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A Self-Normalized Central Limit Theorem for Markov Random Walks

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  04 January 2016

Cheng-Der Fuh  and
Tian-Xiao Pang
Cheng-Der Fuh*
Affiliation:
National Central University
Tian-Xiao Pang*
Affiliation:
Zhejiang University
*
∗ Postal address: Graduate Institute of Statistics, National Central University, Jhongli, Taiwan. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China. Email address: [email protected]
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Abstract

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Motivated by the study of the asymptotic normality of the least-squares estimator in the (autoregressive) AR(1) model under possibly infinite variance, in this paper we investigate a self-normalized central limit theorem for Markov random walks. That is, let {Xn, n ≥ 0} be a Markov chain on a general state space X with transition probability P and invariant measure π. Suppose that an additive component Sn takes values on the real line , and is adjoined to the chain such that {Sn, n ≥ 1} is a Markov random walk. Assume that Sn = ∑k=1nξk, and that {ξn, n ≥ 1} is a nondegenerate and stationary sequence under π that belongs to the domain of attraction of the normal law with zero mean and possibly infinite variance. By making use of an asymptotic variance formula of Sn / √n, we prove a self-normalized central limit theorem for Sn under some regularity conditions. An essential idea in our proof is to bound the covariance of the Markov random walk via a sequence of weight functions, which plays a crucial role in determining the moment condition and dependence structure of the Markov random walk. As illustrations, we apply our results to the finite-state Markov chain, the AR(1) model, and the linear state space model.

Keywords

Self-normalizedcentral limit theoremMarkov random walkPoisson equationdomain of attraction of the normal law

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J25: Continuous-time Markov processes on general state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 452 - 478
Copyright
© Applied Probability Trust 

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