Sojourn in an elliptical domain
type: Item
instance of: scholarly article
title: Sojourn in an elliptical domain
zbMATH Open document ID: 0598.60037
DOI: 10.1016/0304-4149(86)90103-1
author: Murad S. Taqqu
published in: Stochastic Processes and their Applications
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://hdl.handle.net/1813/8515
review text: Let \(X(t)=(X_ 1(t),X_ 2(t))\), \(t\geq 0\), be a two-dimensional stationary Gaussian process with zero mean. \(X_ 1(t)\) and \(X_ 2(t)\) are correlated with long-range dependence and the cross-relations between \(X_ 1(t)\) and \(X_ 2(t)\) are allowed to have the same order of magnitude as the auto-correlations. Let \(D=\{(x_ 1,x_ 2):\) \(a^ 2x^ 2_ 1+b^ 2x^ 2_ 2\leq 1\}\), \(0<a\leq b\), be an elliptical domain. Define \(M(t)=\int^{t}_{0}I[X(s)\in D]ds\), \(t\geq 0\), which denotes the amount of time that the process X(\(\cdot)\) has spent in the domain D by time t. The main theorem concerns the asymptotic behaviour as \(t\to \infty\) of the standardized process \[ Z_ t(\tau)=\{M(t\tau)- EM(t\tau)\}/[Var M(t)]^{1/2},\quad \tau \geq 0. \] It is shown that as \(t\to \infty\) the process \(Z_ t(\tau)\) converges weakly in C[0,\(\infty)\) to a limiting process \(\bar Z(\tau)\) which admits a Wiener- Itô double integral representation.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60K99
zbMATH DE Number: 3965093
zbMATH Keywords: self-similar process
zbMATH Keywords: weak convergence
zbMATH Keywords: two-dimensional stationary Gaussian process
zbMATH Keywords: long-range dependence
zbMATH Keywords: cross-relations
zbMATH Keywords: Wiener-Itô double integral representation
MaRDI profile type: Publication
OpenAlex ID: W2103807271
cites work: Multiple stochastic integrals with dependent integrators
cites work: Q3738334
description: scientific article; zbMATH DE number 3965093