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

TL;DR

This paper introduces a new probabilistic model for multi-player sports leagues with ergodic strength dynamics, proving quenched laws of large numbers and central limit theorems for team victories conditioned on initial strengths.

Contribution

It generalizes existing models by incorporating ergodic processes for team strengths and establishes a novel quenched CLT for sums involving non-stationary, ergodic processes.

Findings

Proved a quenched law of large numbers for team victories.

Established a quenched central limit theorem for sums of non-stationary processes.

Developed a new CLT for sums of functions of stationary and i.i.d. sequences.

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