Subexponential densities of compound Poisson sums and the supremum of a random walk

We characterize the subexponential densities on (0, oo) for compound Pois-son distributions on [0, oo) with absolutely continuous Le ' vy measures. In particular, we show that the class of all subexponential probability density functions on [0, oo) is closed under generalized convolution roots for compound Poisson sums. Moreover, we give an application to the subexponential density on (0, oo) for the distribution of the supre-mum of a random walk.