Persistence probabilities and a decorrelation inequality for the Rosenblatt process and Hermite processes

TL;DR

This paper investigates the persistence probabilities of Hermite processes, introduces a decorrelation inequality for the Rosenblatt process, and provides bounds for the persistence exponents of general Hermite processes.

Contribution

It derives a new decorrelation inequality for the Rosenblatt process and applies it to compute and bound persistence exponents for Hermite processes.

Findings

Derived a decorrelation inequality for the Rosenblatt process.

Computed the persistence exponent for the Rosenblatt process.

Provided bounds for the persistence probabilities of general Hermite processes.

Abstract

We study persistence probabilities of Hermite processes. As a tool, we derive a general decorrelation inequality for the Rosenblatt process, which is reminiscent of Slepian's lemma for Gaussian processes or the FKG inequality and which may be of independent interest. This allows to compute the persistence exponent for the Rosenblatt process. For general Hermite processes, we derive upper and lower bounds for the persistence probabilites with the conjectured persistence exponent, but with non-matching boundaries.