Multidimensional Dickman distribution and operator selfdecomposability

TL;DR

This paper extends the multidimensional Dickman distribution to vector-valued random elements, demonstrating their infinite divisibility and operator selfdecomposability, with applications in limit theorems.

Contribution

It introduces a new class of vector-valued distributions based on affine transformations involving random matrices, expanding the theoretical framework of the multidimensional Dickman distribution.

Findings

Distributions are infinitely divisible and operator selfdecomposable.

Identifies cases where these distributions appear as limits.

Provides a characterization of the extended multidimensional Dickman distribution.

Abstract

The one-dimensional Dickman distribution arises in various stochastic models across number theory, combinatorics, physics, and biology. Recently, a definition of the multidimensional Dickman distribution has appeared in the literature, together with its application to approximating the small jumps of multidimensional L\'evy processes. In this paper, we extend this definition to a class of vector-valued random elements, which we characterise as fixed points of a specific affine transformation involving a random matrix obtained from the matrix exponential of a uniformly distributed random variable. We prove that these new distributions possess the key properties of infinite divisibility and operator selfdecomposability. Furthermore, we identify several cases where this new distribution arises as a limiting distribution.