literature (see references
given above) – in addition to moment and mixing conditions – stationarity is
assumed, and so we cannot directly apply these results. An exception are
[PRW97, Eks14], where stationarity is not assumed but it is assumed a stronger
version333This is assumption (B3) in [Eks14] that is also assumed in [PRW97].
Since in our case all moments of $g\left(\bm{\xi}_{j},s_{j}\right)$ are finite
(because $g$ is a bounded function), we can take the parameter $\delta$ in
[Eks14] equal to $+\infty$ and thus condition (B3) in [Eks14] requires that
$\sum_{n=1}^{\infty}n^{2}\alpha_{n}<+\infty$. of our condition in Eq. 6.
Therefore, to the best of our knowledge, 1.6 does not follow from known
results in the literature.

###  1.5 Calibration of the parameters of the model and some examples

An interesting feature of our model consists in the fact that the main
parameters that characterize the evolution of the league can be statistically
calibrated in order for the model to describe real-life tournaments. Such
parameters are:

  * •

The function $f:\mathbb{R}_{+}^{2}\to[0,1]$ that controls the winning
probability of the matches.

  * •

The distribution $\mu$ of the initial strengths $(\bm{s}_{i})_{i}$ of the
teams.

  * •

The marginal distribution $\nu$ of the tilting process $\bm{\xi}$.

We end this section with two examples. The first one is more theoretical and
the second one more related to the statistical calibration.

######  Example 1.7.

We assume that:

  * •

$f(x,y)=\frac{x}{x+y}$;

  * •

for all $i\in\mathbb{N}$, the initial strengths $\bm{s}_{i}$ are uniformly
distributed in $[0,1]$, i.e. $\mu$ is the Lebesgue measure on $[0,1]$;

  * •

the tilting process $\bm{\xi}$ is a Markov chain with state space $\\{a,b\\}$
for some $a\in(0,1),b\in(1,\infty)$, with transition matrix
$\begin{pmatrix}p_{a}&1-p_{a}\\\ 1-p_{b}&p_{b}\end{pmatrix}$ for some
$p_{a},p_{b}\in(0,1)$. Note that the invariant measure $\nu=(\nu_{a},\nu_{b})$
is equal to
$\left(\frac{1-p_{b}}{2-p_{a}-p_{b}},\frac{1-p_{a}}{2-p_{a}-p_{b}}\right)$.

Under this assumptions, 1.3 guarantees that for any fixed $s\in\mathbb{R}_{+}$
and for almost every realization $\vec{s}=(s_{i})_{{i\in\mathbb{N}}}$ of the
random sequence $(\bm{s}_{i})_{{i\in\mathbb{N}}}\in[0,1]^{\mathbb{N}}$, the
following convergence holds under $\mathbb{P}_{\vec{s}}$:

$\frac{\bm{W}_{n}(s)}{2n}\xrightarrow[n\to\infty]{P}\ell(s),$ (16)

where

$\ell(s)=\int_{\mathbb{R}^{3}_{+}}\frac{sv}{sv+uv^{\prime}}d\nu(v)d\nu(v^{\prime})d\mu(u)=\sum_{i,j\in\\{a,b\\}}\frac{s\cdot
i}{j}\log\left(1+\frac{j}{s\cdot i}\right)\nu_{i}\cdot\nu_{j}.$ (17)

In particular if $a=\frac{1}{2},b=2,p_{a}=\frac{1}{2},p_{b}=\frac{1}{2}$ then

$\ell(s)=\frac{s}{16}\log\left(\frac{(1+s)^{8}(4+s)(1+4s)^{16}}{2^{32}\cdot
s^{25}}\right),$ (18)

whose graph is plotted in Fig. 1.

Figure 1: The graph of the function $\ell(s)$ in Eq. 18 for $s\in[0,1]$.

In addition, 1.4 implies that for any $s\in\mathbb{R}_{+}$ and for almost
every realization $\vec{s}=(s_{i})_{{i\in\mathbb{N}}}$ of the random sequence
$(\bm{s}_{i})_{{i\in\mathbb{N}}}\in[0,1]^{\mathbb{N}}$, the following
convergence also holds under $\mathbb{P}_{\vec{s}}$:

$\frac{\bm{W}_{n}(s)-\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]}{\sqrt{2n}}\xrightarrow{d}\bm{\mathcal{N}}(0,\sigma(s)^{2}+\rho(s)^{2}),$
(19)

where $\sigma(s)^{2}$ and $\rho(s)^{2}$ can be computed as follows. From Eqs.
10 and 11 we know that

$\sigma(s)^{2}=\ell(s)-\mathbb{E}\left[\left(F_{s}(\bm{V},\bm{U})\right)^{2}\right]$
(20)

and that

$\rho(s)^{2}=\mathbb{E}\left[\tilde{F}_{s}(\bm{V},\bm{U})^{2}\right]+2\cdot\sum_{k=1}^{\infty}\mathbb{E}\left[\tilde{F}_{s}(\bm{\xi}_{1},\bm{U})\tilde{F}_{s}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right].$
(21)

Recall that $\ell(s)$ was computed in Eq. 17. Note that

$F_{s}(x,y)=\mathbb{E}\left[f\left(s\cdot
x,y\cdot\bm{V}^{\prime}\right)\right]=\sum_{v^{\prime}\in\\{a,b\\}}f\left(s\cdot
x,y\cdot v^{\prime}\right)\nu_{v^{\prime}}$

and that

$\tilde{F}_{s}(x,y)=F_{s}(x,y)-\mathbb{E}\left[F_{s}(\bm{V},y)\right]=\sum_{v^{\prime}\in\\{a,b\\}}f\left(s\cdot
x,y\cdot
v^{\prime}\right)\nu_{v^{\prime}}-\sum_{(v,v^{\prime})\in\\{a,b\\}^{2}}f\left(s\cdot
v,y\cdot v^{\prime}\right)\nu_{v^{\prime}}\nu_{v}.$

Therefore

$\mathbb{E}\left[\left(F_{s}(\bm{V},\bm{U})\right)^{2}\right]=\sum_{v\in\\{a,b\\}}\left(\int_{0}^{1}\left(\sum_{v^{\prime}\in\\{a,b\\}}f\left(s\cdot
v,u\cdot v^{\prime}\right)\nu_{v^{\prime}}\right)^{2}du\right)\nu_{v}$ (22)

and

$\mathbb{E}\left[\left(\tilde{F}_{s}(\bm{V},\bm{U})\right)^{2}\right]=\sum_{w\in\\{a,b\\}}\left(\int_{0}^{1}\left(\sum_{v^{\prime}\in\\{a,b\\}}f\left(s\cdot
w,u\cdot
v^{\prime}\right)\nu_{v^{\prime}}-\sum_{(v,v^{\prime})\in\\{a,b\\}^{2}}f\left(s\cdot
v,u\cdot v^{\prime}\right)\nu_{v^{\prime}}\nu_{v}\right)^{2}du\right)\nu_{v}.$
(23)

Moreover,

$\sum_{k=1}^{\infty}\mathbb{E}\left[\tilde{F}_{s}(\bm{\xi}_{1},\bm{U})\tilde{F}_{s}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right]=\sum_{k=1}^{\infty}\sum_{(i,j)\in\\{a,b\\}^{2}}\mathbb{P}(\bm{\xi}_{1}=i,\bm{\xi}_{1+k}=j)\int_{0}^{1}\tilde{F}_{s}(i,u)\;du\int_{0}^{1}\tilde{F}_{s}(j,u^{\prime})\;du^{\prime}.$
(24)

It remains to compute $\mathbb{P}(\bm{\xi}_{1}=i,\bm{\xi}_{1+k}=j)$ for
$i,j\in\\{a,b\\}$. Note that

$\begin{pmatrix}p_{a}&1-p_{a}\\\
1-p_{b}&p_{b}\end{pmatrix}=\begin{pmatrix}1&\frac{1-p_{a}}{p_{b}-1}\\\
1&1\end{pmatrix}\begin{pmatrix}1&0\\\
0&(p_{a}+p_{b}-1)\end{pmatrix}\begin{pmatrix}\frac{p_{b}-1}{p_{a}+p_{b}-2}&\frac{p_{a}-1}{p_{a}+p_{b}-2}\\\
\frac{1-p_{b}}{p_{a}+p_{b}-2}&\frac{p_{b}-1}{p_{a}+p_{b}-2}\end{pmatrix}\eqqcolon
SJS^{-1}.$ (25)

Therefore,

$\displaystyle\mathbb{P}(\bm{\xi}_{1}=a,\bm{\xi}_{1+k}=a)=\frac{p_{b}-1+(p_{a}-1)(p_{a}+p_{b}-1)^{k}}{p_{a}+p_{b}-2}\cdot\nu_{a},$
(26)
$\displaystyle\mathbb{P}(\bm{\xi}_{1}=a,\bm{\xi}_{1+k}=b)=\frac{(p_{a}-1)((p_{a}+p_{b}-1)^{k}-1)}{p_{a}+p_{b}-2}\cdot\nu_{a},$
(27)
$\displaystyle\mathbb{P}(\bm{\xi}_{1}=b,\bm{\xi}_{1+k}=a)=\frac{(p_{b}-1)((p_{a}+p_{b}-1)^{k}-1)}{p_{a}+p_{b}-2}\cdot\nu_{b},$
(28)
$\displaystyle\mathbb{P}(\bm{\xi}_{1}=b,\bm{\xi}_{1+k}=b)=\frac{p_{a}-1+(p_{b}-1)(p_{a}+p_{b}-1)^{k}}{p_{a}+p_{b}-2}\cdot\nu_{b}.$
(29)

With some tedious but straightforward computations444We developed a
_Mathematica_ software to quickly make such computations for various choices
of the function $f(x,y)$, and of the parameters $a,b,p_{a},p_{b}$. The
software is available at the following link., we can explicitly compute
$\sigma(s)^{2}$ and $\rho(s)^{2}$. The graphs of the two functions
$\sigma(s)^{2}$ and $\rho(s)^{2}$ for $s\in[0,1]$ are plotted in Fig. 2 for
three different choices of the parameters $a,b,p_{a},p_{b}$. It is interesting
to note that $\sigma(s)^{2}$ is much larger than $\rho(s)^{2}$ when $p_{a}$
and $p_{b}$ are small, $\sigma(s)^{2}$ is comparable with $\rho(s)^{2}$ when
$p_{a}$ and $p_{b}$ are around 0.9, and $\rho(s)^{2}$ is much larger than
$\sigma(s)^{2}$ when $p_{a}$ and $p_{b}$ are very close to 1.

Figure 2: In green the graph of $\sigma(s)^{2}$. In red the graph of
$\rho(s)^{2}$. In blue the graph of $\sigma(s)^{2}+\rho(s)^{2}$. Left: The
parameters of the model are $a=1/2,b=2,p_{a}=2/5,p_{b}=2/5$. Middle: The
parameters of the model are $a=1/2,b=2,p_{a}=92/100,p_{b}=92/100$. Right: The
parameters of the model are $a=1/2,b=2,p_{a}=99/100,p_{b}=99/100$.

######  Example 1.8.

We collect here some data related to the Italian national basketball league in
order to compare some real data with our theoretical results. We believe that
it would be interesting to develop a more accurate and precise analysis of
real data in some future projects.

In Fig. 3, the rankings of the last 22 national leagues played among exactly
16 teams in the league are shown (some leagues, like the 2011-12 league, are
not tracked since in those years the league was not formed by 16 teams).

