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

We characterize the subexponential densities on $(0,\mathrm{\infty })$ for compound Poisson distributions on $[0,\mathrm{\infty })$ with absolutely continuous L'evy 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 $(0,\mathrm{\infty })$ for the distribution of the supremum of a random walk.