The Martin boundary and ratio limit theorems for killed random walks

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 1:1 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(tau* = n)/P-0{tau* = n}, where tau* is the lifetime of S*, in the case when S drifts to -infinity, and the right-hand tail of its step distribution is 'locally sub-exponential'.