Limit behavior of quadratic forms of moving averages and statistical solutions of the Burgers' equation
type: Item
instance of: scholarly article
title: Limit behavior of quadratic forms of moving averages and statistical solutions of the Burgers' equation
zbMATH Open document ID: 0817.60030
author: Yiming Hu
author: Wojbor A. Woyczyński
published in: Journal of Multivariate Analysis
publication date: 1995-07-17T00:00:00Z
review text: Let \(\dots, X_{-1}, X_ 0, X_ 1, \dots\) be symmetric i.i.d. random variables belonging to the domain of normal attraction of an \(\alpha\)- stable distribution with \(1 < \alpha < 2\), and let \(c_ j = j^{-\beta} \Lambda (j)\) with \(\alpha \beta > 1\) and slowly varying \(\Lambda (j)\). The limit behaviour of the partial sum processes \(D_ n(t) = \sum^{[nt]}_{k=1} Q(\sum^ k_{j=-\infty}c_{k-j} X_ j)\), where \(Q(x)\) is a second order polynomial, is studied. In the second part of the paper the problem of asymptotic behaviour of solutions of the Burgers' equation \(u_ t + uu_ x = \nu u_{xx}\), where \(\nu > 0\), with initial datum \(u(x,0) = u_ 0(x)\) being a strictly stationary process, is discussed. It is shown how the results of the first part can be applied here.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 718683
zbMATH Keywords: quadratic forms
zbMATH Keywords: moving averages
zbMATH Keywords: convergence of finite-dimensional distributions
zbMATH Keywords: domain of normal attraction
zbMATH Keywords: Burgers' equation
reviewed by: T. Inglot
MaRDI profile type: Publication
full work available at URL: jmva.1995.1002
OpenAlex ID: W2090242976
DOI: 10.1006/JMVA.1995.1002
description: scientific article; zbMATH DE number 718683