Open Access
November 1997 On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics
Yosef Rinott, Vladimir Rotar
Ann. Appl. Probab. 7(4): 1080-1105 (November 1997). DOI: 10.1214/aoap/1043862425

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)=Xi(t)iϵV,t=0,1,, where V is the vertex set of an n-vertex regular graph, and Xi(t)=+1or1. The chain evolves from time t to t+1 by choosing a random vertex i, and a random neighbor of it j, and setting Xi(t+1)=Xj(t) and Xk(t+1)=Xk(t) for all ki. For a stationary antivoter chain, we study the normal approximation of Un=Un(t)=ΣiXi(t) 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.

Citation

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Yosef Rinott. Vladimir Rotar. "On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted -statistics." Ann. Appl. Probab. 7 (4) 1080 - 1105, November 1997. https://doi.org/10.1214/aoap/1043862425

Information

Published: November 1997
First available in Project Euclid: 29 January 2003

zbMATH: 0890.60019
MathSciNet: MR1484798
Digital Object Identifier: 10.1214/aoap/1043862425

Subjects:
Primary: 60F05 , 60K35
Secondary: 60J10 , 62E20

Keywords: distance regularity , Markov chains , Random graphs , Stein's method

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.7 • No. 4 • November 1997
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