Asymptotics for the local time of a strongly dependent vector-valued Gaussian random field
type: Item
instance of: scholarly article
title: Asymptotics for the local time of a strongly dependent vector-valued Gaussian random field
zbMATH Open document ID: 0874.60046
DOI: 10.1007/BF02187395
author: Paul Doukhan
author: José R. León
published in: Acta Mathematica Hungarica
publication date: 1997-11-10T00:00:00Z
review text: The authors study the local time of normalized vector-valued Gaussian random fields \(X=(X_t; t\in\mathbb{R}^d)\), \(X_t\in\mathbb{R}^p\). Extending a result of \textit{S. M. Berman} [Stochastic Processes Appl. 12, 1-26 (1981; Zbl 0471.60082)] they obtain a series expansion of the local time \(\ell_X\) in terms of multidimensional Hermite polynomials. Moreover, for \(X\) stationary with a covariance matrix of long range dependence type, the authors apply a theorem of \textit{M. V. Sanchez de Naranjo} [J. Multivariate Anal. 44, No. 2, 227-255 (1993; Zbl 0770.60025)] to deduce, from this series expansion, the asymptotic behaviour of the local time \(\ell_X([0,\tau]^d,x)\) \((x\in\mathbb{R}^p)\) as \(\tau\to\infty\), in terms of random spectral measures associated to \(X\). Finally, the authors consider a sufficiently regular bijective transform \(Y\) of \(X\) (\(X\) stationary) and give series expansions of the kernel estimates for the marginal density of \(Y\). This extends a result of \textit{M. Rosenblatt} [NSF-CBMS Regional Conference Series in Probability and Statistics 3 (1991)].
reviewed by: M.Dozzi
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 878138
zbMATH Keywords: local time
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: Edgeworth expansion
zbMATH Keywords: long range dependence
zbMATH Keywords: Gaussian random fields
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: stationary
zbMATH Keywords: random spectral measures
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02187395
OpenAlex ID: W1985814741
description: scientific article; zbMATH DE number 878138