Figure 3: The rankings of the last 22 national italian basket leagues played
with exactly 16 teams in the league. In the Italian basket league every team
plays two matches against every other team and every victory gives two points.

The mean number of points of the 22 collected leagues are (from the team
ranked first to the team ranked 16-th):

  1. 1.

47,45

  2. 2.

42,27

  3. 3.

40,18

  4. 4.

37,59

  5. 5.

35,91

  6. 6.

34,09

  7. 7.

31,73

  8. 8.

30,55

  9. 9.

29,18

  10. 10.

27,77

  11. 11.

26,09

  12. 12.

24,36

  13. 13.

22,00

  14. 14.

19,59

  15. 15.

17,82

  16. 16.

12,41

The diagram of this ranking is given in the left-hand side of Fig. 4.

As mentioned above, an interesting question consists in assessing whether it
is possible to describe the behaviour of these leagues by using our model.
More precisely, we looked for a function $f(x,y)$ and two distributions $\mu$
and $\nu$ such that the graph of $\ell(s)$ for $s\in[0,1]$ can well
approximate the graph in the left-hand side of Fig. 4. We find out that
choosing $\mu$ to be the uniform measure on the interval $[0.1,0.999]$,
$\nu=0.6\cdot\delta_{0.25}+0.9\cdot\delta_{1.3},$ and

$f(x,y)\coloneqq\frac{g(x)}{g(x)+g(y)},\quad\text{with}\quad
g(x)\coloneqq\log\left(1-\min\left\\{\frac{x}{1.3},0.999\right\\}\right),$
(30)

then the graph of $\ell(s)$ for $s\in[0.1,0.999]$, is the one given in the
right-hand side of Fig. 4. Note that the two graphs have a similar convexity.

Figure 4: Left: The diagram of the mean number of points of the 22 leagues
collected in Fig. 3. On the $x$-axis the teams are ranked from the weakest to
the strongest; on the $y$-axis the mean number of points are plotted. Right:
The graph of $\ell(s)$ for $s\in[0.1,0.999]$ for the specific $\mu,\nu$ and
$f(x,y)$ given in (and before) Eq. 30.

###  1.6 Open problems

We collect some open questions and conjectures that we believe might be
interesting to investigate in future projects:

  * •

Conditioning on the initial strengths of the teams, how many times do we need
to run the league in order to guarantee that the strongest team a.s. wins the
league? We point out that a similar question was investigated in [BNH07] for
the model considered by the authors.

  * •

In the spirit of large deviations results, we believe it would be interesting
to study the probability that the weakest team wins the league, again
conditioning on the initial strengths of the teams.

  * •

Another natural question is to investigate the whole final ranking of the
league. We conjecture the following.

For a sequence of initial strengths $(s_{i})_{i\in[2n-1]_{0}}$, we denote by
$(\tilde{T}_{i})_{i\in[2n-1]_{0}}$ the sequence of teams reordered according
to their initial strengths $(s_{i})_{i\in[2n-1]_{0}}$ (from the weakest to the
strongest). Set

$\tilde{\bm{W}}_{n}(i)\coloneqq\text{Number of wins of the team
}\tilde{T}_{i}\text{ at the end of a league with $2n$ players}.$ (31)

Let $\tilde{\bm{\mathcal{W}}}_{n}(x):[0,1]\to\mathbb{R}$ denote the piece-wise
linear continuous function obtained by interpolating the values
$\tilde{\bm{W}}_{n}(i/n)$ for all $i\in[2n-1]_{0}$.

Denote by $H_{\mu}(y)$ the cumulative distribution function of $\mu$ and by
$H_{\mu}^{-1}(x)$ the generalized inverse distribution function, i.e.
$H_{\mu}^{-1}(x)=\inf\\{y\in\mathbb{R}_{+}:H_{\mu}(y)\geq x\\}$.

######  Conjecture 1.9.

Suppose that the assumptions in Section 1.3.1 hold with the additional
requirement that the function $f$ is continuous555The assumption that $f$ is
continuous might be relaxed.. For $\mu^{\mathbb{N}\cup\\{0\\}}$-almost every
sequence
$\vec{s}=(s_{i})_{{i\in\mathbb{N}\cup\\{0\\}}}\in\mathbb{R}^{\mathbb{N}\cup\\{0\\}}_{+}$
and for every choice of the calendar of the league, under
$\mathbb{P}_{\vec{s}}$ the following convergence of càdlàg processes holds

$\frac{\tilde{\bm{\mathcal{W}}}_{n}(x)}{2n}\xrightarrow[n\to\infty]{P}\ell(H_{\mu}^{-1}(x)),$
(32)

where $\ell(s)$ is defined as in 1.3 by

$\ell(s)=\mathbb{E}\left[f\left(s\cdot\bm{V},\bm{U}\cdot\bm{V^{\prime}}\right)\right]=\int_{\mathbb{R}^{3}_{+}}f\left(s\cdot
v,u\cdot v^{\prime}\right)d\nu(v)d\nu(v^{\prime})d\mu(u).$ (33)

Note that the correlations between various teams in the league strongly depend
on the choice of the calendar but we believe that this choice does not affect
the limiting result in the conjecture above. We refer the reader to Fig. 5 for
some simulations that support our conjecture.

We also believe that the analysis of the local limit (as defined in [Bor20])
of the whole ranking should be an interesting but challenging question (and in
this case we believe that the choice of the calendar will affect the limiting
object). Here, with local limit we mean the limit of the ranking induced by
the $k$ teams – for every fixed $k\in\mathbb{N}$ – in the neighborhood (w.r.t.
the initial strengths) of a distinguished team (that can be selected uniformly
at random or in a deterministic way, say for instance the strongest team).

  * •

As mentioned above, we believe that it would be interesting to develop a more
accurate and precise analysis of real data in order to correctly calibrate the
parameters of our model collected at the beginning of Section 1.5.

Figure 5: Two simulations of the diagrams of
$(\tilde{\bm{W}}_{n}(i))_{i\in[999]_{0}}$ in the setting of 1.7 with different
choices of $a,b,p_{a}$ and $p_{b}$. The values
$(\tilde{\bm{W}}_{n}(i))_{i\in[999]_{0}}$ are plotted in blue. The limiting
functions $1000\cdot\ell(H_{\mu}^{-1}(x/1000))$ for $x\in[0,1000]$ are plotted
in green and red respectively. Left: In this simulation the parameters are
$a=\frac{1}{2},b=2,p_{a}=\frac{1}{2},p_{b}=\frac{1}{2}$. Middle: In this
simulation the parameters are
$a=\frac{1}{10},b=10,p_{a}=\frac{1}{10},p_{b}=\frac{1}{10}$. Right: The two
limiting functions are overlapped in the same diagram to highlight the
different slopes.

##  2 Proof of the law of large numbers

The proof of 1.3 follows from the following result using standard second
moment arguments.

######  Proposition 2.1.

We have that

$\mathbb{E}\left[\frac{\bm{W}_{n}(s)}{2n}\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s.}\ell(s),\qquad\text{and}\qquad\mathbb{E}\left[\left(\frac{\bm{W}_{n}(s)}{2n}\right)^{2}\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s.}\ell(s)^{2}.$
(34)

The rest of this section is devoted to the proof of 2.1.

We recall (see Section 1.4.2) that we assumed that, for every $j\in[2n-1]$,
the team $T_{0}$ plays against the team $T_{j}$ the $j$-th day and we denoted
by $W_{j}=W_{j}(s)$ the event that the team $T_{0}$ wins the match. In
particular, $\bm{W}_{n}(s)=\sum_{j=1}^{2n-1}\mathds{1}_{W_{j}}$ and

$\mathbb{P}\left(W_{j}\middle|\bm{s}_{j},\bm{\xi}^{0}_{j},\bm{\xi}^{j}_{j}\right)=f(s\cdot\bm{\xi}^{0}_{j},\bm{s}_{j}\cdot\bm{\xi}^{j}_{j}).$
(35)

###### Proof of 2.1.

We start with the computations for the first moment. From Eq. 35 we have that

$\mathbb{E}\left[\bm{W}_{n}(s)\middle|(\bm{s}_{i})_{i}\right]=\sum_{j=1}^{2n-1}\mathbb{P}(W_{j}|(\bm{s}_{i})_{i})=\sum_{j=1}^{2n-1}\mathbb{E}\left[f\left(s\cdot\bm{\xi}^{0}_{j},\bm{s}_{j}\cdot\bm{\xi}^{j}_{j}\right)\middle|\bm{s}_{j}\right].$
(36)

Since for all $j\in[2n-1]$, $\bm{\xi}^{j}_{j}$, $\bm{\xi}^{0}_{j}$ and
$\bm{s}_{j}$ are independent, $\bm{\xi}^{0}_{j}\stackrel{{\scriptstyle
d}}{{=}}\bm{V}$, and $\bm{\xi}^{j}_{j}\stackrel{{\scriptstyle
d}}{{=}}\bm{V}^{\prime}$, we have

$\mathbb{E}\left[f\left(s\cdot\bm{\xi}^{0}_{j},\bm{s}_{j}\cdot\bm{\xi}^{j}_{j}\right)\middle|\bm{s}_{j}\right]=G_{s}\left(\bm{s}_{j}\right),$
(37)

where
$G_{s}(x)=\mathbb{E}\left[f\left(s\cdot\bm{V},x\cdot\bm{V}^{\prime}\right)\right]$.
By the Law of large numbers, we can conclude that

$\mathbb{E}\left[\frac{\bm{W}_{n}(s)}{2n}\middle|(\bm{s}_{i})_{i}\right]=\frac{1}{2n}\sum_{j=1}^{2n-1}G_{s}\left(\bm{s}_{j}\right)\xrightarrow[n\to\infty]{a.s.}\mathbb{E}\left[G_{s}\left(\bm{U}\right)\right]=\ell(s).$
(38)

We now turn to the second moment. Since for all $i,j\in[2n-1]$ with $i\neq j$,
conditioning on
$(\bm{s}_{i},\bm{s}_{j},\bm{\xi}^{0}_{i},\bm{\xi}^{i}_{i},\bm{\xi}^{0}_{j},\bm{\xi}^{j}_{j})$
the events $W_{i}$ and $W_{j}$ are independent, we have that, using Eq. 35,

$\mathbb{E}\left[\bm{W}_{n}(s)^{2}\middle|(\bm{s}_{i})_{i}\right]=\mathbb{E}\left[\bm{W}_{n}(s)\middle|(\bm{s}_{i})_{i}\right]+\sum_{\begin{subarray}{c}\ell,j=1\\\
\ell\neq j\end{subarray}}^{2n-1}\mathbb{P}(W_{\ell}\cap
W_{j}|\bm{s}_{\ell},\bm{s}_{j})\\\
=\mathbb{E}\left[\bm{W}_{n}(s)\middle|(\bm{s}_{i})_{i}\right]+\sum_{\begin{subarray}{c}\ell,j=1\\\
\ell\neq
j\end{subarray}}^{2n-1}\mathbb{E}\left[f\left(s\cdot\bm{\xi}^{0}_{\ell},\bm{s}_{\ell}\cdot\bm{\xi}^{\ell}_{\ell}\right)f\left(s\cdot\bm{\xi}^{0}_{j},\bm{s}_{j}\cdot\bm{\xi}^{j}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right].$
(39)

