Abstract: For a random walk on the real line with negative mean, we obtain a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk when the conditions for the exponential estimate are not satisfied. The results are applied to the Sparre Andersen model and some new results on the probability of ruin are presented.

Key words: Random walk, Ruin probability, Subexponential distributions, Class S(ν), Ladder height, Wiener-Hopf identity.