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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1974, Volume 15, Issue 4, Pages 613–620 (Mi mz7385)

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

A. T. Semenov

Novosibirsk State University

Abstract: Let $\xi_1,\xi_2,…$ be a sequence of independent identically distributed random variables with zero means. We consider the functional
$$\eta_n=\sum_{k=0}^n\theta(S_k)$$
where $S_1=0$, $S_k=\sum_{i=1}^k\xi_i$ ($k\ge1$ and $\theta(x)=1$ for $x\ge0$, $\theta(x)=0$ for $x<0$. It is readily seen that $\eta_n$ is the time spent by the random walk $S_n$, $n\ge0$, on the positive semi-axis after $n$ steps.
For the simplest walk the asymptotics of the distribution $P(\eta_n=k)$ for $n\to\infty$ and $k\to\infty$, 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 $P(\eta_n=nx)$ and $P(nx_1\le\eta_n\le nx_2)$ for $0<\delta_1\le x=k/n\le\delta_2<1$, $0<\delta_1\le x_1<x_2\le\delta_2<1$.

Full text: PDF file (467 kB)

English version:
Mathematical Notes, 1974, 15:4, 362–366

Bibliographic databases:

UDC: 519.2

Citation: A. T. Semenov, “Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis”, Mat. Zametki, 15:4 (1974), 613–620; Math. Notes, 15:4 (1974), 362–366

Citation in format AMSBIB
\Bibitem{Sem74} \by A.~T.~Semenov \paper Asymptotic behavior of the dwell time distribution for a~random walk on a~positive semi-axis \jour Mat. Zametki \yr 1974 \vol 15 \issue 4 \pages 613--620 \mathnet{http://mi.mathnet.ru/mz7385} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=353457} \zmath{https://zbmath.org/?q=an:0309.60041} \transl \jour Math. Notes \yr 1974 \vol 15 \issue 4 \pages 362--366 \crossref{https://doi.org/10.1007/BF01095129}