On directional convolution equivalent densities

We propose a definition of directional multivariate subexponential and convolution equivalent densities and find a useful characterization of these notions for a class of integrable and almost radial decreasing functions. We apply this result to show that the density of the absolutely continuous part of the compound Poisson measure built on a given density $f$ is directionally convolution equivalent and inherits its asymptotic behaviour from $f$ if and only if $f$ is directionally convolution equivalent. We also extend this characterization to the densities of more general infinitely divisible distributions on $\mathbb{R}^d$, $d \geq 1$, which are not pure compound Poisson.