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