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We consider a random walk Sn = Pn i=1 Xi with i.i.d. Xi. We assume that the Xi take values in Zd, have bounded support and zero mean. For A‰ Zd;A6=;, we deflne ¿A = inffn‚ 0 : Sn 2 Ag. We prove that there exists a constant C, depending on the common distribution of the Xi and d only, such that sup;6=A‰Zd Pf¿A = ngC=n;n‚ 1.

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