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

Kovtun, Anastasiia 2026. Dickman type stochastic processes with short- and long-range dependence. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

This thesis introduces and studies stationary stochastic processes with Dickman one-dimensional distributions and adaptable correlation structures, primarily using the frameworks of Ornstein–Uhlenbeck processes and their superpositions. In the one-dimensional setting, this approach establishes a natural connection between the Dickman distribution and the homogeneous Poisson process. Furthermore, it yields a previously unknown series representation of the Dickman distribution and enables us to introduce generalisations of the Dickman distribution related to Urbanik’s Lm classes. We additionally demonstrate the flexibility of the dependence structure of the constructed models by studying their higher-order characteristics, specifically, higher-order spectral densities and cumulants. This analysis leads to the introduction of the concept of higher-order long-range dependence for a stationary stochastic process, in addition to the widely accepted second-order theory. In higher dimensions, we propose a novel definition of the multivariate Dickman distribution, which we call the operator Dickman distribution. We provide a detailed study of its properties, and show, in particular, that it belongs to the important class of operator selfdecomposable distributions. We further investigate the conditions and specific instances under which convergence to the operator Dickman distribution occurs, illustrating several of its key applications. This definition also allows us to use the Ornstein–Uhlenbeck processes and their superpositions to study multidimensional stochastic models associated with the Dickman distribution. For these processes, we study their weak scaling limits under both short- and long-range dependence conditions. Additionally, this work provides an in-depth historical perspective on the Dickman distribution, its corresponding constructions, and key applications.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Date of First Compliant Deposit:	4 August 2026
Date of Acceptance:	30 July 2026
Last Modified:	05 Aug 2026 09:18
URI:	https://orca.cardiff.ac.uk/id/eprint/188729

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