For all $\ell,j\in[2n-1]$ with $\ell\neq j$, $\bm{s}_{\ell}$, $\bm{s}_{j}$,
$\bm{\xi}^{\ell}_{\ell}$ and $\bm{\xi}^{j}_{j}$ are mutually independent and
independent of $(\bm{\xi}^{0}_{\ell},\bm{\xi}^{0}_{j})$. In addition,
$(\bm{\xi}^{0}_{\ell},\bm{\xi}^{0}_{j})\stackrel{{\scriptstyle
d}}{{=}}(\bm{\xi}_{\ell},\bm{\xi}_{j})$ and
$\bm{\xi}^{\ell}_{\ell}\stackrel{{\scriptstyle
d}}{{=}}\bm{\xi}^{j}_{j}\stackrel{{\scriptstyle d}}{{=}}\bm{V}^{\prime}$.
Thus, we have that

$\mathbb{E}\left[f\left(s\cdot\bm{\xi}^{0}_{\ell},\bm{s}_{\ell}\cdot\bm{\xi}^{\ell}_{\ell}\right)f\left(s\cdot\bm{\xi}^{0}_{j},\bm{s}_{j}\cdot\bm{\xi}^{j}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right]=\mathbb{E}\left[F_{s}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)F_{s}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right],$
(40)

where we recall that $F_{s}(x,y)=\mathbb{E}\left[f\left(s\cdot
x,y\cdot\bm{V}^{\prime}\right)\right]$. Simple consequences of the
computations done for the first moment are that

$\frac{\mathbb{E}\left[\bm{W}_{n}(s)\middle|(\bm{s}_{i})_{i}\right]}{n^{2}}\xrightarrow[n\to\infty]{a.s.}0\quad\text{and}\quad\frac{\mathbb{E}\left[\sum_{j=1}^{2n-1}F_{s}^{2}(\bm{\xi}_{j},\bm{s}_{j})\middle|(\bm{s}_{i})_{i}\right]}{n^{2}}\xrightarrow[n\to\infty]{a.s.}0.$

Therefore, we can write

$\mathbb{E}\left[\left(\frac{\bm{W}_{n}(s)}{2n}\right)^{2}\middle|(\bm{s}_{i})_{i}\right]=\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}F_{s}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)F_{s}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]+\bm{o}(1)\\\
=\mathbb{E}\left[\left(\frac{1}{2n}\sum_{j=1}^{2n-1}F_{s}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\right)^{2}\middle|(\bm{s}_{i})_{i}\right]+\bm{o}(1),$
(41)

where $\bm{o}(1)$ denotes a sequence of random variables that a.s. converges
to zero. We now need the following result, whose proof is postponed at the end
of this section.

######  Proposition 2.2.

For all bounded, measurable functions $g:\mathbb{R}^{2}_{+}\to\mathbb{R}_{+}$,
we have that

$\mathbb{E}\left[\left(\frac{1}{2n}\sum_{j=1}^{2n-1}g\left(\bm{\xi}_{j},\bm{s}_{j}\right)\right)^{2}\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s.}\mathbb{E}\left[g\left(\bm{V},\bm{U}\right)\right]^{2}.$
(42)

From Eq. 41 and the proposition above, we conclude that

$\mathbb{E}\left[\left(\frac{\bm{W}_{n}(s)}{2n}\right)^{2}\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s.}\mathbb{E}\left[F_{s}\left(\bm{V},\bm{U}\right)\right]^{2}=\ell(s)^{2}.$
(43)

This concludes the proof of 2.1. ∎

It remains to prove 2.2. We start with the following preliminary result.

######  Lemma 2.3.

For every quadruple $(A,A^{\prime},B,B^{\prime})$ of Borel subsets of
$\mathbb{R}_{+}$, we have that

$\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{A\times
B}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s.}\mathbb{E}\left[\mathds{1}_{A\times
B}(\bm{V},\bm{U})\right]\mathbb{E}\left[\mathds{1}_{A^{\prime}\times
B^{\prime}}(\bm{V},\bm{U})\right].$ (44)

###### Proof.

Note that since the process $\bm{\xi}$ is independent of $(\bm{s}_{i})_{i}$,

$\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{A\times
B}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\\\
=\frac{1}{2n}\sum_{\ell=1}^{2n-1}\mathds{1}_{B}(\bm{s}_{\ell})\cdot\frac{1}{2n}\sum_{j=1}^{2n-1}\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)\mathds{1}_{B^{\prime}}(\bm{s}_{j}).$ (45)

For all $\ell\in[2n-1]$, we can write

$\displaystyle\frac{1}{2n}\sum_{j=1}^{2n-1}\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)$
$\displaystyle\mathds{1}_{B^{\prime}}(\bm{s}_{j})$ (46)
$\displaystyle=\mathbb{P}\left(\bm{V}\in A\right)$
$\displaystyle\mathbb{P}\left(\bm{V}\in
A^{\prime}\right)\frac{1}{2n}\sum_{j=1}^{2n-1}\mathds{1}_{B^{\prime}}(\bm{s}_{j})$
(47)
$\displaystyle+\frac{1}{2n}\sum_{j=1}^{2n-1}\left(\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in
A^{\prime}\right)\right)\mathds{1}_{B^{\prime}}(\bm{s}_{j}).$ (48)

First, from the Law of large numbers we have that

$\frac{1}{2n}\sum_{j=1}^{2n-1}\mathds{1}_{B^{\prime}}(\bm{s}_{j})\xrightarrow[n\to\infty]{a.s.}\mu(B^{\prime}).$
(49)

Secondly, we show that the second sum in the right-hand side of Eq. 46 is
negligible. We estimate

$\left|\frac{1}{2n}\sum_{j=1}^{2n-1}\left(\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in
A^{\prime}\right)\right)\mathds{1}_{B^{\prime}}(\bm{s}_{j})\right|\\\
\leq\frac{1}{2n}\sum_{j=1}^{2n-1}\left|\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in A^{\prime}\right)\right|\\\
=\frac{1}{2n}\sum_{j=1}^{\ell-1}\left|\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in A^{\prime}\right)\right|\\\
+\frac{1}{2n}\sum_{j=\ell}^{2n-1}\left|\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in A^{\prime}\right)\right|$ (50)

Using the stationarity assumption in Eq. 3, the right-hand side of the
equation above can be rewritten as follows

$\frac{1}{2n}\sum_{j=2}^{\ell}\left|\mathbb{P}\left(\bm{\xi}_{j}\in
A,\bm{\xi}_{1}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in A^{\prime}\right)\right|\\\
+\frac{1}{2n}\sum_{j=1}^{2n-\ell}\left|\mathbb{P}\left(\bm{\xi}_{1}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in A^{\prime}\right)\right|.$ (51)

Therefore, we obtain that

$\left|\frac{1}{2n}\sum_{j=1}^{2n-1}\left(\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in
A^{\prime}\right)\right)\mathds{1}_{B^{\prime}}(\bm{s}_{j})\right|\\\
\leq\frac{1}{2n}\sum_{j=1}^{2n}\left|\mathbb{P}\left(\bm{\xi}_{1}\in
A^{\prime},\bm{\xi}_{j}\in A\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in A^{\prime}\right)\right|\\\
+\frac{1}{2n}\sum_{j=1}^{2n}\left|\mathbb{P}\left(\bm{\xi}_{1}\in
A,\bm{\xi}_{j}\in A^{\prime}\right)-\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in A^{\prime}\right)\right|.$ (52)

The upper bound above is independent of $\ell$ and tends to zero because the
process $\bm{\xi}$ is weakly-mixing (see Eq. 4); we can thus deduce from Eqs.
46 and 49 that, uniformly for all $\ell\in[2n-1]$,

$\frac{1}{2n}\sum_{j=1}^{2n-1}\mathbb{P}\left(\bm{\xi}_{\ell}\in
A,\bm{\xi}_{j}\in
A^{\prime}\right)\mathds{1}_{B^{\prime}}(\bm{s}_{j})\xrightarrow[n\to\infty]{a.s.}\mathbb{P}\left(\bm{V}\in
A\right)\mathbb{P}\left(\bm{V}\in A^{\prime}\right)\mu(B^{\prime}).$ (53)

Hence, from Eqs. 45 and 53 and the Law of large numbers, we can conclude that

$\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{A\times
B}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\\\
\xrightarrow[n\to\infty]{a.s.}\mathbb{P}\left(\bm{V}\in
A\right)\mu(B)\mathbb{P}\left(\bm{V}\in
A^{\prime}\right)\mu(B^{\prime})=\mathbb{E}\left[\mathds{1}_{A\times
B}(\bm{V},\bm{U})\right]\mathbb{E}\left[\mathds{1}_{A^{\prime}\times
B^{\prime}}(\bm{V},\bm{U})\right].\qed$ (54)

We now show that we can extend the result of 2.3 to all Borel subsets of
$\mathbb{R}_{+}^{2}$, denoted by $\mathcal{B}(\mathbb{R}_{+}^{2})$. We also
denote by $\mathcal{R}(\mathbb{R}_{+}^{2})$ the sets of rectangles $A\times B$
of $\mathbb{R}_{+}^{2}$ with $A,B\in\mathcal{B}(\mathbb{R}_{+})$.

We recall that we denote by $\sigma\left(\mathcal{A}\right)$ and
$\lambda\left(\mathcal{A}\right)$ the sigma algebra and the monotone class
generated by a collection of sets $\mathcal{A}$, respectively.

######  Lemma 2.4.

For all $C,C^{\prime}\in\mathcal{B}(\mathbb{R}_{+}^{2})$, we have that

$\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{C}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{C^{\prime}}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s.}\mathbb{E}\left[\mathds{1}_{C}(\bm{V},\bm{U})\right]\mathbb{E}\left[\mathds{1}_{C^{\prime}}(\bm{V},\bm{U})\right].$
(55)

###### Proof.

We first fix a rectangle $A\times B\in\mathcal{R}(\mathbb{R}_{+}^{2})$ and we
consider the set

$\mathcal{A}_{A\times
B}\coloneqq\left\\{C\in\mathcal{B}(\mathbb{R}_{+}^{2})\middle|\text{Eq.
}\eqref{eq:second_mom_ind_bor}\text{ holds with }C^{\prime}=A\times
B\right\\}.$ (56)

By 2.3, we have $\mathcal{R}(\mathbb{R}_{+}^{2})\subseteq\mathcal{A}_{A\times
B}\subseteq\mathcal{B}(\mathbb{R}_{+}^{2})$. If we show that
$\mathcal{A}_{A\times B}$ is a monotone class, then we can conclude that
$\mathcal{A}_{A\times B}=\mathcal{B}(\mathbb{R}_{+}^{2})$. Indeed, by the
monotone class theorem (note that $\mathcal{R}(\mathbb{R}_{+}^{2})$ is closed
under finite intersections), we have that

$\mathcal{B}(\mathbb{R}_{+}^{2})=\sigma\left(\mathcal{R}(\mathbb{R}_{+}^{2})\right)=\lambda\left(\mathcal{R}(\mathbb{R}_{+}^{2})\right)\subseteq\mathcal{A}_{A\times
B}.$ (57)

The equality $\mathcal{A}_{A\times B}=\mathcal{B}(\mathbb{R}_{+}^{2})$ implies
that Eq. 55 holds for every set in
$\mathcal{B}(\mathbb{R}_{+}^{2})\times\mathcal{R}(\mathbb{R}_{+}^{2})$.
Finally, if we also show that for any fixed Borel set
$C^{*}\in\mathcal{B}(\mathbb{R}_{+}^{2})$, the set

$\mathcal{A}_{C^{*}}\coloneqq\left\\{C^{\prime}\in\mathcal{B}(\mathbb{R}_{+}^{2})\middle|\text{Eq.
}\eqref{eq:second_mom_ind_bor}\text{ holds with }C=C^{*}\right\\}$ (58)

is a monotone class, then using again the same arguments that we used in Eq.
57 (note that $\mathcal{R}(\mathbb{R}_{+}^{2})\subseteq\mathcal{A}_{C^{*}}$
thanks to the previous step) we can conclude that Eq. 55 holds for every pair
of sets in
$\mathcal{B}(\mathbb{R}_{+}^{2})\times\mathcal{B}(\mathbb{R}_{+}^{2})$,
proving the lemma. Therefore, in order to conclude the proof, it is sufficient
to show that $\mathcal{A}_{A\times B}$ and $\mathcal{A}_{C^{*}}$ are monotone
classes.

