Dickman type stochastic processes with short- and long- range dependence

TL;DR

This paper explores the properties of Dickman distributions and their connection to Ornstein-Uhlenbeck processes driven by Poisson noise, including their dependence structures and simulation methods.

Contribution

It introduces a new analysis of Dickman distributions as stationary solutions of Poisson-driven Ornstein-Uhlenbeck processes and develops a numerical simulation algorithm.

Findings

The marginal distribution of the Ornstein-Uhlenbeck process is Dickman.

Superpositions can exhibit short- or long-range dependence with Dickman marginals.

A numerical algorithm for simulating these processes is provided.

Abstract

We study properties of the (generalized) Dickman distribution with two parameters and the stationary solution of the Ornstein-Uhlenbeck stochastic differential equation driven by a Poisson process. In particular, we show that the marginal distribution of this solution is the Dickman distribution. Additionally, we investigate superpositions of Ornstein-Uhlenbeck processes which may have short- or long-range dependencies and marginal distribution of the form of the Dickman distribution. The numerical algorithm for simulation of these processes is presented.