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Teoriya Veroyatnostei i ee Primeneniya, 1981, Volume 26, Issue 4, Pages 769–783
(Mi tvp3506)
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This article is cited in 34 scientific papers (total in 34 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 $\nu_k$.
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
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https://www.mathnet.ru/eng/tvp3506 https://www.mathnet.ru/eng/tvp/v26/i4/p769
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