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

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