Fluctuations of Multi-Dimensional Kingman-L\'Evy Processes

TL;DR

This paper introduces k-dimensional Kingman-Lévy processes, explores their fluctuation properties, and derives key decompositions and series representations, extending the understanding of multi-dimensional stochastic processes related to Kingman convolutions.

Contribution

It extends the study of multi-dimensional Kingman convolutions by defining k-dimensional Kingman-Lévy processes and establishing their fundamental fluctuation properties and decompositions.

Findings

Derived the Lévy-Itô decomposition for k-dimensional KL-processes.

Established series representation of Rosiński type for these processes.

Demonstrated fluctuation properties analogous to k-symmetric Lévy processes.

Abstract

In the recent paper \cite{Ng5} we have introduced a method of studying the multi-dimensional Kingman convolutions and their associated stochastic processes by embedding them into some multi-dimensional ordinary convolutions which allows to study multi-dimensional Bessel processes in terms of the cooresponding Brownian motions. Our further aim in this paper is to introduce k-dimensional Kingman-L\'evy (KL) processes and prove some of their fluctuation properties which are analoguous to that of k-symmetric L\'evy processes. In particular, the L\'evy-It\^o decomposition and the series representation of Rosi\'nski type for k-dimensional KL-processes are obtained.