On Coupling Constructions and Rates in the Clt for Dependent Summands with Applications to the Antivoter Model and Weighted U-statistics

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 t i i∈ t = 0 1 where is the vertex set of an n-vertex regular graph, and X t i = +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 t+1 i = −X t j and X t+1 k = X t k for all k = i. For a stationary antivoter chain, we study the normal approximation of Un = U t n = ∑ i X t i for large n and consider some conditions on sequences of graphs such that Un 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.