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 Teor. Veroyatnost. i Primenen., 1981, Volume 26, Issue 4, Pages 769–783 (Mi tvp3506)

This article is cited in 6 scientific papers (total in 6 papers)

An asymptotic behaviour of local times of a recurrent random walk with finite variance

A. N. Borodin

Leningrad

Abstract: The paper deals with the asymptotic behaviour (as $n\to\infty$) of the number $\varphi(n,r)$ of times the recurrent random walk $\nu_k$ hits the point $r$ till time $n$. We prove that if the random walk has a finite variance then the processes
$$t_n(t,x)=n^{-1/2}\varphi([nt],[x\sqrt n]),\qquad(t,x)\in[0,\infty)\times\mathbf R^1$$
(where $[a]$ is the integer part of $a$), converge weakly to the process $\mathbf t(t,x)$ – the Brownian local time at the point $x$ after time $t$. This result is applied to the investigation of a limit behaviour of a number of processes generated by a recurrent random walk $\nu_k$.

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English version:
Theory of Probability and its Applications, 1982, 26:4, 758–772

Bibliographic databases:

Received: 16.04.1980

Citation: A. N. Borodin, “An asymptotic behaviour of local times of a recurrent random walk with finite variance”, Teor. Veroyatnost. i Primenen., 26:4 (1981), 769–783; Theory Probab. Appl., 26:4 (1982), 758–772

Citation in format AMSBIB
\Bibitem{Bor81} \by A.~N.~Borodin \paper An asymptotic behaviour of local times of a~recurrent random walk with finite variance \jour Teor. Veroyatnost. i Primenen. \yr 1981 \vol 26 \issue 4 \pages 769--783 \mathnet{http://mi.mathnet.ru/tvp3506} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=636771} \zmath{https://zbmath.org/?q=an:0488.60078|0474.60056} \transl \jour Theory Probab. Appl. \yr 1982 \vol 26 \issue 4 \pages 758--772 \crossref{https://doi.org/10.1137/1126082} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1982PM42700008} 

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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. A. N. Borodin, “Brownian local time”, Russian Math. Surveys, 44:2 (1989), 1–51
2. Khoshnevisan D., Levin D.A., Mendez-Hernandez P.J., “Exceptional times and invariance for dynamical random walks”, Probability Theory and Related Fields, 134:3 (2006), 383–416
3. V. I. Afanasyev, “Convergence to the local time of Brownian meander”, Discrete Math. Appl., 29:3 (2019), 149–158
4. Andrey Pilipenko, Vladislav Khomenko, “On a limit behavior of a random walk with modifications upon each visit to zero”, Theory Stoch. Process., 22(38):1 (2017), 71–80
5. V. I. Afanasyev, “Functional limit theorem for the local time of stopped random walk”, Discrete Math. Appl., 30:3 (2020), 147–157
6. O. O. Prykhodko, “The limit behaviour of random walks with arrests”, Theory Stoch. Process., 24(40):2 (2019), 79–88
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