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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 4, Pages 789–796 (Mi tvp135)

Short Communications

On a probability distribution of some random walk functionals

A. S. Mishchenko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: In the theory of Brownian motion many related processes have been considered for a long time and have already been studied. Among them there are such Brownian motion functionals as a local time, an occupation time above some fixed level, a value of the maximum on a segment, and the argument of that maximum. One-dimensional distributions of them and some joint distributions are explicitly calculated, and many other relations are established. In this paper we consider a simple symmetric random walk, i.e., a random walk with a Bernoulli step. Based on it we define discrete analogues of the functional mentioned above. As the main result we prove a certain equality of two conditional distributions which includes all those discrete random variables. The proof is based upon a rather interesting transform on the set of all random walk paths which rearranges in some way its positive and negative excursions. Further we perform a limit passage to obtain the analogous equality between the conditional distributions of Brownian motion functionals. Both the discrete and continuous variants of this equality have never been mentioned before.

Keywords: Brownian motion, random walk, local time, occupation time, maximum, distribution, excursions.

DOI: https://doi.org/10.4213/tvp135

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English version:
Theory of Probability and its Applications, 2006, 50:4, 710–717

Bibliographic databases:

Citation: A. S. Mishchenko, “On a probability distribution of some random walk functionals”, Teor. Veroyatnost. i Primenen., 50:4 (2005), 789–796; Theory Probab. Appl., 50:4 (2006), 710–717

Citation in format AMSBIB
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