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.


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)
Full Text: DOI EuDML


[1] Boas, R. Ph.: Entire Functions. Academic Press Inc., New York, 1954. · Zbl 0058.30201
[2] Bourekh, Y.: Quelques résultats sur le probl‘ eme de Skorokhod. PhD thesis, Université de Paris VI, 1996.
[3] Falkner, N. and Fitzsimmons, P. J.: Stopping distributions of right processes. Probab. Theory Related Fields 89 (1991), 301-318. · Zbl 0729.60071 · doi:10.1007/BF01198789
[4] Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. I. Third Edition. John Wiley & Sons, Inc., New York, 1968. · Zbl 0155.23101
[5] Kummer, E. E.: Über Ergänzungssätze zu den allgemeinen Reziprozitäts- gesetzen. Journal für die reine und angewandte Mathematik [J. Reine Angew. Math.] 44 (1852), 93-146.
[6] Lucacs, E.: Characteristic Functions, 2nd edition. Griffin, 1970.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.