On a probability distribution of some random walk functionals
A. S. Mishchenko
M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
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.
Brownian motion, random walk, local time, occupation time, maximum, distribution, excursions.
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Theory of Probability and its Applications, 2006, 50:4, 710–717
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
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\paper On a probability distribution of some random walk functionals
\jour Teor. Veroyatnost. i Primenen.
\jour Theory Probab. Appl.
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