## On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $$U$$-statistics.(English)Zbl 0890.60019

Summary: 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 $${\mathbf X}^{(t)}$$, and a function $$U= U({\mathbf X}^{(t)})$$, 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 $${\mathbf X}^{(t)}= \{X^{(t)}_i\}_{i\in{\mathcal V}}$$, $$t= 0,1,\dots$$, where $${\mathcal V}$$ 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 $$k\neq i$$. For a stationary antivoter chain, we study the normal approximation of $$U_n= U^{(t)}_n= \sum_i X^{(t)}_i$$ 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.

### MSC:

 60F05 Central limit and other weak theorems 60K35 Interacting random processes; statistical mechanics type models; percolation theory 62E20 Asymptotic distribution theory in statistics 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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