Multidimensional Dickman distribution and operator selfdecomposability

TL;DR

This work extends the one-dimensional Dickman distribution to a broad class of multivariate, operator-valued targets by defining operator Dickman distributions $\mathcal{D}(Q,\nu)$ as fixed points of $X\overset{d}{=}U^{Q}(X'+W)$. It proves that these distributions are infinitely divisible and operator selfdecomposable, provides a tractable characteristic-function representation, and derives a multivariate density expression in the isotropic case $\mathcal{D}(\tfrac{1}{\theta}I,\nu)$ with uniform on the sphere, together with an integro-differential equation for the density. The authors connect these distributions to limit theorems and small-jump approximations for multidimensional Lévy processes, and offer a simulation scheme based on a weakly convergent infinite-series representation. The results open avenues for modeling multidimensional stochastic systems with random affine dynamics and for practical computation via explicit densities and simulation algorithms.

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 Lévy 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.

Figures (3)

• Figure 1: The random samples of size 500 from the 2-dimensional operator Dickman distribution $\mathcal{D}(Q,\nu)$ with $\nu$ being the uniform distribution on the $\mathbb{S}^1$ and various $Q$.
• Figure 2: The first and second coordinates of random samples of size 500 from the operator Dickman distribution $\mathcal{D}(I, \nu)$ on $\mathbb{R}^d$, where $\nu$ is the uniform distribution on the unit sphere $\mathbb{S}^{d-1}$, for $d=3,6,9$.
• Figure 3: The random samples of size 500 from the $2$-dimensional Dickman distribution $\mathcal{D}(I, \nu_{\mu,\kappa})$ with various values of parameters $\mu$ and $\kappa$.

Theorems & Definitions (45)

• Definition 3.1: Multidimensional Dickman distribution
• Definition 3.2
• Remark 1
• Remark 2
• Remark 3
• Lemma 3.3
• proof
• Proposition 3.4
• proof
• Corollary 3.5