The multifaceted behavior of integrated supOU processes: The infinite variance case

TL;DR

This paper investigates the limit behavior of integrated infinite variance supOU processes, revealing diverse possible limit processes including fractional Brownian motion, Lévy stable processes, and even finite-moment processes, depending on specific conditions.

Contribution

It provides a comprehensive analysis of the asymptotic behavior of infinite variance supOU processes, highlighting their diverse limit distributions and conditions for convergence.

Findings

Limit processes include fractional Brownian motion and Lévy stable processes.

Infinite variance supOU processes can converge to finite-moment processes.

The behavior depends on specific process parameters and normalization.

Abstract

The so-called "supOU" processes, namely the superpositions of Ornstein-Uhlenbeck type processes are stationary processes for which one can specify separately the marginal distribution and the dependence structure. They can have finite or infinite variance. We study the limit behavior of integrated infinite variance supOU processes adequately normalized. Depending on the specific circumstances, the limit can be fractional Brownian motion but it can also be a process with infinite variance, a L\'evy stable process with independent increments or a stable process with dependent increments. We show that it is even possible to have infinite variance integrated supOU processes converging to processes whose moments are all finite. A number of examples are provided.