On directional convolution equivalent densities

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