THE MARTIN BOUNDARY AND RATIO LIMIT THEOREMS FOR KILLED RANDOM WALKS

Abstract

It is shown that if S is an aperiodic random walk on the integers, S* is the Markov chain that arises when S is killed when it leaves the non-negative integers, and H⁺ is the renewal process of weak increasing ladder heights in S, then there is a 1[ratio ]1 correspondence between functions which are non-negative and superregular for S* and H⁺. This allows all the regular functions for S* to be described, and thus a result due to Spitzer to be completed for the recurrent case. This result is then applied to give a ratio limit theorem for P_x(τ* =n)/P₀{τ*=n}, where τ* is the lifetime of S*, in the case when S drifts to −∞, and the right-hand tail of its step distribution is ‘locally sub-exponential’.