CONCENTRATION OF MEASURE FOR BROWNIAN PARTICLE SYSTEMS INTERACTING THROUGH THEIR RANKS

• 成果类型: Article
• 署名作者: Pal, Soumik d; Shkolnikov, Mykhaylo
• 署名单位: University of Washington; University of Washington Seattle; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP954
• 发表日期: 2014
• 页码: 1482-1508
• 关键词: information inequalities; transportation cost; skorokhod problem; convex duality

摘要

We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using transportation cost inequalities for stochastic processes we provide uniform fluctuation bounds for the ordered particles, their local time of collisions and various associated statistics over intervals of time. For example, such processes, when exponentiated and rescaled, exhibit power law decay under stationarity; we derive concentration bounds for the empirical estimates of the index of the power law over large intervals of time. A key ingredient in our proofs is a novel upper bound on the Lipschitz constant of the Skorokhod map that transforms a multidimensional Brownian path to a path which is constrained not to leave the positive orthant.