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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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Theory of Probability and its Applications, 1982, 26:4, 758–772
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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
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\by A.~N.~Borodin
\paper An asymptotic behaviour of local times of a~recurrent random walk with finite variance
\jour Teor. Veroyatnost. i Primenen.
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\pages 769--783
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\jour Theory Probab. Appl.
\yr 1982
\vol 26
\issue 4
\pages 758--772
\crossref{https://doi.org/10.1137/1126082}
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http://mi.mathnet.ru/eng/tvp3506 http://mi.mathnet.ru/eng/tvp/v26/i4/p769
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This publication is cited in the following articles:
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A. N. Borodin, “Brownian local time”, Russian Math. Surveys, 44:2 (1989), 1–51
  -
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
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V. I. Afanasyev, “Convergence to the local time of Brownian meander”, Discrete Math. Appl., 29:3 (2019), 149–158
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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
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V. I. Afanasyev, “Functional limit theorem for the local time of stopped random walk”, Discrete Math. Appl., 30:3 (2020), 147–157
-
O. O. Prykhodko, “The limit behaviour of random walks with arrests”, Theory Stoch. Process., 24(40):2 (2019), 79–88
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