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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.
Received: 24.11.2004
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
Linking options:
https://www.mathnet.ru/eng/tvp135https://doi.org/10.4213/tvp135 https://www.mathnet.ru/eng/tvp/v50/i4/p789
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