QUENCHED LAW OF LARGE NUMBERS AND QUENCHED CENTRAL LIMIT THEOREM FOR MULTIPLAYER LEAGUES WITH ERGODIC STRENGTHS

• 成果类型: Article
• 署名作者: Borga, Jacopo; Cavalli, Benedetta
• 署名单位: University of Zurich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1790
• 发表日期: 2022
• 页码: 4398-4425
• 关键词: dependent random-variables; sums

摘要

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 accord-ing 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 4 = (4i)iEZ>0 satisfying a mixing condition and an independent sequence of i.i.d. random variables (si)iEZ>0, we prove a quenched-conditioning on (si)iEZ>0-central limit theorem for sums of the form sigma ni=1 g(4i, si), where g is a bounded measurable function. We high-light that the random variables g(4i, si) are not stationary conditioning on (si)iEZ>0.