Intermittency and multiscaling in limit theorems

TL;DR

This paper explores how intermittent stochastic processes can satisfy limit theorems through multiscale behaviors, revealing complex growth patterns and secondary scales, especially in supOU processes.

Contribution

It provides a detailed analysis of multiscale growth in intermittent processes and extends understanding of their limit behaviors, focusing on large deviations and supOU processes.

Findings

Intermittent processes can have multiple growth scales.

Secondary scales decrease as a power function of time.

The approach applies broadly, including to supOU processes.

Abstract

It has been recently discovered that some random processes may satisfy limit theorems even though they exhibit intermittency, namely an unusual growth of moments. In this paper we provide a deeper understanding of these intricate limiting phenomena. We show that intermittent processes may exhibit a multiscale behavior involving growth at different rates. To these rates correspond different scales. In addition to a dominant scale, intermittent processes may exhibit secondary scales. The probability of these scales decreases to zero as a power function of time. For the analysis, we consider large deviations of the rate of growth of the processes. Our approach is quite general and covers different possible scenarios with special focus on the so-called supOU processes.