A new look at Bergström's theorem on convergence in distribution for sums of dependent random variables
type: Item
instance of: scholarly article
title: A new look at Bergström's theorem on convergence in distribution for sums of dependent random variables
zbMATH Open document ID: 0536.60032
DOI: 10.1007/BF02760561
author: Henry A. Krieger
published in: Israel Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: In the paper Period. Math. Hung. 2, 173-190 (1972; Zbl 0252.60009), the reviewer used a comparison method to prove a limit theorem for sums of \(\phi\)-mixing random variables. The author of the paper under review claims that there is a gap in my proof. By an analogous comparison but dealing with characteristic functions instead of Gaussian transforms, used by me, he proves that the theorem is true. Now he has misunderstood my definition of a certain associated sequence of random variables. To a given truncated sequence \(\{\xi_{j}^{(n)}(\epsilon)\}^{k_ n}_{j=1}\) of random variables I have associated a sequence \(\{\zeta_ j^{(n)}(\epsilon)\}^{k_ n}_{j=1}\) such that \(E[\zeta_ j^{(n)}(\epsilon)\zeta^{(n)}_{j-\nu}(\epsilon)]=E[\xi_ j^{(n)}(\epsilon)\xi^{(n)}_{j+\nu}(\epsilon)]\) for 0\(\leq \nu \leq k\), \(j+\nu \leq k_ n\) and \(E[\zeta_ j^{(n)}(\epsilon)\zeta^{(j)}_{j-\nu}(\epsilon)]=E[\xi_ j^{(n)}(\epsilon)\xi^{(n)}_{j-\nu}(\epsilon)]\) otherwise where k is a given positive integer. Hence in the first relation it is required that also \(j\geq k\) and ''otherwise'' includes the case \(j<k\). It is also required that the vector \((\zeta_ 1^{(n)}(\epsilon),...,\zeta^{(n)}_{k_ n}(\epsilon))\) is Gaussian. The present author correctly points out that such a vector may not exist since its covariance matrix may not be positive definite. However I consider arbitrarily large k and it follows by the \({\mathcal I}\)-condition and my condition \(C_ 5\) that the covariances to \(E[\xi_ j^{(n)}(\epsilon)\xi^{(n)}_{j+\nu}(\epsilon)]\) converge to 0 uniformly with respect to j and \(\nu\) as \(n\to \infty\). Hence we may assume that the covariance matrix is positive definite for sufficiently large k and \(n>k\). The author has chosen the Gaussian vector \((\zeta_ 1^{(n)}(\epsilon),...\zeta_{k_ n}^{(n)}(\epsilon))\) in the more natural way \(E[\zeta_ i^{(n)}(\epsilon)\zeta_ j^{(n)}(\epsilon)]=E[\xi_ i^{(n)}(\epsilon)\xi_ j^{(n)}(\epsilon)]\) for all i and j. This choice is also …
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 3852118
zbMATH Keywords: comparison method
zbMATH Keywords: sums of \(\phi\)-mixing random variables
MaRDI profile type: Publication
cites work: A Comparison Method for Distribution Functions of Sums of Independent and Dependent Random Variables
cites work: On the convergence of sums of random variables in distribution under mixing condition
cites work: Q4061742
cites work: Q5560061
cites work: Q5829358
cites work: Some Limit Theorems for Stationary Processes
description: scientific article; zbMATH DE number 3852118