RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



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






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

Full text: PDF file (878 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

Citation in format AMSBIB
\Bibitem{Mis05}
\by A.~S.~Mishchenko
\paper On a probability distribution of some random walk functionals
\jour Teor. Veroyatnost. i Primenen.
\yr 2005
\vol 50
\issue 4
\pages 789--796
\mathnet{http://mi.mathnet.ru/tvp135}
\crossref{https://doi.org/10.4213/tvp135}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2331992}
\zmath{https://zbmath.org/?q=an:1110.60047}
\elib{http://elibrary.ru/item.asp?id=9157518}
\transl
\jour Theory Probab. Appl.
\yr 2006
\vol 50
\issue 4
\pages 710--717
\crossref{https://doi.org/10.1137/S0040585X97982104}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000243284300015}


Linking options:
  • http://mi.mathnet.ru/eng/tvp135
  • https://doi.org/10.4213/tvp135
  • http://mi.mathnet.ru/eng/tvp/v50/i4/p789

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Теория вероятностей и ее применения Theory of Probability and its Applications
    Number of views:
    This page:253
    Full text:44
    References:41

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019