#  Quenched law of large numbers and quenched central limit theorem for multi-player leagues with ergodic strengths (with Benedetta Cavalli). Annals of Applied Probability 32 (2022), no. 6, pp. 4398-4425.

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 $$\mathbb{R}_+=\mathbb{R}\cap[0,\infty)$$.

#### The model

We consider a league of $$2n\in2\mathbb{N}$$ teams denoted by $$\{T_{i}\}_{i\in [2n-1]_0}$$ whose initial random strengths are denoted by $$\{S_{i}\}_{i\in [2n-1]_0}\in \mathbb{R}_+^{2n}$$.

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 $$S_i$$ of the team $$T_i$$ is modified every day according to a discrete time $$\mathbb{R}_+$$-valued stochastic process $$\textbf{\xi}^i=(\xi^i_j)_{j\in \mathbb{N}}$$. More precisely, the strength of the team $$T_i$$ on the $$p$$-th day is equal to $$S_i\cdot \xi^i_p\in \mathbb{R}_+$$.

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 $$S_i, S_j, \xi^i_p, \xi^j_p$$, the probability that $$T_i$$ wins is $$f(S_i\cdot \xi^i_p,S_j\cdot \xi^j_p)$$.

Moreover, conditionally on the strengths of the teams, the results of different matches are independent.

#### Assumptions

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. $$S_{0}=s$$ a.s., and we set

$$W_n(s):=\text{Number of wins of the team }T_{0}\text{ at the end of a league with \(2n$$ players}.\)

We investigate a quenched law of large numbers and a quenched central limit theorem for $$W_n(s)$$. In the following two subsections we state some assumptions on the model.

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

We make the following natural assumptions 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 $$\xi=(\xi_\ell)_{\ell\in \mathbb{N}}$$ be a discrete time $$\mathbb{R}_+$$-valued stochastic process such that $$\xi_\ell\stackrel{d}{=}\nu,$$ for all  $$\ell\in \mathbb{N},$$ and (this is a weak-form of the stationarity property for stochastic processes):
$$\left(\xi_\ell, \xi_k\right)\stackrel{d}{=} \left(\xi_{\ell+\tau}, \xi_{k+\tau}\right), \quad\text{for all}\quad \ell,k,\tau\in\mathbb{N}.$$

We further assume that the process $$\xi$$ is weakly-mixing, that is, for every $$A \in \sigma(\xi_1)$$ and every collection of sets $$B_\ell \in \sigma(\xi_\ell)$$, it holds that
$$\frac{1}{n} \sum_{\ell=1}^n \left|\mathbb{P}(A \cap B_\ell)-\mathbb{P}(A)\mathbb{P}(B_\ell)\right|\rightarrow 0.$$

The additional assumptions on our model are the following:

•  For all $$i\in[2n-1]_0$$, the stochastic processes $$\xi^i$$ are independent copies of $$\xi$$.
• The initial random strengths $$\{S_i\}_{i\in[2n-1]}$$ 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 $$\{S_i\}_{i\in[2n-1]}$$ are independent of the processes $$\{ \xi^i\}_{i\in[2n-1]_0}$$ and of the process $$\xi$$.
##### 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 $$\xi$$. For $$k \in \mathbb{N}$$, we introduce the two $$\sigma$$-algebras $$\mathcal{A}_1^{k} = \sigma \left(\xi_1, \dots, \xi_k \right)$$ and $$\mathcal{A}_k^{\infty} = \sigma \left(\xi_k, \dots \right)$$ and we define for all $$n\in\mathbb{N}$$,
$$\alpha_n = \sup_{k \in \mathbb{N}, A \in \mathcal{A}_1^{k},B \in \mathcal{A}_{k+n}^{\infty}} \left| \mathbb{P} \left( A \cap B \right) – \mathbb{P}(A)\mathbb{P}(B) \right|.$$
We assume that $$\sum_{n=1}^{\infty} \alpha_n < \infty.$$
Note that this condition, in particular, implies that the process $$\xi$$ 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:

• $$p\to +\infty$$ and $$q\to+\infty$$,
• $$q=o(p)$$ and $$p=o(n)$$ as $$n \to \infty$$,
• $$n p^{-1 } \alpha_q =o(1)$$,
• $$\frac{p}{n} \cdot \sum_{j=1}^p j \alpha_j = o(1)$$.

