Quenched law of large numbers and quenched central limit theorem for multiplayer leagues with ergodic strengths

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 $\mathit{\xi }={({\mathit{\xi }}_{\mathit{i}})}_{\mathit{i}\in {\mathbb{Z}}_{>0}}$ satisfying a mixing condition and an independent sequence of i.i.d. random variables ${({\mathit{s}}_{\mathit{i}})}_{\mathit{i}\in {\mathbb{Z}}_{>0}}$, we prove a quenched—conditioning on ${({\mathit{s}}_{\mathit{i}})}_{\mathit{i}\in {\mathbb{Z}}_{>0}}$—central limit theorem for sums of the form ${\sum }_{\mathit{i}=1}^{\mathit{n}}\mathit{g}({\mathit{\xi }}_{\mathit{i}},{\mathit{s}}_{\mathit{i}})$, where g is a bounded measurable function. We highlight that the random variables $\mathit{g}({\mathit{\xi }}_{\mathit{i}},{\mathit{s}}_{\mathit{i}})$ are not stationary conditioning on ${({\mathit{s}}_{\mathit{i}})}_{\mathit{i}\in {\mathbb{Z}}_{>0}}$.