# Independence of time and position for a random walk

### Christophe Ackermann

Université Henri Poincaré, Vandoeuvre lès Nancy, France### Gérard Lorang

Université du Luxembourg, Luxembourg### Bernard Roynette

Université Henri Poincaré, Vandoeuvre lès Nancy, France

## Abstract

Given a real-valued random variable $X$ whose Laplace transform is analytic in a neighbourhood of 0, we consider a random walk $(S_{n},n≥0)$, 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. $(S_{n∧T},n≥0)$ is uniformly integrable. The underlying filtration $(F_{n},n≥0)$ 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 $minT≤5$. 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 $a∈N_{∗}$. Finally, we are concerned with a related problem in two dimensions. Namely, given two independent random walks $(S_{n},n≥0)$ and $(S_{n},n≥0)$ with the same incremental distribution, we search for stopping times $T$ such that $S_{T}$ and $S_{T}$ are independent.

## Cite this article

Christophe Ackermann, Gérard Lorang, Bernard Roynette, Independence of time and position for a random walk. Rev. Mat. Iberoam. 20 (2004), no. 3, pp. 893–952

DOI 10.4171/RMI/410