Persistence probabilities in centered, stationary, Gaussian processes in discrete time

TL;DR

This paper investigates how the probability that a centered, stationary Gaussian process remains positive over a long discrete time interval decays, revealing conditions under which this decay is faster than exponential and establishing bounds on the decay rate.

Contribution

It provides new lower bounds for persistence probabilities based on spectral measure conditions and demonstrates that the decay can be faster than exponential, with quadratic bounds when the spectral measure is not singular.

Findings

Persistence probability can decay faster than exponential.

If spectral measure is not singular, decay rate is at most quadratic.

Numerical evidence supports the quadratic decay bound.

Abstract

Lower bounds for persistence probabilities of stationary Gaussian processes in discrete time are obtained under various conditions on the spectral measure of the process. Examples are given to show that the persistence probability can decay faster than exponentially. It is shown that if the spectral measure is not singular, then the exponent in the persistence probability cannot grow faster than quadratically. An example that appears (from numerical evidence) to achieve this lower bound is presented.