Spectral representations of quasi-infinitely divisible processes

TL;DR

This paper introduces quasi-infinitely divisible (QID) processes, develops their spectral representations and stochastic integrals, and demonstrates their density among random measures, expanding the framework of infinitely divisible processes.

Contribution

It formulates spectral representations and Lévy-Khintchine formulas for QID processes, a class larger than ID processes, and shows their density in the space of random measures.

Findings

QID processes have spectral representations and Lévy-Khintchine formulas.

QID random measures are dense in the space of all random measures.

The class of QID processes strictly contains ID processes.

Abstract

In this work we first introduce quasi-infinitely divisible (QID) random measures and formulate spectral representations. Then, we introduce QID stochastic integrals and present integrability conditions and continuity properties. Further, we introduce QID stochastic processes, i.e. stochastic processes with QID finite dimensional distributions. For example, a process $X$ is QID if there exist two ID processes $Y$ and $Z$ such that $X+Y\stackrel{d}{=}Z$ with $Y$ independent of $X$. The class of QID processes is strictly larger than the class of ID processes. We provide spectral representations and L\'{e}vy-Khintchine formulations for potentially all QID processes. Finally, we prove that QID random measures are dense in the space of random measures under convergence in distribution. Throughout this work we present many examples.