The purpose of this note is to establish a uniform estimate for the mass function P(S-m = y) of an integer-valued random walk when y --> infinity and (y - m mu)/rootm --> infinity where mu is the mean of the step distribution. (The local central limit theorem provides such an estimate when (y - m mu)/ rootm is bounded.) The assumptions are that the mass function p of the step distribution is regularly varying at infinity with index -kappa, where kappa > 3, and that Sigma (infinity)(n=0)n(x ')p(-n) < infinity for some kappa ' > 2. From this result, a ratio limit theorem is derived, and this in turn Is applied to yield some new information about the space-time Martin boundary of certain random walks. [References: 10]
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