ON COUPLING CONSTRUCTIONS AND RATES IN THE CLT FOR DEPENDENT SUMMANDS WITH APPLICATIONS TO THE ANTIVOTER MODEL AND WEIGHTED U-STATISTICS

• 成果类型: Article
• 署名作者: Rinott, Yosef; Rotar, Vladimir
• 署名单位: University of California System; University of California San Diego; Russian Academy of Sciences; Central Economics & Mathematics Institute RAS
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 1997
• 页码: 1080-1105
• 关键词: CENTRAL-LIMIT-THEOREM; asymptotic distributions; random-variables

摘要

This paper deals with rates of convergence in the CLT for certain types of dependency. The main idea is to combine a modification of a theorem of Stein, requiring a coupling construction, with a dynamic set-up provided by a Markov structure that suggests natural coupling variables. More specifically, given a stationary Markov chain X-(t) and a function U=U(X-(t)), we propose a way to study the proximity of U to a normal random variable when the state space is large. We apply the general method to the study of two problems. In the first, we consider the antivoter chain X-(t)=X-i((t))}(i is an element of V), t=0, 1, , where V is the vertex set of an n-vertex regular graph, and X-i((t)) = +1 or -1. The chain evolves from time t to t+1 by choosing a random vertex i, and a random neighbor of it j, and setting X-i(t+1) = -X-j((t)) and X-k((t+1)) = X-k((t)) for all k not equal i. For a stationary antivoter chain, we study the normal approximation of U-n=U-n((t)) = Sigma X-i(i)(t) for large n and consider some conditions on sequences of graphs such that U-n is asymptotically normal, a problem posed by Aldous and Fill. The same approach may also be applied in situations where a Markov chain does not appear in the original statement of a problem but is constructed as an auxiliary device. This is illustrated by considering weighted U-statistics. In particular we are able to unify and generalize some results on normal convergence for degenerate weighted U-statistics and provide rates.