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 $X^{(t)} = {X_i^{(t)}} _{i \epsilon \mathscr{V}}, t = 0, 1, \dots,$ where $\mathscr{V}$ is the vertex set of an n-vertex regular graph, and $X_i^{(t)} = +1 \text{or} -1$. 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 $k \not= i$. For a stationary antivoter chain, we study the normal approximation of $U_n = U_n^{(t)} = \Sigma_i X_i^{(t)}$ 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.