Central limit theorems for the local times of certain Markov processes and the squares of Gaussian processes
type: Item
instance of: scholarly article
title: Central limit theorems for the local times of certain Markov processes and the squares of Gaussian processes
zbMATH Open document ID: 0722.60078
author: Robert J. Adler
author: Michael B. Marcus
author: Joel Zinn
published in: The Annals of Probability
publication date: 1990-01-01T00:00:00Z
review text: Let \(\{X_ t:\) \(t\geq 0\}\) be an \({\mathbb{R}}^ d\)-valued, symmetric right Markov process with stationary transition density, and let \(\hat X_ t\), \(t\geq 0\), denote the process killed at an exponential random time. A theorem of Dynkin relating the local time \(\{L_ x:\) \(x\in {\mathbb{R}}^ d\}\) of \(\hat X_ t\) and the Gaussian process \(\{G_ x:\) \(x\in {\mathbb{R}}^ d\}\) with covariance the Green's function of \(\hat X_ t\), is used to show that \(L_ x\) satisfies the central limit theorem in \(C({\mathbb{R}}^ d)\) if and only if G(x) is sample continuous. The proof requires a very nice lemma, namely that the square of a sample continuous Gaussian process satisfies the CLT on the space of continuous functions. In later work, \textit{M. Marcus} and \textit{J. Rosen} [Ann. Probab., to appear] obtain very interesting results for local times of symmetric Markov processes via Dynkin's isomorphism.
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4190865
zbMATH Keywords: symmetric right Markov process
zbMATH Keywords: local time
zbMATH Keywords: Gaussian process
zbMATH Keywords: central limit theorem
zbMATH Keywords: sample continuous
reviewed by: Evarist Giné
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990738
OpenAlex ID: W1980738660
DOI: 10.1214/AOP/1176990738
description: scientific article; zbMATH DE number 4190865