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Superpositions of CARMA processes

Danijel Grahovac, Magdalena Mikić

TL;DR

This work extends the supOU framework to the CARMA setting by introducing supCARMA processes, defined as Lévy-basis-driven superpositions of CARMA processes. Focusing on the $p=2$ case, the paper shows that three structurally distinct types arise (I, II, III) according to the eigenstructure of the underlying CAR$(2)$ matrix, and provides precise existence conditions, moment formulas, and explicit second-order properties for each type. The resulting correlation structures can exhibit long-range dependence and oscillatory behavior, enabling flexible modeling of time series with complex dependence patterns and infinite-divisibility marginals. The framework generalizes supOU models to higher-order dynamics and offers tunable dependence via the mixing measure across eigenstructures, with concrete examples demonstrating long-memory behavior under Gamma/Beta specifications and complex oscillations under conjugate-dominant mixing.

Abstract

We introduce supCARMA processes, defined as superpositions of Lévy-driven CARMA processes with respect to a Lévy 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.

Superpositions of CARMA processes

TL;DR

This work extends the supOU framework to the CARMA setting by introducing supCARMA processes, defined as Lévy-basis-driven superpositions of CARMA processes. Focusing on the case, the paper shows that three structurally distinct types arise (I, II, III) according to the eigenstructure of the underlying CAR matrix, and provides precise existence conditions, moment formulas, and explicit second-order properties for each type. The resulting correlation structures can exhibit long-range dependence and oscillatory behavior, enabling flexible modeling of time series with complex dependence patterns and infinite-divisibility marginals. The framework generalizes supOU models to higher-order dynamics and offers tunable dependence via the mixing measure across eigenstructures, with concrete examples demonstrating long-memory behavior under Gamma/Beta specifications and complex oscillations under conjugate-dominant mixing.

Abstract

We introduce supCARMA processes, defined as superpositions of Lévy-driven CARMA processes with respect to a Lévy basis, as a natural extension of the superpositions of Ornstein-Uhlenbeck type processes. We then focus on supCAR processes and show that they can be classified into three distinct types determined by the eigenstructure of the underlying CAR 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 processes a flexible class for modeling time series with oscillatory correlations or strong dependence.
Paper Structure (12 sections, 11 theorems, 218 equations, 1 figure)

This paper contains 12 sections, 11 theorems, 218 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. CARMA processes
  5. Lévy bases
  6. SupCARMA processes
  7. Existence from the state-space representation
  8. Existence from the mixed moving average representation
  9. SupCAR$(2)$ processes
  10. SupCAR$(2)$ of type I
  11. SupCAR$(2)$ of type II
  12. SupCAR$(2)$ of type III

Key Result

Theorem 1

Suppose that $\int_{|x|>1} \log |x| \mu(dx)<\infty$ and that there exist measurable functions $\rho \colon \mathcal{A}_p \to (0,\infty)$ and $\eta \colon \mathcal{A}_p \to [1,\infty]$ such that and Then the supCARMA process e:def:supCARMA is well-defined and stationary.

Figures (1)

  • Figure 1: Correlation functions of supCAR$(2)$-III processes for different choices of $\pi_r$ and with $\pi_{\psi}$ given by \ref{['eq:exIII:pipsi']}. For shape parameters $\alpha+3 \in (3,4]$, the correlation function exhibits non-oscillatory long-range dependence, while for larger values of the shape parameter oscillations arise.

Theorems & Definitions (28)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Definition 2
  • Definition 3
  • Theorem 2
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 18 more