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 whose Laplace transform is analytic in a neighbourhood of 0, we consider a random walk , starting from the origin and with increments distributed as . We investigate the class of stopping times which are independent of and standard, i.e. is uniformly integrable. The underlying filtration 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 remains an open problem in the discrete setting, even though we manage to identify the solutions in the special case where is a stopping time in the natural filtration of a Bernoulli random walk and . Some examples illustrate our general theorems, in particular the first time where (resp. the age of the walk or Pitman's process) reaches a given level . Finally, we are concerned with a related problem in two dimensions. Namely, given two independent random walks and with the same incremental distribution, we search for stopping times such that and 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