Revista Matemática Iberoamericana

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Volume 20, Issue 3, 2004, pp. 893–952
DOI: 10.4171/RMI/410

Published online: 2004-12-31

Independence of time and position for a random walk

Christophe Ackermann[1], Gérard Lorang[2] and Bernard Roynette[3]

(1) Université Henri Poincaré, Vandoeuvre lès Nancy, France
(2) Université du Luxembourg, Luxembourg
(3) Université Henri Poincaré, Vandoeuvre lès Nancy, France

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.

Keywords: Independence, random walk, stopping time, Wald’s identity, Khinchine’s inequalities, Pitman’s process, age process

Ackermann Christophe, Lorang Gérard, Roynette Bernard: Independence of time and position for a random walk. Rev. Mat. Iberoamericana 20 (2004), 893-952. doi: 10.4171/RMI/410