Almost sure growth of integrated supOU processes

TL;DR

This paper investigates the almost sure growth rates of integrated supOU processes, revealing how their complex dependence structures influence asymptotic behavior and establishing strong laws of large numbers for these processes.

Contribution

It provides new almost sure growth rate results for integrated supOU processes, linking asymptotics to the Lévy measure and dependence structure, and introduces a Marcinkiewicz--Zygmund type SLLN.

Findings

Growth rates depend on Lévy measure behavior at zero and infinity

Established a Marcinkiewicz--Zygmund type strong law of large numbers

Characterized asymptotics for integrated supOU processes

Abstract

Superpositions of Ornstein-Uhlenbeck processes allow a flexible dependence structure, including long range dependence for OU-type processes. Their complex asymptotics are governed by three effects: the behavior of the L\'evy measure both at infinity and at zero, and the behavior at zero of the measure governing the dependence. We establish almost sure rates of growth depending on the characteristics of the process and prove a Marcinkiewicz--Zygmund type SLLN for the integrated process.