Divergent sums over excursions
type: Item
instance of: scholarly article
title: Divergent sums over excursions
zbMATH Open document ID: 0841.60060
DOI: 10.1016/0304-4149(94)00014-X
author: K. Bruce Erickson
published in: Stochastic Processes and their Applications
publication date: 1996-07-18T00:00:00Z
review text: Let \(X\) be a canonical standard Markov process, and let 0 be a recurrent point of the state space which is regular for itself (but is neither trap nor a holding point). The author proves criteria for the almost sure convergence or divergence of sums of functions of excursions away from 0. These criteria are in terms of the expectation of the integrated excursion measure, either in the natural time scale or in the scale of local time of \(X\) at 0. For certain diffusions these results lead, together with the explicit calculation of the excursion measure, to interesting 0-1 type laws for the small time behaviour of the sample paths of \(X\). If \(X\) is reflecting Brownian motion, the author obtains 0-1 laws for the maxima of \(X\) over excursion intervals and for the areas under the excursions.
reviewed by: M.Dozzi
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 713923
zbMATH Keywords: Markov process
zbMATH Keywords: excursion
zbMATH Keywords: zero-one law
zbMATH Keywords: excursion measure
zbMATH Keywords: reflecting Brownian motion
MaRDI profile type: Publication
cites work: Q4265385
cites work: Q4772511
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cites work: Q4187050
description: scientific article; zbMATH DE number 713923