On the asymptotic behaviour of first passage times for transient random walk
type: Item
instance of: scholarly article
title: On the asymptotic behaviour of first passage times for transient random walk
zbMATH Open document ID: 0643.60053
DOI: 10.1007/BF00319553
publication date: 1989-01-01T00:00:00Z
review text: Let \(\tau _ x\) denote the time at which a random walk with finite positive mean first passes into (x,\(\infty)\), where \(x\geq 0\). This paper establishes the asymptotic behaviour of \(\Pr \{\tau _ x>n\}\) as \(n\to \infty\) for fixed x in two cases. In the first case the left hand tail of the step-distribution is regularly varying, and in the second the step- distribution satisfies a one-sided Cramér type condition. As a corollary, it follows that in the first case \(\lim _{n\to \infty}\Pr \{\tau _ n>n\}/\Pr \{\tau _ 0>n\}\) coincides with the limit of the same quantity for recurrent random walks satisfying Spitzer's condition, but in the second case the limit is more complicated.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 4048790
zbMATH Keywords: first passage times
zbMATH Keywords: regularly varying
zbMATH Keywords: recurrent random walk
author: Ronald A. Doney
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4048790