A Local Limit Theorem for Moderate Deviations

Abstract

The purpose of this note is to establish a uniform estimate for the mass function P(Sm = y) of an integer-valued random walk when y → ∞ and ( y − m μ ) / m → ∞ where μ isthe mean of the step distribution. (The local central limit theorem provides such an estimate when ( ( y − m μ ) / 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 ∑ n = 0 ∞ n 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.