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A Self-Normalized Central Limit Theorem for Markov Random Walks
| Content Provider | Scilit |
|---|---|
| Author | Fuh, Cheng-Der Pang, Tian-Xiao |
| Copyright Year | 2012 |
| Description | 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. |
| Related Links | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2B84C2AB8F5FBF8B564FC172332DD3E9/S0001867800005681a.pdf/div-class-title-a-self-normalized-central-limit-theorem-for-markov-random-walks-div.pdf |
| Ending Page | 478 |
| Page Count | 27 |
| Starting Page | 452 |
| ISSN | 00018678 |
| e-ISSN | 14756064 |
| DOI | 10.1017/s0001867800005681 |
| Journal | Advances in Applied Probability |
| Issue Number | 02 |
| Volume Number | 44 |
| Language | English |
| Publisher | Cambridge University Press (CUP) |
| Publisher Date | 2012-06-01 |
| Access Restriction | Open |
| Subject Keyword | Advances in Applied Probability Mathematical Physics Central Limit Theorem Markov Random Walk Domain of Attraction of the Normal Law |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics Statistics and Probability |
