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Independence of time and position for a random walk. (English) Zbl 1068.60068

Given a real-valued random variable \(X\) whose Laplace transform is analytic in a neighbourhood of \(0\), a random walk \((S_n,\, n\geq0)\), starting from the origin and with increments distributed as \(X\), is considered. This paper investigates the class of stopping times \(T\) which are independent of \(S_T\) and standard, i.e. \((S_{n\wedge T},\;n\geq0)\) is uniformly integrable. The underlying filtration \(({\mathcal F}_n,\,n\geq0)\) is not supposed to be natural. 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 \(\min T\leq5\). Some examples illustrate the general theorems, in particular the first time where \(| S_n| \) (resp. the age of the walk or Pitman’s process) reaches a given level \(a\in N^*\). Finally, this paper is concerned with a related problem in two dimensions. Namely, given two independent random walks \((S^{\prime}_n,\,n\geq0)\) and \((S^{\prime\prime}_n,\,n\geq0)\) with the same incremental distribution, the class of stopping times \(T\) is investigated such that \(S^{\prime}_T\) and \(S^{\prime\prime}_T\) are independent.

MSC:

60G50 Sums of independent random variables; random walks
60G40 Stopping times; optimal stopping problems; gambling theory
30D20 Entire functions of one complex variable (general theory)
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References:

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