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
Mots-clés :
occupation time, maximum, distribution, excursions.
A. S. Mishchenko. On a probability distribution of some random walk functionals. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 4, pp. 789-796. http://geodesic.mathdoc.fr/item/TVP_2005_50_4_a12/
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author = {A. S. Mishchenko},
title = {On a probability distribution of some random walk functionals},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {789--796},
year = {2005},
volume = {50},
number = {4},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2005_50_4_a12/}
}
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AU - A. S. Mishchenko
TI - On a probability distribution of some random walk functionals
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