We start by proving that $\mathcal{A}_{A\times B}$ is a monotone class:

  * •

Obviously $\mathbb{R}_{+}^{2}\in\mathcal{A}_{A\times B}$.

  * •

If $C,D\in\mathcal{A}_{A\times B}$ and $C\subseteq D$, then $D\setminus
C\in\mathcal{A}_{A\times B}$ because $\mathds{1}_{D\setminus
C}=\mathds{1}_{D}-\mathds{1}_{C}$.

  * •

Let now $(C_{m})_{m\in\mathbb{N}}$ be a sequence of sets in
$\mathcal{A}_{A\times B}$ such that $C_{m}\subseteq C_{m+1}$ for all
$m\in\mathbb{N}$. We want to show that
$C\coloneqq\bigcup_{m}C_{m}\in\mathcal{A}_{A\times B}$, i.e. that

$\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{C}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s.}\mathbb{E}\left[\mathds{1}_{C}(\bm{V},\bm{U})\right]\mathbb{E}\left[\mathds{1}_{A\times
B}(\bm{V},\bm{U})\right].$ (59)

Since $\mathds{1}_{C}=\lim_{m\to\infty}\mathds{1}_{C_{m}}$, then by monotone
convergence we have for all $n\in\mathbb{N}$,

$\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{C_{m}}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[m\to\infty]{a.s.}\\\
\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathbb{E}\left[\mathds{1}_{C}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right].$
(60)

We also claim that:

    * (a)

For all $m\in\mathbb{N}$,

$\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathbb{E}\left[\mathds{1}_{C_{m}}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right]\xrightarrow[n\to\infty]{a.s.}\mathbb{E}\left[\mathds{1}_{C_{m}}(\bm{V},\bm{U})\right]\mathbb{E}\left[\mathds{1}_{A\times
B}(\bm{V},\bm{U})\right].$ (61)

    * (b)

The convergence in Eq. 60 holds uniformly for all $n\in\mathbb{N}$.

Item (a) holds since $C_{m}\in\mathcal{A}_{A\times B}$. Item (b) will be
proved at the end. Items (a) and (b) allow us to exchange the following
a.s.-limits as follows

$\displaystyle\lim_{n\to\infty}\mathbb{E}$
$\displaystyle\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{C}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]$ (62)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:lim_of_indicator}}}{{=}}$
$\displaystyle\lim_{n\to\infty}\lim_{m\to\infty}\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathbb{E}\left[\mathds{1}_{C_{m}}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right]$
(63) $\displaystyle=$
$\displaystyle\lim_{m\to\infty}\lim_{n\to\infty}\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathbb{E}\left[\mathds{1}_{C_{m}}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right]$
(64) $\displaystyle=$
$\displaystyle\lim_{m\to\infty}\mathbb{E}\left[\mathds{1}_{C_{m}}(\bm{V},\bm{U})\right]\mathbb{E}\left[\mathds{1}_{A\times
B}(\bm{V},\bm{U})\right]=\mathbb{E}\left[\mathds{1}_{C}(\bm{V},\bm{U})\right]\mathbb{E}\left[\mathds{1}_{A\times
B}(\bm{V},\bm{U})\right],$ (65)

where the last equality follows by monotone convergence. This proves Eq. 59
and concludes the proof (up to proving item (b)) that $\mathcal{A}_{A\times
B}$ is a monotone class.

Proof of item (b). Since $\mathds{1}_{C\setminus
C_{m}}=\mathds{1}_{C}-\mathds{1}_{C_{m}}$, it is enough to show that

$\sup_{n}\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{C\setminus
C_{m}}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[m\to\infty]{a.s.}0.$
(66)

We set $D_{m}\coloneqq C\setminus C_{m}$, and define

$(D_{m})^{s}\coloneqq\\{x\in\mathbb{R}_{+}|(x,s)\in D_{m}\\},\quad\text{for
all}\quad s\in\mathbb{R},$

$\pi_{Y}(D_{m})\coloneqq\\{y\in\mathbb{R}_{+}|\exists x\in\mathbb{R}_{+}\text{
s.t. }(x,y)\in D_{m}\\}.$

Since $\bm{\xi}_{\ell}\stackrel{{\scriptstyle d}}{{=}}\bm{V}$ for all
$\ell\in[2n-1]$, then for all $\ell,j\in[2n-1]$, a.s.

$\displaystyle\mathbb{E}\left[\mathds{1}_{D_{m}}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right]$
$\displaystyle=\mathbb{P}\left(\bm{\xi}_{\ell}\in(D_{m})^{\bm{s}_{\ell}},\bm{\xi}_{j}\in
A\middle|\bm{s}_{\ell}\right)\mathds{1}_{\pi_{Y}(D_{m})}(\bm{s}_{\ell})\mathds{1}_{B}(\bm{s}_{j})$
(67)
$\displaystyle\leq\mathbb{P}\left(\bm{\xi}_{\ell}\in(D_{m})^{\bm{s}_{\ell}}\middle|\bm{s}_{\ell}\right)=\mathbb{P}\left(\bm{V}\in(D_{m})^{\bm{s}_{\ell}}\middle|\bm{s}_{\ell}\right).$
(68)

Therefore a.s.

$\sup_{n}\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathds{1}_{C\setminus
C_{m}}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A\times
B}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\leq\mathbb{P}\left(\bm{V}\in(C\setminus
C_{m})^{\bm{s}_{\ell}}\middle|\bm{s}_{\ell}\right).$ (69)

Now the sequence $\mathbb{P}\left(\bm{V}\in(C\setminus
C_{m})^{\bm{s}_{\ell}}\middle|\bm{s}_{\ell}\right)$ is a.s. non-increasing
(because $C_{m}\subseteq C_{m+1}$) and hence has an a.s. limit. The limit is
non-negative and its expectation is the limit of expectations which is $0$
because $C\coloneqq\bigcup_{m}C_{m}$. This completes the proof of item (b).

It remains to prove that $\mathcal{A}_{C*}$ is also a monotone class. The
proof is similar to the proof above, replacing the bound in Eq. 68 by

$\mathbb{E}\left[\mathds{1}_{C^{*}}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{D_{m}}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right]\\\
=\mathbb{P}\left(\bm{\xi}_{\ell}\in(C^{*})^{\bm{s}_{\ell}},\bm{\xi}_{j}\in(D_{m})^{\bm{s}_{j}}\middle|\bm{s}_{\ell},\bm{s}_{j}\right)\mathds{1}_{\pi_{Y}(C^{*})}(\bm{s}_{\ell})\mathds{1}_{\pi_{Y}(D_{m})}(\bm{s}_{j})\leq\mathbb{P}\left(\bm{\xi}_{j}\in(D_{m})^{\bm{s}_{j}}\middle|\bm{s}_{j}\right).$
(70)

This completes the proof of the lemma. ∎

We now generalize the result in 2.4 to all bounded and measurable functions,
hereby proving 2.2.

###### Proof of 2.2.

We further assume that $g:\mathbb{R}^{2}_{+}\to\mathbb{R}_{+}$ is non-
negative, the general case following by stardand arguments. Fubini’s theorem,
together with the fact that
$g(x,y)=\int_{0}^{\|g\|_{\infty}}\mathds{1}_{\left\\{z\leq
g(x,y)\right\\}}dz$, yields

$\mathbb{E}\left[\left(\frac{1}{2n}\sum_{j=1}^{2n-1}g\left(\bm{\xi}_{j},s_{j}\right)\right)^{2}\middle|(\bm{s}_{i})_{i}\right]=\mathbb{E}\left[\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\int_{0}^{\|g\|_{\infty}}\mathds{1}_{\left\\{z\leq
g\left(\bm{\xi}_{\ell},s_{\ell}\right)\right\\}}dz\int_{0}^{\|g\|_{\infty}}\mathds{1}_{\left\\{t\leq
g\left(\bm{\xi}_{j},s_{j}\right)\right\\}}dt\middle|(\bm{s}_{i})_{i}\right]\\\
=\int_{0}^{\|g\|_{\infty}}\int_{0}^{\|g\|_{\infty}}\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathbb{E}\left[\mathds{1}_{A(z)}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A(t)}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|\bm{s}_{\ell},\bm{s}_{j}\right]dz\;dt,$
(71)

where $A(s)=\left\\{(x,y)\in\mathbb{R}_{+}^{2}\middle|g(x,y)\geq s\right\\}$.
By 2.4, for almost every $(z,t)\in\mathbb{R}^{2}_{+}$

$\frac{1}{4n^{2}}\sum_{\ell,j=1}^{2n-1}\mathbb{E}\left[\mathds{1}_{A(z)}\left(\bm{\xi}_{\ell},\bm{s}_{\ell}\right)\mathds{1}_{A(t)}\left(\bm{\xi}_{j},\bm{s}_{j}\right)\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s.}\mathbb{E}\left[\mathds{1}_{A(z)}\left(\bm{V},\bm{U}\right)\right]\mathbb{E}\left[\mathds{1}_{A(t)}\left(\bm{V},\bm{U}\right)\right].$
(72)

Since the left-hand side is bounded by $1$, we can conclude by dominated
convergence that

$\mathbb{E}\left[\left(\frac{1}{2n}\sum_{j=1}^{2n-1}g\left(\bm{\xi}_{j},s_{j}\right)\right)^{2}\middle|(\bm{s}_{i})_{i}\right]\xrightarrow[n\to\infty]{a.s}\mathbb{E}\left[g\left(\bm{V},\bm{U}\right)\right]^{2},$
(73)

completing the proof of the proposition. ∎

##  3 Proof of the central limit theorems

In this section, we start by proving 1.4 using 1.6 and then we prove the
latter result.

###### Proof of 1.4.

We set
$\bm{X}_{n}\coloneqq\frac{\bm{W}_{n}(s)-2n\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]}{\sqrt{2n}}$.
In order to prove the convergence in Eq. 9, it is enough to show that, for
every $t\in\mathbb{R}$,

$\mathbb{E}_{\vec{s}}\left[e^{it\bm{X}_{n}}\right]\xrightarrow{n\to\infty}e^{-\frac{t^{2}}{2}\left(\sigma(s)^{2}+\rho(s)^{2}\right)},$
(74)

where we recall that
$\sigma(s)^{2}=\mathbb{E}\left[F_{s}(\bm{V},\bm{U})-F^{2}_{s}(\bm{V},\bm{U})\right]$
and $\rho(s)^{2}=\rho_{F_{s}}^{2}$. Note that $\sigma(s)^{2}+\rho(s)^{2}$ is
finite thanks to 1.6 and the fact that $F_{s}-F_{s}^{2}$ is a bounded and
measurable function.

