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 ${(S_{n},n\geq 0)}$, 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.