Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis |
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Authors: | A T Semenov |
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Institution: | (1) Novosibirsk State University, USSR |
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Abstract: | Let 1, 2, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional
n
=
k=o
n
(S
k
) where S1=0, Sk=
i=1
k
i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that n is the time spent by the random walk Sn, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in 1]. In this paper we obtain the asymptotic expansions in powers of n–1 of the probabilities P(hn = nx) and P(nx1 n nx2) for 0<1, x = k/n 2<1, 0<1x122<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work. |
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Keywords: | |
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