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

A. T. Semenov

Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 613-620

Résumé

Let $\xi_1,\xi_2,\dots$ 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 $0<\delta_1\le x=k/n\le\delta_2<1$, $0<\delta_1\le x_1<x_2\le\delta_2<1$.