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October, 2004 Independence of time and position for a random walk
Christophe Ackermann, Gérard Lorang, Bernard Roynette
Rev. Mat. Iberoamericana 20(3): 893-952 (October, 2004).

Abstract

Given a real-valued random variable $X$ whose Laplace transform is analytic in a neighbourhood of 0, we consider a random walk ${(S_{n},n\geq 0)}$, starting from the origin and with increments distributed as $X$. We investigate the class of stopping times $T$ which are independent of $S_{T}$ and standard, i.e. $(S_{n\wedge T},n\geq 0)$ is uniformly integrable. The underlying filtration $(\mathcal{F}_{n},n\geq 0)$ is not supposed to be natural. Our research has been deeply inspired by \cite{De Meyer-Roynette-Vallois-Yor 2002}, where the analogous problem is studied, but not yet solved, for the Brownian motion. Likewise, 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 $\min T \le 5$. Some examples illustrate our general theorems, in particular the first time where $|S_{n}|$ (resp. the age of the walk or Pitman's process) reaches a given level $a\in\mathbb{N}^{\ast}$. Finally, we are concerned with a related problem in two dimensions. Namely, given two independent random walks $(S_{n}^{\prime},n\geq 0)$ and $(S_{n}^{\prime\prime},n\geq 0)$ with the same incremental distribution, we search for stopping times $T$ such that $S_{T}^{\prime}$ and $S_{T}^{\prime\prime}$ are independent.

Citation

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Christophe Ackermann. Gérard Lorang. Bernard Roynette. "Independence of time and position for a random walk." Rev. Mat. Iberoamericana 20 (3) 893 - 952, October, 2004.

Information

Published: October, 2004
First available in Project Euclid: 27 October 2004

zbMATH: 1068.60068
MathSciNet: MR2124492

Subjects:
Primary: 60C05 , 60E10 , 60G40 , 60G42 , 60G50 , 60J10

Keywords: age process , independence , Khinchine's inequalities , Pitman's process , Random walk , stopping time , Wald's identity

Rights: Copyright © 2004 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.20 • No. 3 • October, 2004
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