Recalling that $\bm{W}_{n}(s)=\sum_{j=1}^{2n-1}\mathds{1}_{W_{j}}$, $W_{j}$
being the event that the team $T_{0}$ wins against the team $T_{j}$, and that,
conditioning on $\left(\bm{\xi}^{0}_{r}\right)_{{}_{r\in[2n-1]}}$, the results
of different matches are independent, we have that

$\mathbb{E}_{\vec{s}}\left[e^{it\bm{X}_{n}}\right]=e^{-\sqrt{2n}\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]\cdot
it}\cdot\mathbb{E}_{\vec{s}}\left[e^{\frac{it}{\sqrt{2n}}\sum_{j=1}^{2n-1}\mathds{1}_{W_{j}}}\right]\\\
=e^{-\sqrt{2n}\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]\cdot
it}\cdot\mathbb{E}_{\vec{s}}\left[\mathbb{E}_{\vec{s}}\left[e^{\frac{it}{\sqrt{2n}}\sum_{j=1}^{2n-1}\mathds{1}_{W_{j}}}\middle|\left(\bm{\xi}^{0}_{r}\right)_{{}_{r\in[2n-1]}}\right]\right]\\\
=e^{-\sqrt{2n}\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]\cdot
it}\cdot\mathbb{E}_{\vec{s}}\left[\prod_{j=1}^{2n-1}\mathbb{E}_{\vec{s}}\left[e^{\frac{it}{\sqrt{2n}}\mathds{1}_{W_{j}}}\middle|\bm{\xi}^{0}_{j}\right]\right].$
(75)

Since, by assumption, we have that
$\mathbb{P}_{\vec{s}}\left(W_{j}\middle|\bm{\xi}^{0}_{j},\bm{\xi}^{j}_{j}\right)=f(s\cdot\bm{\xi}^{0}_{j},s_{j}\cdot\bm{\xi}^{j}_{j})$
and, for all $j\in[2n-1]$, $\bm{\xi}^{j}_{j}$ is independent of
$\bm{\xi}^{0}_{j}$, $\bm{\xi}^{0}_{j}\stackrel{{\scriptstyle
d}}{{=}}\bm{\xi}_{j}$ and $\bm{\xi}^{j}_{j}\stackrel{{\scriptstyle
d}}{{=}}\bm{V}^{\prime}$, we have that

$\mathbb{E}_{\vec{s}}\left[e^{it\bm{X}_{n}}\right]=e^{-\sqrt{2n}\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]\cdot
it}\cdot\mathbb{E}\left[\prod_{j=1}^{2n-1}\mathbb{E}\left[1+\left(e^{\frac{it}{\sqrt{2n}}}-1\right)f(s\cdot\bm{\xi}^{0}_{j},s_{j}\cdot\bm{\xi}^{j}_{j})\middle|\bm{\xi}^{0}_{j}\right]\right]\\\
=e^{-\sqrt{2n}\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]\cdot
it}\cdot\mathbb{E}\left[\prod_{j=1}^{2n-1}\left(1+\left(e^{\frac{it}{\sqrt{2n}}}-1\right)\cdot
F_{s}(\bm{\xi}_{j},s_{j})\right)\right],$ (76)

where we recall that $F_{s}(x,y)=\mathbb{E}\left[f\left(s\cdot
x,y\cdot\bm{V}^{\prime}\right)\right]$. Rewriting the last term as

$e^{-\sqrt{2n}\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]\cdot
it}\cdot\mathbb{E}\left[e^{\sum_{j=1}^{2n-1}\log\left(1+(e^{it/\sqrt{2n}}-1)\cdot
F_{s}(\bm{\xi}_{j},s_{j})\right)}\right],$ (77)

and observing that

$\sum_{j=1}^{2n-1}\log\left(1+\left(e^{it/\sqrt{2n}}-1\right)\cdot
F_{s}\left(\bm{\xi}_{j},s_{j}\right)\right)\\\
=\sum_{j=1}^{2n-1}\left(\frac{it}{\sqrt{2n}}\cdot
F_{s}\left(\bm{\xi}_{j},s_{j}\right)-\frac{t^{2}}{4n}\left(F_{s}\left(\bm{\xi}_{j},s_{j}\right)-F_{s}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right)+O\left(\frac{1}{n\sqrt{n}}\right)\right)\\\
=\frac{it}{\sqrt{2n}}\sum_{j=1}^{2n-1}F_{s}\left(\bm{\xi}_{j},s_{j}\right)-\frac{t^{2}}{2}\cdot\frac{1}{2n}\sum_{j=1}^{2n-1}\left(F_{s}\left(\bm{\xi}_{j},s_{j}\right)-F_{s}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right)+O\left(\frac{1}{\sqrt{n}}\right),$
(78)

we obtain that the characteristic function
$\mathbb{E}_{\vec{s}}\left[e^{it\bm{X}_{n}}\right]$ is equal to

$e^{O\left(\frac{1}{\sqrt{n}}\right)}\cdot
e^{-\frac{t^{2}}{2}\sigma(s)^{2}}\cdot\mathbb{E}\left[e^{\frac{it}{\sqrt{2n}}\left(\sum_{j=1}^{2n-1}F_{s}(\bm{\xi}_{j},s_{j})-2n\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]\right)}\cdot
e^{-\frac{t^{2}}{2}\cdot\left(\frac{1}{2n}\sum_{j=1}^{2n-1}\left(F_{s}\left(\bm{\xi}_{j},s_{j}\right)-F_{s}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right)-\sigma(s)^{2}\right)}\right].$
(79)

Now we set

$\displaystyle\bm{A}_{n}\coloneqq
e^{\frac{it}{\sqrt{2n}}\left(\sum_{j=1}^{2n-1}F_{s}(\bm{\xi}_{j},s_{j})-2n\cdot\mathbb{E}_{\vec{s}}[\bm{W}_{n}(s)]\right)},$
(80) $\displaystyle\bm{B}_{n}\coloneqq
e^{-\frac{t^{2}}{2}\cdot\left(\frac{1}{2n}\sum_{j=1}^{2n-1}\left(F_{s}\left(\bm{\xi}_{j},s_{j}\right)-F_{s}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right)-\sigma(s)^{2}\right)},$
(81)

obtaining that
$\mathbb{E}_{\vec{s}}\left[e^{it\bm{X}_{n}}\right]=e^{O\left(\frac{1}{\sqrt{n}}\right)}\cdot
e^{-\frac{t^{2}}{2}\sigma(s)^{2}}\left(\mathbb{E}\left[\bm{A}_{n}\right]-\mathbb{E}\left[\bm{A}_{n}(1-\bm{B}_{n})\right]\right)$.
Hence, Eq. 74 holds if we show that

  1. 1.

$\mathbb{E}\left[\bm{A}_{n}\right]\to e^{-\frac{t^{2}}{2}\rho(s)^{2}}$,

  2. 2.

$\mathbb{E}\left[\bm{A}_{n}(1-\bm{B}_{n})\right]\to 0$ .

Item 1 follows from 1.6. For Item 2, since $|\bm{A}_{n}|=1$, we have that

$\left|\mathbb{E}\left[\bm{A}_{n}(1-\bm{B}_{n})\right]\right|\leq\mathbb{E}\left[|1-\bm{B}_{n}|\right].$
(82)

Recalling that
$\sigma(s)^{2}=\mathbb{E}\left[F_{s}(\bm{V},\bm{U})-F^{2}_{s}(\bm{V},\bm{U})\right]$,
and that $\bm{\xi}_{j}\stackrel{{\scriptstyle d}}{{=}}\bm{V}$ for all
$j\in[2n-1]$, we have that

$\frac{1}{{2n}}\sum_{j=1}^{2n-1}\left(F_{s}\left(\bm{\xi}_{j},s_{j}\right)-F_{s}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right)-\sigma(s)^{2}=\\\
\frac{1}{{2n}}\sum_{j=1}^{2n-1}\left(F_{s}\left(\bm{\xi}_{j},s_{j}\right)-F_{s}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right)-\frac{1}{{2n}}\sum_{j=1}^{2n-1}\mathbb{E}\left[F_{s}\left(\bm{V},s_{j}\right)-F_{s}^{2}\left(\bm{V},s_{j}\right)\right]\\\
+\frac{1}{{2n}}\sum_{j=1}^{2n-1}\mathbb{E}\left[F_{s}\left(\bm{V},s_{j}\right)-F_{s}^{2}\left(\bm{V},s_{j}\right)\right]-\mathbb{E}\left[F_{s}(\bm{V},\bm{U})-F^{2}_{s}(\bm{V},\bm{U})\right]\xrightarrow{P}0,$
(83)

where for the limit we used once again 1.6 and similar arguments to the ones
already used in the proof of 2.1. Since the function $e^{-t^{2}x/2}$ is
continuous and the random variable
$\frac{1}{{2n}}\left(\sum_{j=1}^{2n-1}F_{s}\left(\bm{\xi}_{j},s_{j}\right)-F_{s}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right)-\sigma(s)^{2}$
is bounded, we can conclude that $\mathbb{E}\left[|1-\bm{B}_{n}|\right]\to 0$.
This ends the proof of 1.4. ∎

The rest of this section is devoted to the proof of 1.6. We start by stating a
lemma that shows how the coefficients $\alpha_{n}$ defined in Eq. 5 control
the correlations of the process $\bm{\xi}$.

######  Lemma 3.1 (Theorem 17.2.1 in [IL71]).

Fix $\tau\in\mathbb{N}$ and let $\bm{X}$ be a random variable measurable
w.r.t. $\mathcal{A}_{1}^{k}$ and $\bm{Y}$ a random variable measurable w.r.t.
$\mathcal{A}^{\infty}_{k+\tau}$. Assume, in addition, that $|\bm{X}|<C_{1}$
almost surely and $|\bm{Y}|<C_{2}$ almost surely. Then

$\left|\mathbb{E}\left[\bm{X}\bm{Y}\right]-\mathbb{E}[\bm{X}]\mathbb{E}[\bm{Y}]\right|\leq
4\cdot C_{1}\cdot C_{2}\cdot\alpha_{\tau}.$ (84)

We now focus on the behaviour of the random variables
$\sum_{j=1}^{2n-1}g\left(\bm{\xi}_{j},s_{j}\right)$ appearing in the statement
of 1.6. It follows directly from 2.2 and Chebyshev’s inequality that, for
$\mu^{\mathbb{N}}$-almost every sequence
$(s_{i})_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$,

$\frac{1}{2n}\sum_{j=1}^{2n-1}g\left(\bm{\xi}_{j},s_{j}\right)\xrightarrow{P}\mathbb{E}\left[g\left(\bm{V},\bm{U}\right)\right].$
(85)

We aim at establishing a central limit theorem. Recalling the definition of
the function $\tilde{g}$ in the statement of 1.6, we note that, for all
$j\in\mathbb{N}$,

$\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)=g\left(\bm{\xi}_{j},s_{j}\right)-\mathbb{E}[g\left(\bm{V},s_{j}\right)],$
(86)

and so $\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\right]=0$.
Define

$\rho_{g,n}^{2}\coloneqq\operatorname{Var}\left(\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\right)=\operatorname{Var}\left(\sum_{j=1}^{2n-1}g\left(\bm{\xi}_{j},s_{j}\right)\right).$
(87)

The following lemma shows that the variance $\rho_{g,n}^{2}$ is asymptotically
linear in $n$ and proves the first part of 1.6.

