On directional convolution equivalent densities

TL;DR

This paper introduces a new concept of directional convolution equivalent densities, characterizes them for certain functions, and applies these results to analyze the asymptotic behavior of densities in compound Poisson and more general infinitely divisible distributions.

Contribution

It defines directional convolution equivalent densities, provides a characterization for integrable decreasing functions, and extends these results to a broader class of infinitely divisible distributions.

Findings

Density of compound Poisson measure is directionally convolution equivalent if and only if the original density is.

Asymptotic behavior of densities is inherited from the base density under the new definition.

The characterization applies to a wide class of infinitely divisible distributions, not just compound Poisson.

Abstract

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.