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 Diskr. Mat., 2019, Volume 31, Issue 3, Pages 3–16 (Mi dm1567)

Two-sided problem for the random walk with bounded maximal increment

V. I. Afanasyev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We consider a random walk with zero drift and finite positive variance $\sigma^{2}$. For positive numbers $y, z$ we find the limit as $n\rightarrow\infty$ of the probability that the first exit of the walk from interval $(-z\sigma\sqrt{n}, y\sigma\sqrt{n})$ occurs through its left end, while the maximum increment of the walk until the exit is smaller than $x\sigma\sqrt{n}$, where $x$ is a positive number. The limit theorem is established for the moment of the first exit of the walk from the indicated interval under the condition that this exit occurs through its left end and the value of the maximum walk increment is bounded.

Keywords: random walks with zero drift, boundary problems, limit theorems

DOI: https://doi.org/10.4213/dm1567

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English version:
Discrete Mathematics and Applications, 2021, 31:2, 79–89

Bibliographic databases:

UDC: 519.217.31

Citation: V. I. Afanasyev, “Two-sided problem for the random walk with bounded maximal increment”, Diskr. Mat., 31:3 (2019), 3–16; Discrete Math. Appl., 31:2 (2021), 79–89

Citation in format AMSBIB
\Bibitem{Afa19} \by V.~I.~Afanasyev \paper Two-sided problem for the random walk with bounded maximal increment \jour Diskr. Mat. \yr 2019 \vol 31 \issue 3 \pages 3--16 \mathnet{http://mi.mathnet.ru/dm1567} \crossref{https://doi.org/10.4213/dm1567} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4010387} \elib{https://elibrary.ru/item.asp?id=46024508} \transl \jour Discrete Math. Appl. \yr 2021 \vol 31 \issue 2 \pages 79--89 \crossref{https://doi.org/10.1515/dma-2021-0008} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000640071300001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85104568302} 

• http://mi.mathnet.ru/eng/dm1567
• https://doi.org/10.4213/dm1567
• http://mi.mathnet.ru/eng/dm/v31/i3/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Afanasyev, “On the times of attaining high levels by a random walk in a random environment”, Theory Probab. Appl., 65:3 (2020), 359–374
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