Superpositions of CARMA processes

TL;DR

This paper introduces supCARMA processes, a new class of models formed by superimposing Lévy-driven CARMA processes, which can exhibit complex dependence structures like long-range dependence and oscillations.

Contribution

The paper extends CARMA processes to superpositions driven by Lévy bases, classifies supCAR(2) processes into three types, and derives their correlation structures and conditions for existence.

Findings

supCAR(2) processes can exhibit long-range dependence.

They can have non-monotone correlation functions.

The classification depends on the eigenstructure of the CAR(2) matrix.

Abstract

We introduce supCARMA processes, defined as superpositions of L\'evy-driven CARMA processes with respect to a L\'evy basis, as a natural extension of the superpositions of Ornstein-Uhlenbeck type processes. We then focus on supCAR$(2)$ processes and show that they can be classified into three distinct types determined by the eigenstructure of the underlying CAR$(2)$ matrix. For each type we provide conditions for existence and derive explicit expressions for the correlation function. The resulting correlation structures may exhibit long-range dependence and can be non-monotone. These features make supCAR$(2)$ processes a flexible class for modeling time series with oscillatory correlations or strong dependence.