## 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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