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

Antal A Jarai (Carleton University, Canada)
Harry Kesten (Cornell University)


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 $\sup_{\emptyset \ne A \subset \Bbb Z^d} P\{\tau_A =n\} \le C/n, n \ge 1$.

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Pages: 152-161

Publication Date: November 17, 2004

DOI: 10.1214/ECP.v9-1119


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