Subexponential distributions and integrated tails

Let F be a distribution function on [0,∞) with finite expectation. In terms of the hazard rate of F several conditions are given which simultaneously imply subexponentiality of F and of its integrated tail distribution F 1 . These conditions apply to a wide class of longtailed distributions, and they can also be used in connection with certain random walks which occur in risk theory and queueing theory.