######  Lemma 3.2.

The quantity $\rho_{g}^{2}$ defined in Eq. 14 is finite. Moreover, for
$\mu^{\mathbb{N}}$-almost every sequence
$(s_{i})_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$, we have that

$\rho_{g,n}^{2}=2n\cdot\rho_{g}^{2}\cdot(1+o(1)).$ (88)

###### Proof.

We have that

$\rho_{g,n}^{2}=\operatorname{Var}\left(\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\right)=\mathbb{E}\left[\left(\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\right)^{2}\right]\\\
=\mathbb{E}\left[\sum_{j=1}^{2n-1}\tilde{g}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right]+2\cdot\mathbb{E}\left[\sum_{i=1}^{2n-2}\sum_{j=i+1}^{2n-1}\tilde{g}\left(\bm{\xi}_{i},s_{i}\right)\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\right]\\\
=\sum_{j=1}^{2n-1}\mathbb{E}\left[\tilde{g}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right]+2\cdot\sum_{i=1}^{2n-2}\sum_{k=1}^{2n-1-i}\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{i},s_{i}\right)\tilde{g}\left(\bm{\xi}_{i+k},s_{i+k}\right)\right].$
(89)

Using similar arguments to the ones already used in the proof of 2.1, we have
that

$\sum_{j=1}^{2n-1}\mathbb{E}\left[\tilde{g}^{2}\left(\bm{\xi}_{j},s_{j}\right)\right]=2n\cdot\mathbb{E}\left[\tilde{g}^{2}(\bm{V},\bm{U})\right]+o(n).$
(90)

We now show that

$\lim_{n\to\infty}\frac{1}{2n}\sum_{i=1}^{2n-2}\sum_{k=1}^{2n-1-i}\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{i},s_{i}\right)\tilde{g}\left(\bm{\xi}_{i+k},s_{i+k}\right)\right]=\sum_{k=1}^{\infty}\mathbb{E}\left[\tilde{g}(\bm{\xi}_{1},\bm{U})\tilde{g}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right].$
(91)

First we show that the right-hand side is convergent. We start by noting that
from Fubini’s theorem and 3.1,

$\mathbb{E}\left[\mathbb{E}\left[\tilde{g}(\bm{\xi}_{1},\bm{U})\tilde{g}(\bm{\xi}_{1+k},\bm{U}^{\prime})\middle|\bm{\xi}_{1},\bm{\xi}_{1+k}\right]\right]=\int_{\mathbb{R}^{2}}\mathbb{E}\left[\tilde{g}(\bm{\xi}_{1},x)\tilde{g}(\bm{\xi}_{1+k},y)\right]d\mu(x)d\mu(y)\leq
4\cdot\alpha_{k}.$

Therefore, thanks to the assumption in Eq. 6, we have that

$\sum_{k=1}^{\infty}\mathbb{E}\left[\tilde{g}(\bm{\xi}_{1},\bm{U})\tilde{g}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right]=\sum_{k=1}^{\infty}\mathbb{E}\left[\mathbb{E}\left[\tilde{g}(\bm{\xi}_{1},\bm{U})\tilde{g}(\bm{\xi}_{1+k},\bm{U}^{\prime})\middle|\bm{\xi}_{1},\bm{\xi}_{1+k}\right]\right]\leq\sum_{k=1}^{\infty}4\cdot\alpha_{k}<\infty.$
(92)

Now we turn to the proof of the limit in Eq. 91. Using the stationarity
assumption for the process $\bm{\xi}$ in Eq. 3, we can write

$\frac{1}{2n}\sum_{i=1}^{2n-2}\sum_{k=1}^{2n-1-i}\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{i},s_{i}\right)\tilde{g}\left(\bm{\xi}_{i+k},s_{i+k}\right)\right]=\sum_{k=1}^{2n-2}\frac{1}{2n}\sum_{i=1}^{2n-1-k}\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{1},s_{i}\right)\tilde{g}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right].$
(93)

Using a monotone class argument similar to the one used for the law of large
numbers, we will show that the right-hand side of the equation above converges
to

$\sum_{k=1}^{\infty}\mathbb{E}\left[\tilde{g}(\bm{\xi}_{1},\bm{U})\tilde{g}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right].$
(94)

We start, as usual, from indicator functions. We have to prove that for all
quadruplets $(A,A^{\prime},B,B^{\prime})$ of Borel subsets of
$\mathbb{R}_{+}$, it holds that

$\sum_{k=1}^{2n-2}\frac{1}{2n}\sum_{i=1}^{2n-1-k}\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}\left(\bm{\xi}_{1},s_{i}\right)\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right]\\\
\to\sum_{k=1}^{\infty}\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}(\bm{\xi}_{1},\bm{U})\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right]<\infty,$ (95)

where for every rectangle $R$ of $\mathbb{R}_{+}^{2}$,

$\displaystyle\tilde{\mathds{1}}_{R}\left(x,y\right)\coloneqq\mathds{1}_{R}\left(x,y\right)-\mathbb{E}\left[\mathds{1}_{R}\left(\bm{\xi}_{1},y\right)\right].$
(96)

Setting
$S_{n}\coloneqq\sum_{k=1}^{n}\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}(\bm{\xi}_{1},\bm{U})\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right]$, we estimate

$\left|S_{\infty}-\sum_{k=1}^{2n-2}\frac{1}{2n}\sum_{i=1}^{2n-1-k}\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}\left(\bm{\xi}_{1},s_{i}\right)\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right]\right|\\\
\leq\left|S_{\infty}-S_{2n-2}\right|+\left|S_{2n-2}-\sum_{k=1}^{2n-2}\frac{1}{2n}\sum_{i=1}^{2n-2}\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}\left(\bm{\xi}_{1},s_{i}\right)\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right]\right|\\\
+\left|\sum_{k=1}^{2n-2}\frac{1}{2n}\sum_{i=2n-1-k}^{2n-2}\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}\left(\bm{\xi}_{1},s_{i}\right)\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right]\right|.$ (97)

Clearly, the first term in the right-hand side of the inequality above tends
to zero, being the tail of a convergent series (the fact that
$S_{\infty}<\infty$ follows via arguments already used for Eq. 92). For the
last term, we notice that, using 3.1,

$\left|\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}\left(\bm{\xi}_{1},s_{i}\right)\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right]\right|\leq
4\cdot\alpha_{k},$ (98)

and thus

$\left|\sum_{k=1}^{2n-2}\frac{1}{2n}\sum_{i=2n-1-k}^{2n-2}\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}\left(\bm{\xi}_{1},s_{i}\right)\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right]\right|\leq\frac{1}{2n}\sum_{k=1}^{2n-2}4k\cdot\alpha_{k},$
(99)

which converges to $0$ as $n$ goes to infinity by the assumption in Eq. 6 and
the same arguments used in 1.2. It remains to bound the second term. Expanding
the products and recalling that
$\bm{V},\bm{V}^{\prime},\bm{U},\bm{U}^{\prime}$ are independent random
variables such that $\bm{\xi}_{1}\stackrel{{\scriptstyle
d}}{{=}}\bm{\xi}_{1+k}\stackrel{{\scriptstyle
d}}{{=}}\bm{V}\stackrel{{\scriptstyle d}}{{=}}\bm{V}^{\prime}$ and
$\bm{U}\stackrel{{\scriptstyle d}}{{=}}\bm{U}^{\prime}\stackrel{{\scriptstyle
d}}{{=}}\mu$, we have that

$S_{2n-2}=\sum_{k=1}^{2n-2}\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}(\bm{\xi}_{1},\bm{U})\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right]\\\
=\sum_{k=1}^{2n-2}\mathbb{E}\left[\mathds{1}_{A\times
B}(\bm{\xi}_{1},\bm{U})\mathds{1}_{A^{\prime}\times
B^{\prime}}(\bm{\xi}_{1+k},\bm{U}^{\prime})\right]-\mathbb{E}\left[\mathds{1}_{A\times
B}(\bm{V},\bm{U})\mathds{1}_{A^{\prime}\times
B^{\prime}}(\bm{V}^{\prime},\bm{U}^{\prime})\right]\\\
=\sum_{k=1}^{2n-2}\mu\left(B\right)\cdot\mu\left(B^{\prime}\right)\cdot\left(\mathbb{E}\left[\mathds{1}_{A\times
A^{\prime}}(\bm{\xi}_{1},\bm{\xi}_{1+k})\right]-\mathbb{E}\left[\mathds{1}_{A\times
A^{\prime}}(\bm{V},\bm{V}^{\prime})\right]\right).$ (100)

Similarly, we obtain

$\mathbb{E}\left[\tilde{\mathds{1}}_{A\times
B}\left(\bm{\xi}_{1},s_{i}\right)\tilde{\mathds{1}}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right]\\\
=\mathbb{E}\left[\mathds{1}_{A\times
B}\left(\bm{\xi}_{1},s_{i}\right)\mathds{1}_{A^{\prime}\times
B^{\prime}}\left(\bm{\xi}_{1+k},s_{i+k}\right)\right]-\mathbb{E}\left[\mathds{1}_{A\times
B}\left(\bm{V},s_{i}\right)\right]\mathbb{E}\left[\mathds{1}_{A^{\prime}\times
B^{\prime}}\left(\bm{V}^{\prime},s_{i+k}\right)\right]\\\ =\mathds{1}_{B\times
B^{\prime}}(s_{i},s_{i+k})\cdot\left(\mathbb{E}\left[\mathds{1}_{A\times
A^{\prime}}\left(\bm{\xi}_{1},\bm{\xi}_{1+k}\right)\right]-\mathbb{E}\left[\mathds{1}_{A\times
A^{\prime}}\left(\bm{V},\bm{V}^{\prime}\right)\right]\right).$ (101)

Therefore the second term in the right-hand side of Eq. 97 is bounded by

$\sum_{k=1}^{2n-2}\left|\mathbb{E}\left[\mathds{1}_{A\times
A^{\prime}}\left(\bm{\xi}_{1},\bm{\xi}_{1+k}\right)\right]-\mathbb{E}\left[\mathds{1}_{A\times
A^{\prime}}\left(\bm{V},\bm{V}^{\prime}\right)\right]\right|\cdot\left|\mu(B)\mu(B^{\prime})-\frac{1}{2n}\sum_{i=1}^{2n-2}\mathds{1}_{B\times
B^{\prime}}(s_{i},s_{i+k})\right|.$ (102)

Using A.1 we have that

$\sup_{k\in[2n-2]}\left|\mu(B)\mu(B^{\prime})-\frac{1}{2n}\sum_{i=1}^{2n-2}\mathds{1}_{B\times
B^{\prime}}(\bm{s}_{i},\bm{s}_{i+k})\right|\xrightarrow[n\to\infty]{a.s.}0.$

In addition, using once again 3.1 and the assumption in Eq. 6 we have that

$\sum_{k=1}^{2n-2}\left|\mathbb{E}\left[\mathds{1}_{A\times
A^{\prime}}\left(\bm{\xi}_{1},\bm{\xi}_{1+k}\right)\right]-\mathbb{E}\left[\mathds{1}_{A\times
A^{\prime}}\left(\bm{V},\bm{V}^{\prime}\right)\right]\right|<\infty.$

The last two equations imply that the bound in Eq. 102 tends to zero as $n$
tends to infinity for $\mu^{\mathbb{N}}$-almost every sequence
$(s_{i})_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$, completing the
proof of Eq. 95.

