A Bound for the Distribution of the Hitting Time of Arbitrary Sets by Random Walk

Abstract

We consider a random walk $S_n = \sum_{i=1}^n X_i$ with i.i.d. $X_i$. We assume that the $X_i$ take values in $\Bbb Z^d$, have bounded support and zero mean. For $A \subset \Bbb Z^d, A \ne \emptyset$ we define $\tau_A = \inf{n \ge 0: S_n \in A}$. We prove that there exists a constant $C$, depending on the common distribution of the $X_i$ and $d$ only, such that $\sup_{\emptyset \ne A \subset \Bbb Z^d} P\{\tau_A =n\} \le C/n, n \ge 1$.