Central limit theorem for weakly dependent variables
type: Item
instance of: scholarly article
title: Central limit theorem for weakly dependent variables
zbMATH Open document ID: 0645.60026
DOI: 10.1007/BF01159533
author: N. K. Bakirov
published in: Mathematical Notes
publication date: 1987-01-01T00:00:00Z
review text: Let \(X_ n\), \(n=0,\pm 1,..\). be a stationary random sequence with E \(X_ 1=0\), Var \(X_ 1<\infty\), \(S_ n=X_ 1+...+X_ n\), \(S_ n^*=S_ n/\sqrt{Var S_ n}\), \({\mathcal M}_ k=\sigma \{X_ i\), \(| i| \geq k\}\), satisfying the following conditions: 1) for all integers i, \(\rho_ i(n)=E| E(X_ 0X_ i| {\mathcal M}_ n)-E X_ 0X_ i| \to 0\), \(n\to \infty,\) 2) \(\sum^{\infty}_{k=-\infty}E X_ 0X_ k<\infty\) uniformly over all \(\bar X{}_ i\), \(i=0,\pm 1,....\) If \(\sigma^ 2=\sum^{\infty}_{k=-\infty}E X_ 0X_ k\neq 0\), then \[ \lim_{n\to \infty}P\{S^ x_ n<x\}\to (2\pi)^{-}\int^{x}_{- \infty}e^{-u^ 2/2}du. \]
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 4052676
zbMATH Keywords: mixing condition
zbMATH Keywords: central limit theorem
zbMATH Keywords: stationary random sequence
MaRDI profile type: Publication
cites work: Q5535438
cites work: Q3894712
cites work: Q5342183
cites work: Q3266686
full work available at URL: https://doi.org/10.1007/bf01159533
OpenAlex ID: W2023697633
description: scientific article; zbMATH DE number 4052676