The dichotomy of recurrence and transience of semi-Levy processes

TL;DR

This paper establishes the recurrence and transience dichotomy for semi-Levy processes, extending known results from Levy processes, and introduces semi-random walks to analyze their properties.

Contribution

It proves the recurrence-transience dichotomy for semi-Levy processes and introduces semi-random walks for their analysis, along with laws of large numbers.

Findings

Recurrence and transience dichotomy holds for semi-Levy processes.

Semi-random walks are useful for analyzing semi-Levy processes.

Laws of large numbers are established for semi-Levy processes.

Abstract

Semi-Levy process is an additive process with periodically stationary increments. In particular, it is a generalization of Levy process. The dichotomy of recurrence and transience of Levy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Levy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Levy process constructed from two independent Levy processes is investigated. Finally, we prove the laws of large numbers for semi-Levy processes.