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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 4, Pages 797–806 (Mi tvp136)

Short Communications

Discrete Bessel process and its properties

A. S. Mishchenko

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

Abstract: This paper considers a discrete analogue of a three-dimensional Bessel process — a certain discrete random process, which converges to a continuous Bessel process in the sense of the Donsker–Prokhorov invariance principle, and which has an elementary path structure such as in the case of a simple random walk.
The paper introduces four equivalent definitions of a discrete Bessel process, which describe this process from different points of view. The study of this process shows that its relationship to the simple random walk repeats the well-known properties which connect the continuous three-dimensional Bessel process with the standard Brownian motion. Thus, hereby we state and prove discrete versions of Pitman's theorem, Williams theorem on Brownian path decomposition, and some other statements related to these two processes.

Keywords: Bessel process, random walk, discrete analogues, Pitman theorem, Lévy theorem, Williams theorem.

DOI: https://doi.org/10.4213/tvp136

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English version:
Theory of Probability and its Applications, 2006, 50:4, 700–709

Bibliographic databases:

Citation: A. S. Mishchenko, “Discrete Bessel process and its properties”, Teor. Veroyatnost. i Primenen., 50:4 (2005), 797–806; Theory Probab. Appl., 50:4 (2006), 700–709

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp136
• https://doi.org/10.4213/tvp136
• http://mi.mathnet.ru/eng/tvp/v50/i4/p797

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This publication is cited in the following articles:
1. Debs P., Gradinaru M., “Penalization for birth and death processes”, Journal of Theoretical Probability, 21:3 (2008), 745–771
2. Ya. A. Lyulko, “On the distribution of time spent by Markov chain at different levels until achieving a fixed state”, Theory Probab. Appl., 56:1 (2012), 140–149
3. Ya. A. Lyulko, A. N. Shiryaev, “Sharp maximal inequalities for stochastic processes”, Proc. Steklov Inst. Math., 287:1 (2014), 155–173
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