Persistence probabilities of two-sided (integrated) sums of correlated stationary Gaussian sequences

TL;DR

This paper investigates the likelihood that certain two-sided discrete-time Gaussian sequences, which are analogs of fractional Brownian motion and its integrated form, remain positive over time, extending continuous-time results to discrete settings.

Contribution

It extends known continuous-time persistence probability results to a broad class of discrete-time Gaussian processes, providing new insights into their behavior.

Findings

Persistence probabilities for discrete-time Gaussian sequences are characterized.

Results generalize continuous-time findings to discrete analogs.

The study broadens understanding of Gaussian process behaviors in discrete settings.

Abstract

We study the persistence probability for some two-sided discrete-time Gaussian sequences that are discrete-time analogs of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous-time in [Molchan, Commun. Math. Phys., 1999] and [Molchen, J. Stat. Phys., 2017] to a wide class of discrete-time processes.