In order to conclude the proof of the lemma, it remains to generalize the
result in Eq. 95 to all bounded and measurable functions. This can be done
using the same techniques adopted to prove the law of large numbers, therefore
we skip the details. ∎

We now complete the proof of 1.6.

###### Proof of 1.6.

Recalling that $\tilde{g}(x,y)\coloneqq
g(x,y)-\mathbb{E}\left[g\left(\bm{V},y\right)\right]$ and thanks to 3.2, it is
enough to show that

$\frac{1}{\rho_{g,n}}\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\xrightarrow{d}\bm{\mathcal{N}}(0,1).$
(103)

The difficulty in establishing this convergence lies in the fact that we are
dealing with a sum of random variables that are neither independent nor
identically distributed. We proceed in two steps. First, we apply the the
Bernstein’s method, thus we reduce the problem to the study of a sum of
“almost” independent random variables. More precisely, we use the decay of the
correlations for the process $\bm{\xi}$ to decompose
$\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)$ into two distinct
sums, in such a way that one of them is a sum of “almost” independent random
variables and the other one is negligible (in a sense that will be specified
in due time). After having dealt with the lack of independence, we settle the
issue that the random variables are not identically distributed using the
Lyapounov’s condition.

We start with the first step. Recall that we assume the existence of two
sequences $p=p(n)$ and $q=q(n)$ such that:

  * •

$p\xrightarrow{n\to\infty}+\infty$ and $q\xrightarrow{n\to\infty}+\infty$,

  * •

$q=o(p)$ and $p=o(n)$ as $n\to\infty$,

  * •

$np^{-1}\alpha_{q}=o(1)$,

  * •

$\frac{p}{n}\cdot\sum_{j=1}^{p}j\alpha_{j}=o(1)$.

As said above, we represent the sum
$\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)$ as a sum of nearly
independent random variables (the _big blocks_ of size $p$, denoted
$\bm{\beta}_{i}$ below) alternating with other terms (the _small blocks_ of
size $q$, denoted $\bm{\gamma}_{i}$ below) whose sum is negligible. We define
$k=\lfloor(2n-1)/(p+q)\rfloor$. We can thus write

$\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)=\sum_{i=0}^{k-1}\bm{\beta}_{i}+\sum_{i=0}^{k}\bm{\gamma}_{i},$
(104)

where, for $0\leq i\leq k-1$,

$\bm{\beta}_{i}=\bm{\beta}_{i}(\tilde{g},n)\coloneqq\sum_{j=ip+iq+1}^{(i+1)p+iq}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right),\quad\quad\bm{\gamma}_{i}=\bm{\gamma}_{i}(\tilde{g},n)\coloneqq\sum_{j=(i+1)p+iq+1}^{(i+1)p+(i+1)q}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right),$
(105)

and

$\quad\quad\bm{\gamma}_{k}=\bm{\gamma}_{k}(\tilde{g},n)\coloneqq\sum_{j=kp+kq+1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right).$
(106)

Henceforth, we will omit the dependence on $\tilde{g}$ and on $n$ simply
writing $\bm{\beta}_{i}$ and $\bm{\gamma}_{i}$, in order to simplify the
notation (whenever it is clear). Setting
$\bm{H}_{n}^{\prime}\coloneqq\frac{1}{\rho_{g,n}}\sum_{i=0}^{k-1}\bm{\beta}_{i}$
and
$\bm{H}_{n}^{\prime\prime}\coloneqq\frac{1}{\rho_{g,n}}\sum_{i=0}^{k}\bm{\gamma}_{i}$,
we can write

$\frac{1}{\rho_{g,n}}\sum_{j=1}^{2n-1}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)=\bm{H}_{n}^{\prime}+\bm{H}_{n}^{\prime\prime}.$
(107)

The proof of Eq. 103 now consists of two steps. First, we show that
$\bm{H}_{n}^{\prime\prime}\xrightarrow{P}0$, and secondly we show that the
characteristic function of $\bm{H}_{n}^{\prime}$ converges to the
characteristic function of a standard Gaussian random variable. Then, we can
conclude using standard arguments.

We start by proving that $\bm{H}_{n}^{\prime\prime}\xrightarrow{P}0$. By
Chebyshev’s inequality and the fact that
$\mathbb{E}[\bm{H}_{n}^{\prime\prime}]=0$, it is enough to show that
$\mathbb{E}\left[\left(\bm{H}_{n}^{\prime\prime}\right)^{2}\right]\to 0$ as
$n\to\infty$. We can rewrite
$\mathbb{E}\left[\left(\bm{H}_{n}^{\prime\prime}\right)^{2}\right]$ as

$\frac{1}{\rho_{g,n}^{2}}\cdot\mathbb{E}\left[\left(\sum_{i=0}^{k}\bm{\gamma}_{i}\right)^{2}\right]=\frac{1}{\rho_{g,n}^{2}}\left(\mathbb{E}\left[\sum_{i=0}^{k-1}\bm{\gamma}_{i}^{2}\right]+\mathbb{E}\left[\bm{\gamma}_{k}^{2}\right]+\mathbb{E}\left[\sum_{\begin{subarray}{c}i,j=0\\\
i\neq
j\end{subarray}}^{k-1}\bm{\gamma}_{i}\bm{\gamma}_{j}\right]+2\mathbb{E}\left[\sum_{i=0}^{k-1}\bm{\gamma}_{i}\bm{\gamma}_{k}\right]\right).$
(108)

Note that, by definition of $\bm{\gamma}_{i}$ and using 3.1 once again, for
$i\neq j$, we have the bounds

$\mathbb{E}\left[\bm{\gamma}_{i}\bm{\gamma}_{j}\right]\leq
q^{2}\cdot\alpha_{p(i-j)},$ (109)

and

$\mathbb{E}\left[\bm{\gamma}_{i}\bm{\gamma}_{k}\right]\leq
q\cdot(p+q)\cdot\alpha_{p(k-i)}.$ (110)

Moreover, by the same argument used in 3.2, we have that

$\mathbb{E}\left[\bm{\gamma}_{i}^{2}\right]=\rho_{g}^{2}\cdot
q\cdot(1+o(1))=O(q)$ (111)

and

$\mathbb{E}\left[\bm{\gamma}_{k}^{2}\right]=O(p+q)=O(p).$ (112)

Hence, using 3.2,

$\frac{1}{\rho_{g,n}^{2}}\mathbb{E}\left[\sum_{i=0}^{k-1}\bm{\gamma}_{i}^{2}\right]=O\left(\frac{kq}{n}\right)=O\left(\frac{q}{p}\right)=o(1)$
(113)

and

$\frac{1}{\rho_{g,n}^{2}}\mathbb{E}\left[\bm{\gamma}_{k}^{2}\right]=o(1).$
(114)

Using Eq. 109, we also have that

$\frac{1}{\rho_{g,n}^{2}}\mathbb{E}\left[\sum_{i,j=1,i\neq
j}^{k-1}\bm{\gamma}_{i}\bm{\gamma}_{j}\right]\leq\frac{2}{\rho_{g,n}^{2}}\sum_{j=0}^{k-1}\sum_{i=j+1}^{k-1}q^{2}\cdot\alpha_{p(i-j)}=\frac{2}{\rho_{g,n}^{2}}\sum_{j=0}^{k-1}\sum_{m=1}^{k-j}q^{2}\cdot\alpha_{pm}\\\
\leq\frac{2}{\rho_{g,n}^{2}}\cdot
kq^{2}\sum_{m=1}^{k}\alpha_{pm}\leq\frac{2kq^{2}}{\rho_{g,n}^{2}\cdot
p}\sum_{m=1}^{\infty}\alpha_{m}=o(1),$ (115)

where in the last inequality we used that, since $\alpha_{n}$ is decreasing,

$\sum_{m=1}^{k}\alpha_{pm}\leq\sum_{m=1}^{k}\frac{1}{p}\cdot\sum_{s=(m-1)p+1}^{mp}\alpha_{s}\leq\frac{1}{p}\cdot\sum_{s=1}^{\infty}\alpha_{s}.$
(116)

Analogously, we can prove that

$\frac{2}{\rho_{g,n}^{2}}\mathbb{E}\left[\sum_{i=0}^{k-1}\bm{\gamma}_{i}\bm{\gamma}_{k}\right]=o(1),$
(117)

concluding the proof that $\bm{H}_{n}^{\prime\prime}\xrightarrow{P}0$.

Now we turn to the study of the limiting distribution of
$\bm{H}_{n}^{\prime}$. We have that, for $t\in\mathbb{R}$,

$\exp\left\\{it\bm{H}_{n}^{\prime}\right\\}=\exp\left\\{\frac{it}{\rho_{g,n}}\sum_{i=0}^{k-1}\bm{\beta}_{i}\right\\}.$
(118)

We now look at
$\exp\left\\{\frac{it}{\rho_{g,n}}\sum_{i=1}^{k-2}\bm{\beta}_{i}\right\\}$ and
$\exp\left\\{\frac{it}{\rho_{g,n}}\bm{\beta}_{k-1}\right\\}$. We have that the
first random variable is measurable with respect to
$\mathcal{A}_{1}^{(k-1)p+(k-2)q}$ and the second one is measurable with
respect to $\mathcal{A}_{(k-1)p+(k-1)q+1}^{\infty}$. So, by 3.1,

$\left|\mathbb{E}\left[\exp\left\\{\frac{it}{\rho_{g,n}}\sum_{i=0}^{k-1}\bm{\beta}_{i}\right\\}\right]-\mathbb{E}\left[\exp\left\\{\frac{it}{\rho_{g,n}}\sum_{i=0}^{k-2}\bm{\beta}_{i}\right\\}\right]\mathbb{E}\left[\exp\left\\{\frac{it}{\rho_{g,n}}\bm{\beta}_{k-1}\right\\}\right]\right|\leq
4\cdot\alpha_{q}.$ (119)

Iterating, we get

$\left|\mathbb{E}\left[\exp\left\\{\frac{it}{\rho_{g,n}}\sum_{i=0}^{k-1}\bm{\beta}_{i}\right\\}\right]-\prod_{i=0}^{k-1}\mathbb{E}\left[\exp\left\\{\frac{it}{\rho_{g,n}}\bm{\beta}_{i}\right\\}\right]\right|\leq
4\cdot(k-1)\cdot\alpha_{q},$ (120)

the latter quantity tending to $0$ as $k\to\infty$ thanks to the assumptions
on the sequences $p$ and $q$. The last step of the proof consists in showing
that, as $n\to\infty$,

$\prod_{i=0}^{k-1}\mathbb{E}\left[\exp\left\\{\frac{it}{\rho_{g,n}}\bm{\beta}_{i}\right\\}\right]\to
e^{\frac{t^{2}}{2}}.$ (121)

Consider a collection of independent random variables $\bm{X}_{n,i}$,
$n\in\mathbb{N},i\in[k-1]_{0}$, such that $\bm{X}_{n,i}$ has the same
distribution as $\frac{1}{\rho_{g,n}}\bm{\beta}_{i}(n)$. By [Bil95, Theorem
27.3], a sufficient condition to ensure that
$\sum_{i=0}^{k-1}\bm{X}_{n,i}\to\bm{\mathcal{N}}(0,1)$ and so verifying Eq.
121, is the well-known Lyapounov’s condition:

$\lim_{n\to\infty}\frac{1}{\rho_{g,n}^{2+\delta}}\sum_{i=0}^{k-1}\mathbb{E}\left[\bm{Y}_{n,i}^{2+\delta}\right]=0,\quad\text{for
some}\quad\delta>0,$ (122)

where $\bm{Y}_{n,i}\coloneqq\bm{X}_{n,i}\cdot\rho_{g,n}\stackrel{{\scriptstyle
d}}{{=}}\sum_{j=ip+iq+1}^{(i+1)p+iq}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)$.
The condition is satisfied with $\delta=2$ thanks to the following lemma.

