The Annals of Applied Probability

On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics

Yosef Rinott and Vladimir Rotar

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Abstract

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.

Article information

Source
Ann. Appl. Probab. Volume 7, Number 4 (1997), 1080-1105.

Dates
First available in Project Euclid: 29 January 2003

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1043862425

Digital Object Identifier
doi:10.1214/aoap/1043862425

Mathematical Reviews number (MathSciNet)
MR1484798

Zentralblatt MATH identifier
0890.60019

Subjects
Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 62E20: Asymptotic distribution theory 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Stein's method random graphs distance regularity Markov chains

Citation

Rinott, Yosef; Rotar, Vladimir. On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics. Ann. Appl. Probab. 7 (1997), no. 4, 1080--1105. doi:10.1214/aoap/1043862425. http://projecteuclid.org/euclid.aoap/1043862425.


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