Two renewal theorems for general random walks tending to infinity

Abstract

Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk Sn into [x, ∞) for fixed x ≧ 0, and the last exit time of the walk from (-∞,x], are given under the condition that Sn → ∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of Sn, are further developed to obtain the "order of magnitude" as x → ∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first lime for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given.