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

Let FF 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 FF. The expansion is based on an expansion for the right Wiener–Hopf factor which we derive first. An application to ruin probabilities is developed.