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∈script V sign}, t = 0, 1,..., where script V sign 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 U_n = U^(t)_n = ∑_i X^(t)_i 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.