On the asymptotic behavior of distributions of first-passage times. II
type: Item
instance of: scholarly article
title: On the asymptotic behavior of distributions of first-passage times. II
zbMATH Open document ID: 1138.60035
DOI: 10.1023/B:MATN.0000023311.52852.25
published in: Mathematical Notes
publication date: 2004-12-06T00:00:00Z
review text: The two-part paper deals with the asymptotic behavior of \(P(\eta_-(x)>n)\) resp. \(P(\eta_+(x)=n),\) \(\eta_+(x)=\min\{k:S_k>x\},\) \(\eta_-(x)=\min\{k:S_k\leq-x\}, S_n=\sum_{j=1}^n\xi_j\), \(\xi,\xi_1,\xi_2,\ldots\) being i.i.d. random variables. While in part I [Math. Notes 75, No. 1, 23--37 (2004); translation from Mat. Zametki 75, No. 1, 24--39 (2004; Zbl 1108.60039)], \(x=0\) and \(n\to\infty\) for some classes of distributions of \(\xi\) is considered, in part II here, \(x\geq0\) or even \(x\to\infty\) is studied. The asymptotic behavior depends on \(E\xi\) and, whether the sums \(D_+=\sum_k{P(S_k>0)\over k},\) \(D_-=\sum_k{P(S_k\leq0)\over k}\) converge or diverge. A rich list of references is given (22 items).
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 2121361
zbMATH Keywords: random walk
zbMATH Keywords: first-passage time
zbMATH Keywords: asymptotics of the distribution of first-passage times
author: A. A. Borovkov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2121361