Two renewal theorems for general random walks tending to infinity

Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S-n into [x, infinity) for fixed x greater than or equal to infinity, and the last exit time of the walk from (-infinity, x], are given under the condition that S-n --> infinity a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S-n, are further developed to obtain the ''order of magnitude'' as x --> infinity 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 time for which the random walk remains above the level n: on K consecutive occasions, which has applications in option pricing, are given.