Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments

Let F be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F . The expansion is based on an expansion for the right Wiener–Hopf factor which we derive first. An application to ruin probabilities is developed.