A central limit theorem for nonhomogeneous semi-Markov random evolutions

We prove a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation. We use a random normalization, which fits non-stationary situations. The theorem applies directly to Markov random fields, including the cases of phase transition and lack of stationarity. One consequence is the asymptotic normality of the maximum pseudo-likelihood estimator for Markov fields in complete generality.

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A Self-Normalized Central Limit Theorem for Markov Random Walks

Advances in Applied Probability ◽

10.1017/s0001867800005681 ◽

2012 ◽

Vol 44 (02) ◽

pp. 452-478

Author(s):

Cheng-Der Fuh ◽

Tian-Xiao Pang

Keyword(s):

Markov Chain ◽

Random Walk ◽

Central Limit Theorem ◽

Limit Theorem ◽

Random Walks ◽

State Space ◽

Central Limit ◽

Infinite Variance ◽

Markov Random ◽

Markov Random Walk

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 {X n , 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 S n takes values on the real line , and is adjoined to the chain such that {S n , n ≥ 1} is a Markov random walk. Assume that S n = ∑ k=1 n ξ k , and that {ξ n , 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 S n / √n, we prove a self-normalized central limit theorem for S n 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.

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A central limit theorem for conditionally centred random fields with an application to Markov fields

Journal of Applied Probability ◽

10.1239/jap/1032265209 ◽

1998 ◽

Vol 35 (3) ◽

pp. 608-621 ◽

Cited By ~ 5

Author(s):

Francis Comets ◽

Martin Janžura

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Random Fields ◽

Markov Random Fields ◽

Moment Condition ◽

Markov Random ◽

Pseudo Likelihood ◽

Strict Positivity ◽

Markov Fields

We prove a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation. We use a random normalization, which fits non-stationary situations. The theorem applies directly to Markov random fields, including the cases of phase transition and lack of stationarity. One consequence is the asymptotic normality of the maximum pseudo-likelihood estimator for Markov fields in complete generality.

Download Full-text

A Self-Normalized Central Limit Theorem for Markov Random Walks

Advances in Applied Probability ◽

10.1239/aap/1339878720 ◽

2012 ◽

Vol 44 (2) ◽

pp. 452-478 ◽

Cited By ~ 1

Author(s):

Cheng-Der Fuh ◽

Tian-Xiao Pang

Keyword(s):

Markov Chain ◽

Random Walk ◽

Central Limit Theorem ◽

Limit Theorem ◽

Random Walks ◽

State Space ◽

Central Limit ◽

Infinite Variance ◽

Markov Random ◽

Markov Random Walk

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.

Download Full-text