Independence of time and position for a random walk
type: Item
instance of: scholarly article
title: Independence of time and position for a random walk
zbMATH Open document ID: 1068.60068
DOI: 10.4171/RMI/410
author: Christophe Ackermann
author: Gérard Lorang
author: Bernard Roynette
published in: Revista Matemática Iberoamericana
publication date: 2005-04-15T00:00:00Z
full work available at URL: https://eudml.org/doc/39724
review text: Given a real-valued random variable \(X\) whose Laplace transform is analytic in a neighbourhood of \(0\), a random walk \((S_n,\, n\geq0)\), starting from the origin and with increments distributed as \(X\), is considered. This paper investigates the class of stopping times \(T\) which are independent of \(S_T\) and standard, i.e. \((S_{n\wedge T},\;n\geq0)\) is uniformly integrable. The underlying filtration \(({\mathcal F}_n,\,n\geq0)\) is not supposed to be natural. The classification of all possible distributions for \(S_T\) remains an open problem in the discrete setting, even though we manage to identify the solutions in the special case where \(T\) is a stopping time in the natural filtration of a Bernoulli random walk and \(\min T\leq5\). Some examples illustrate the general theorems, in particular the first time where \(| S_n| \) (resp. the age of the walk or Pitman's process) reaches a given level \(a\in N^*\). Finally, this paper is concerned with a related problem in two dimensions. Namely, given two independent random walks \((S^{\prime}_n,\,n\geq0)\) and \((S^{\prime\prime}_n,\,n\geq0)\) with the same incremental distribution, the class of stopping times \(T\) is investigated such that \(S^{\prime}_T\) and \(S^{\prime\prime}_T\) are independent.
reviewed by: Yuhu Feng
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 30D20
zbMATH DE Number: 2156861
zbMATH Keywords: stopping time
zbMATH Keywords: Wald's identity
zbMATH Keywords: Khinchin's inequalities
zbMATH Keywords: Pitman's process
zbMATH Keywords: age process
MaRDI profile type: Publication
cites work: Q5833361
cites work: Stopping distributions for right processes
cites work: Q5538132
OpenAlex ID: W1987818153
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description: scientific article; zbMATH DE number 2156861