Independence of time and position for a random walk

Abstract

Given a real-valued random variable $X$ whose Laplace transform is analytic in a neighbourhood of 0, we consider a random walk $(S_n,n\ge 0)$, 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. $(S_{n\wedge T},n\ge 0)$ is uniformly integrable. The underlying filtration $({\mathcal{F}}_n,n\ge 0)$ is not supposed to be natural. Our research has been deeply inspired by [7], 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 $(S_n^{\prime },n\ge 0)$ and $(S_n^{\prime \prime },n\ge 0)$ with the same incremental distribution, we search for stopping times $T$ such that $S_T^{\prime }$ and $S_T^{\prime \prime }$ are independent.