On the first exit time out of a semigroup in $R^m$ for a random walk

A. A. Mogul'skiǐ   ; E. A. Pečerskiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 4, pp. 837-844

Résumé

Let $(S_n)$ be a random walk generated by a sequence of random i. i. d. vectors $(\xi_n)$; $\xi_n\in R^m$. Let $H$ be a subset of $R^m$. In this paper, we study the random variable $$ \eta=\eta_H=\min\{k\colon k\ge 1,S_k\notin H\}. $$ Main results are obtained in the case when $H$ is a semi-group. For $|z|<1$ and $\lambda=(\lambda_1,\dots,\lambda_m)\in R^m$, we prove the formula $$ \sum_{n=0}^{\infty}z^n\mathbf M(e^{i(\lambda,S_n)};\eta_H>n)= \exp\biggl\{\sum_{n=1}^{\infty}\frac{z^n}{n}\mathbf M(e^{i(\lambda,S_n)};E_{0,n})\biggr\} $$ where $E_{0,n}$ is the event: $n$ is not a ladder index for any of $n$ cyclical rearrangements of $\xi_1,\dots,\xi_n$. We find some sufficient conditions for the uniqueness of a solution of the equation $$ (1-z\Phi(\lambda))\psi_1(z,\lambda)=\psi_2(z,\lambda) $$ where $\Phi(\lambda)=\mathbf M\exp\{i(\lambda,\xi_1)\}$. Some estimates for the sequence $(\mathbf P(\eta_H>n))$ are also obtained.