Central limit theorem for functionals of a linear process
type: Item
instance of: scholarly article
title: Central limit theorem for functionals of a linear process
zbMATH Open document ID: 0568.60020
DOI: 10.1007/BF00966294
published in: Lithuanian Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: The central limit theorem for a stationary sequence \(\eta_ t\), \(t\in {\mathbb{Z}}\), where \(\eta_ t=f(X_ t)\), is proved under the conditions \(\sum | r_{\eta}(t)| <\infty\) and \(\sum r_{\eta}(t)>0\). Here \(r_{\eta}(t)=E\eta_ 0\eta_ t\), \(X_ t=\sum_{s\in {\mathbb{Z}}}a(t- s)\xi_ s\) is a linear process, the variables \(\xi_ s\), \(s\in {\mathbb{Z}}\), are independent and identically distributed, \(f: {\mathbb{R}}\to {\mathbb{R}}\) belongs to some class of analytical functions which contains all polynomials. The proof is based on the method of semi-invariants and the expansion of f(x) in Appell polynomials which play the role of Hermite polynomials for the Gaussian process \(X_ t\).
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 3907457
zbMATH Keywords: central limit theorem
zbMATH Keywords: linear process
zbMATH Keywords: method of semi-invariants
author: L. Giraitis
MaRDI profile type: Publication
cites work: Q3865207
cites work: Central limit theorems for non-linear functionals of Gaussian fields
cites work: Non-central limit theorems for non-linear functional of Gaussian fields
cites work: CLT and other limit theorems for functionals of Gaussian processes
cites work: Q5535438
cites work: The Relation of the Classical Orthogonal Polynomials to the Polynomials of Appell
cites work: Q5512473
cites work: Q3662382
cites work: Weak convergence to fractional brownian motion and to the rosenblatt process
cites work: Convergence of integrated processes of arbitrary Hermite rank
cites work: Central Limit Theorems for dependent variables. I
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description: scientific article; zbMATH DE number 3907457