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(τ~* = n)/P_0{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'.
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