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

Abstract

We characterize the subexponential densities on $(0,\infty )$ for compound Poisson distributions on $[0,\infty )$ with absolutely continuous Lévy measures. As a corollary, we show that the class of all subexponential probability density functions on ${\mathbb{R}}_{+}$ is closed under generalized convolution roots of compound Poisson sums. Moreover, we give an application to the subexponential density on $(0,\infty )$ for the distribution of the supremum of a random walk.

arXiv comment15 pages. arXiv admin note: text overlap with arXiv:1912.08366