A general central limit theorem for strong mixing sequences
type: Item
instance of: scholarly article
title: A general central limit theorem for strong mixing sequences
zbMATH Open document ID: 1317.60018
author: Magnus Ekström
published in: Statistics \& Probability Letters
publication date: 2014-11-03T00:00:00Z
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 6363288
zbMATH Keywords: central limit theorem
zbMATH Keywords: strong mixing sequences
zbMATH Keywords: non-stationary sequences
zbMATH Keywords: triangular array
zbMATH Keywords: weakly approaching sequences
reviewed by: Tae Il Jeon
MaRDI profile type: Publication
full work available at URL: j.spl.2014.07.024
OpenAlex ID: W2040738104
cites work: Q4939943
cites work: Weakly approaching sequences of random distributions
cites work: Q5293978
cites work: The Invariance Principle for Stationary Processes
cites work: Mixing: Properties and examples
cites work: Q4335412
cites work: Subsampling for heteroskedastic time series
cites work: Subsampling
cites work: A CENTRAL LIMIT THEOREM AND A STRONG MIXING CONDITION
DOI: 10.1016/J.SPL.2014.07.024
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review text: Author's abtract: A central limit theorem for strong mixing sequences is given that applies to both nonstationary sequences and triangular array settings. The result improves on an earlier central limit theorem for this type of dependence given by \textit{D. N. Politis} et al. in [J. Econom. 81, No. 2, 281--317 (1997; Zbl 0904.62059)].NEWLINENEWLINENEWLINEBelow are some excerpts from the author's introduction:NEWLINENEWLINENEWLINEFor a survey of the literature on CLTs for mixing sequences we refer to, e.g., [\textit{P. Doukhan}, Mixing: Properties and examples. New York: Springer-Verlag (1994; Zbl 0801.60027); \textit{Z.-Y. Lin} and \textit{C.-R. Lu}, Limit theory for mixing dependent random variables. Dordrecht: Kluwer Academic Publishers. New York, NY: Science Press (1996; Zbl 0889.60001); \textit{R. C. Bradley}, Introduction to strong mixing conditions. Vol. 1. Heber City, UT: Kendrick Press (2007; Zbl 1133.60001); Vol. 2. Heber City, UT: Kendrick Press (2007; Zbl 1133.60002); Vol. 3. Heber City, UT: Kendrick Press (2007; Zbl 1134.60004)].NEWLINENEWLINENEWLINEIt should be noted that in addition to moment and mixing conditions, strict stationarity is assumed in many cases. An exception is Politis et al. [loc. cit.], who obtained a CLT which applies to both non-stationary sequences and triangular array settings. Let \(\{ X_{n,i}: 1\leq I \leq d_n \}\) be a triangular array of random variables and denote the strong mixing coefficient corresponding to the \(n\)th row by \(\alpha_{n}(\cdot)\). Define \(X_{n,d_{n}}=d^{-1}_{n}\sum^{d_{n}}_{i=1} X_{n,i}\), \({X}_{n,b,t} =b^{-1}\sum^{t+b-1}_{i=t} X_{n,i}\), and \(\sigma^{2}_{n,b,t}=\text{var}(b^{1/2}={X}_{n,b,t})\). Politis et al. [loc.\,cit., Theorem A.1; Subsampling. New York, NY: Springer (1999; Zbl 0931.62035), Theorem B.0.1] showed that \(d^{1/2}_{n}(=X_{n,d_{n}}-E ={X}_{n,d_{n}}) \) converges weakly to a normal distribution with mean zero and variance \(\sigma^{2}\) under the following conditions: for some \(\delta …
description: scientific article; zbMATH DE number 6363288