Independence of time and position for a random walk

Given a real-valued random variable X whose Laplace transform is analytic in a neighbourhood of 0, we consider a random walk (Sn, n ≥ 0), starting from the origin and with increments distributed as X. We investigate the class of stopping times T which are independent of ST and standard, i.e. (Sn∧T , n ≥ 0) is uniformly integrable. The underlying filtration (Fn, n ≥ 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 ST 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 |Sn| (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.