A LOCAL LIMIT THEOREM FOR MODERATE DEVIATIONS

Abstract

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 → ∞ and (y − mμ)/√m → ∞, where μ is the mean of the step distribution. (The local central limit theorem provides such an estimate when (y − mμ)/√m is bounded.) The assumptions are that the mass function p of the step distribution is regularly varying at ∞ with index −κ, where κ > 3, and that [sum ]^∞_n=0n^κ′p(−n) < ∞ for some κ′ > 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.