A Local Limit Theorem for Moderate Deviations

The purpose of this note is to establish a uniform estimatefor the mass function P(S_m = y) of an integer-valued randomwalk when y and where µ isthe mean of the step distribution. (The local central limittheorem provides such an estimate when ( is bounded.) The assumptions are that the mass functionp of the step distribution is regularly varying at with index–, where > 3, and that for some ' > 2. From this result, a ratio limit theorem isderived, and this in turn is applied to yield some new informationabout the space–time Martin boundary of certain randomwalks.