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


Given a real-valued random variable XX whose Laplace transform is analytic in a neighbourhood of 0, we consider a random walk (Sn,n0){(S_{n},n\geq 0)}, starting from the origin and with increments distributed as XX. We investigate the class of stopping times TT which are independent of STS_{T} and standard, i.e. (SnT,n0)(S_{n\wedge T},n\geq 0) is uniformly integrable. The underlying filtration (Fn,n0)(\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 STS_{T} remains an open problem in the discrete setting, even though we manage to identify the solutions in the special case where TT is a stopping time in the natural filtration of a Bernoulli random walk and minT5\min T \le 5. Some examples illustrate our general theorems, in particular the first time where Sn|S_{n}| (resp. the age of the walk or Pitman's process) reaches a given level aNa\in\mathbb{N}^{\ast}. Finally, we are concerned with a related problem in two dimensions. Namely, given two independent random walks (Sn,n0)(S_{n}^{\prime},n\geq 0) and (Sn,n0)(S_{n}^{\prime\prime},n\geq 0) with the same incremental distribution, we search for stopping times TT such that STS_{T}^{\prime} and STS_{T}^{\prime\prime} 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