######  Lemma 3.3.

Under the assumptions of 1.6, we have that for $\mu^{\mathbb{N}}$-almost every
sequence $(s_{i})_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$, uniformly
for all $i\in[k-1]_{0}$,

$\mathbb{E}\left[\left(\bm{Y}_{n,i}\right)^{4}\right]=O\left(p^{2}\cdot\sum_{j=1}^{p}j\alpha_{j}\right).$
(123)

Before proving the lemma above above, we explain how it implies the condition
in Eq. 122 with $\delta=2$. By 3.2, we have that
$\rho_{g,n}^{2}=2n\cdot\rho^{2}_{g}\cdot(1+o(1))$, thus

$\frac{1}{\rho_{g,n}^{4}}\sum_{i=0}^{k-1}\mathbb{E}\left[\bm{Y}_{n,i}^{4}\right]\leq
C\cdot\frac{k\cdot
p^{2}\cdot\sum_{j=1}^{p}j\alpha_{j}}{4n^{2}\cdot\rho^{4}_{g}}\to 0,$ (124)

where for the limit we used the fact that $\frac{k\cdot p}{n}\to 1$ and that,
by assumption, $\frac{p\cdot\sum_{j=1}^{p}j\alpha_{j}}{n}\to 0$.

We conclude the proof of 1.6 by proving 3.3. Let $A_{0}\coloneqq[p]$ and
$A_{i}\coloneqq[(i+1)p+iq]\setminus[ip+iq]$ for $i\geq 1$. Note that
$|A_{i}|=p$ for all $i\geq 0$. We have that

$\displaystyle\mathbb{E}\left[\left(\bm{Y}_{n,i}\right)^{4}\right]=\mathbb{E}\left[\left(\sum_{j=ip+iq+1}^{(i+1)p+iq}\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\right)^{4}\right]$
(125) $\displaystyle=$ $\displaystyle O\Bigg{(}\sum_{j\in
A_{i}}\mathbb{E}\left[\tilde{g}^{4}\left(\bm{\xi}_{j},s_{j}\right)\right]+\sum_{\begin{subarray}{c}j,k\in
A_{i}\\\ j\neq
k\end{subarray}}\mathbb{E}\left[\tilde{g}^{2}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}^{2}\left(\bm{\xi}_{k},s_{k}\right)\right]+\sum_{\begin{subarray}{c}j,k\in
A_{i}\\\ j\neq
k\end{subarray}}\mathbb{E}\left[\tilde{g}^{3}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\right]$
(126) $\displaystyle+$ $\displaystyle\sum_{\begin{subarray}{c}j,k,l\in
A_{i}\\\ j\neq k\neq
l\end{subarray}}\mathbb{E}\left[\tilde{g}^{2}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\tilde{g}\left(\bm{\xi}_{l},s_{l}\right)\right]+\sum_{\begin{subarray}{c}j,k,l,m\in
A_{i}\\\ j\neq k\neq l\neq
m\end{subarray}}\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\tilde{g}\left(\bm{\xi}_{l},s_{l}\right)\tilde{g}\left(\bm{\xi}_{m},s_{m}\right)\right]\Bigg{)}.$
(127)

The fact that $\tilde{g}$ is bounded and the decay of the correlations will
give us some bounds for each of these terms. First of all, since $\tilde{g}$
is bounded, we have that $\sum_{j\in
A_{i}}\mathbb{E}\left[\tilde{g}^{4}\left(\bm{\xi}_{j},s_{j}\right)\right]=O(p),$
$\sum_{j,k\in A_{i},j\neq
k}\mathbb{E}\left[\tilde{g}^{2}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}^{2}\left(\bm{\xi}_{k},s_{k}\right)\right]=O\left(p^{2}\right),$
and $\sum_{j,k\in A_{i},j\neq
k}\mathbb{E}\left[\tilde{g}^{3}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\right]=O\left(p^{2}\right).$
We now look at the fourth addend. We have that

$\sum_{\begin{subarray}{c}j,k,l\in A_{i}\\\ j\neq k\neq
l\end{subarray}}\mathbb{E}\left[\tilde{g}^{2}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\tilde{g}\left(\bm{\xi}_{l},s_{l}\right)\right]\\\
=O\left(\sum_{\begin{subarray}{c}j,k,l\in A_{i}\\\
j<k<l\end{subarray}}\mathbb{E}\left[\tilde{g}^{2}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\tilde{g}\left(\bm{\xi}_{l},s_{l}\right)\right]\right)=O\left(\sum_{l\in
A_{i}}\sum_{k=ip+iq+1}^{l-1}\sum_{j=ip+iq+1}^{k}\alpha_{l-k}\right)=O(p^{2}),$
(128)

since $\sum_{i=1}^{\infty}\alpha_{i}<+\infty$ by assumption and $|A_{i}|=p$.
Finally, we estimate the last addend. We have that

$\sum_{\begin{subarray}{c}j,k,l,m\in A_{i}\\\ j\neq k\neq l\neq
m\end{subarray}}\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\tilde{g}\left(\bm{\xi}_{l},s_{l}\right)\tilde{g}\left(\bm{\xi}_{m},s_{m}\right)\right]\\\
=O\left(\sum_{\begin{subarray}{c}j,k,l,m\in A_{i}\\\
j<k<l<m\end{subarray}}\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\tilde{g}\left(\bm{\xi}_{l},s_{l}\right)\tilde{g}\left(\bm{\xi}_{m},s_{m}\right)\right]\right)=O\left(\sum_{\begin{subarray}{c}j,k,l,m\in
A_{i}\\\ j<k<l<m\end{subarray}}\min\\{\alpha_{k-j},\alpha_{m-l}\\}\right).$
(129)

We analyse the last expression. Since the sequence
$(\alpha_{n})_{n\in\mathbb{N}}$ is decreasing, we see that

$\sum_{\begin{subarray}{c}j,k,l,m\in A_{i}\\\
j<k<l<m\end{subarray}}\min\\{\alpha_{k-j},\alpha_{m-l}\\}=\sum_{j\in
A_{i}}\sum_{x=1}^{p-j}\sum_{\begin{subarray}{c}l\in A_{i}\\\
l>j+x\end{subarray}}\sum_{y=1}^{p-l}\min\\{\alpha_{x},\alpha_{y}\\}\\\
=\sum_{j\in A_{i}}\sum_{x=1}^{p-j}\sum_{\begin{subarray}{c}l\in A_{i}\\\
l>j+x\end{subarray}}\sum_{y=1}^{x}\alpha_{x}+\sum_{j\in
A_{i}}\sum_{x=1}^{p-j}\sum_{\begin{subarray}{c}l\in A_{i}\\\
l>j+x\end{subarray}}\sum_{y=x+1}^{p-l}\alpha_{y}.$ (130)

Since $|A_{i}|=p$ we have that

$\sum_{j\in A_{i}}\sum_{x=1}^{p-j}\sum_{\begin{subarray}{c}l\in A_{i}\\\
l>j+x\end{subarray}}\sum_{y=1}^{x}\alpha_{x}\leq
p^{2}\cdot\sum_{x=1}^{p}x\alpha_{x},$ (131)

and that

$\sum_{j\in A_{i}}\sum_{x=1}^{p-j}\sum_{\begin{subarray}{c}l\in A_{i}\\\
l>j+x\end{subarray}}\sum_{y=x+1}^{p-l}\alpha_{y}\leq
p^{2}\cdot\sum_{y=1}^{p}y\alpha_{y},$ (132)

from which we conclude that

$\sum_{\begin{subarray}{c}j,k,l,m\in A_{i}\\\ j\neq k\neq l\neq
m\end{subarray}}\mathbb{E}\left[\tilde{g}\left(\bm{\xi}_{j},s_{j}\right)\tilde{g}\left(\bm{\xi}_{k},s_{k}\right)\tilde{g}\left(\bm{\xi}_{l},s_{l}\right)\tilde{g}\left(\bm{\xi}_{m},s_{m}\right)\right]=O\left(p^{2}\cdot\sum_{j=1}^{p}j\alpha_{j}\right).$
(133)

This concludes the proof of 3.3, and hence of 1.6 as well. ∎

##  Appendix A A uniform estimate for central limit theorems of weakly-
correlated random variables

######  Proposition A.1.

Let $(\bm{X}_{i})_{i\in\mathbb{N}}$ be a sequence of i.i.d. real-valued random
variables. Let also $f:\mathbb{R}^{2}\to\mathbb{R}$ be a bounded measurable
function. Then

$\sup_{k\in[n]}\left|\frac{1}{n}\sum_{i=1}^{n}f(\bm{X}_{i},\bm{X}_{i+k})-f(\bm{X}_{1},\bm{X}_{2})\right|\xrightarrow[n\to\infty]{a.s.}0.$

###### Proof.

Let $A>0$ be such that $f(x,y)\leq A$ for all $x,y\in\mathbb{R}^{2}$ and set
$\bm{S}_{n,k}\coloneqq\sum_{i=1}^{n}f(\bm{X}_{i},\bm{X}_{i+k}).$ From666Note
that the assumptions of [FMN20, Proposition 5] are satisfied: indeed by
[FMN20, Theorem 5] we have that for all $r\geq 1$ the $r$-th cumulant of
$S_{n,k}$, denoted by $K^{(r)}(S_{n,k})$ is bounded by,
$|K^{(r)}(S_{n,k})|\leq n\cdot 4^{r-1}\cdot r^{r-2}\cdot A^{r}.$ [FMN20,
Proposition 5], we have that for any $x>0$ and $k\in[n]$,

$P\left(|S_{n,k}-\mathbb{E}[S_{n,k}]|\geq x\right)\leq
2\exp\left\\{-\frac{x^{2}}{18A^{2}\cdot n}\right\\}.$ (134)

Using this bound together with a union bound, we get that for any
$\varepsilon>0$,

$\mathbb{P}\left(\sup_{k\in[n]}\left|\frac{1}{n}\sum_{i=1}^{n}f(\bm{X}_{i},\bm{X}_{i+k})-f(\bm{X}_{1},\bm{X}_{2})\right|\geq\varepsilon\right)\leq
2n\exp\left\\{-\frac{n\cdot\varepsilon^{2}}{18A^{2}}\right\\}.$ (135)

Then the statement of the proposition follows using a standard Borel-Cantelli
argument. ∎

## Acknowledgements

The authors are very grateful to Itai Benjamini, Jean Bertoin, Mathilde Bouvel
and Valentin Féray for many interesting suggestions and discussions. We also
thank Emilio Corso for a careful reading of a preliminary draft of the paper.

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