#### Main results

Let $$V, V’, U, U’$$ be four independent random variables such that $$V\stackrel{d}{=}V’\stackrel{d}{=}\nu$$ and $$U\stackrel{d}{=} U’\stackrel{d}{=}\mu$$. 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 $$\frac{W_n(s)}{2n}$$ 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 $$\frac{W_n(s)}{2n}$$ on the event

$$S_0=s\quad \text{ and }\quad(S_i)_{{i\in[2n-1]}}=(s_i)_{{i\in[2n-1]}}.$$

Theorem (Quenched law of large numbers).
Suppose that the assumptions for the law of large numbers 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 in probability holds

$$\frac{W_n(s)}{2n}\to\ell(s)=\mathbb{E}\left[f\left(s\cdot V, U\cdot V’\right)\right]=\int_{\mathbb{R}^3_+} f\left(s\cdot v,u\cdot v’\right)d\nu(v)d\nu(v’)d\mu(u).$$

Theorm (Quenched central limit theorem).
Suppose that the assumptions for the law of large numbers and the central limit theorem 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 in distribution holds

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

where, for $$F_s(x,y)=\mathbb{E}\left[f\left(s\cdot x,y\cdot V’\right)\right]$$ and $$\tilde F_s \left(x,y\right) = F_s\left(x,y \right) – \mathbb{E}[F_s\left(V, y\right)]$$,
$$\sigma(s)^2=\mathbb{E}\left[F_s(V,U)-\left(F_s(V,U)\right)^2\right]=\ell(s)-\mathbb{E}\left[\left(F_s(V,U)\right)^2\right]$$
and
$$\rho(s)^2= \mathbb{E} \left[ \tilde F_s (V, U)^2 \right] + 2 \cdot \sum_{k=1}^{\infty} \mathbb{E}\left[ \tilde F_s(\xi_1, U) \tilde F_s (\xi_{1+k}, U’) \right],$$
the last series being convergent.

To obtain our results, we also prove the following general result that we believe to be of independent interest.

Theorem.
Let $$\xi=(\xi_\ell)_{\ell\in \mathbb{N}}$$ be a discrete time $$\mathbb{R}_+$$-valued stochastic process such that $$\xi_\ell\stackrel{d}{=}\nu,$$ for all  $$\ell\in \mathbb{N},$$ and $$\left(\xi_\ell, \xi_k\right)\stackrel{d}{=} \left(\xi_{\ell+\tau}, \xi_{k+\tau}\right),$$ for all $$\ell,k,\tau\in\mathbb{N}.$$ Suppose that the assumptions for the central limit theorem 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 $$\tilde g (x,y) = g(x,y) – \mathbb{E} \left[ g\left(V, y\right) \right]$$. Then, the quantity
$$\rho_g^2 = \mathbb{E} \left[ \tilde g (V, U)^2 \right] + 2 \cdot \sum_{k=1}^{\infty} \mathbb{E}\left[ \tilde g(\xi_1, U) \tilde g (\xi_{1+k}, U’) \right]$$
is finite. Moreover, for $$\mu^{\mathbb{N}}$$-almost every sequence $$(s_i)_{{i\in\mathbb{N}}}\in\mathbb{R}^{\mathbb{N}}_{+}$$, the following convergence in distribution holds

$$\frac{\sum_{j=1}^{2n-1}\tilde g\left(\xi_j,s_j\right)}{\sqrt{2n}}\to \mathcal{N}(0,\rho_g^2).$$

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