Persistence of Non-Markovian Gaussian Stationary Processes in Discrete Time

TL;DR

This paper develops an analytical method to calculate the persistence probability of discrete-time Gaussian stationary processes, providing insights into their long-term behavior based on autocorrelation functions.

Contribution

It introduces a modified Independent Interval Approximation to analytically determine persistence probabilities in discrete time, extending understanding beyond large-time asymptotics.

Findings

Analytical expressions for persistence probability in various Gaussian processes.

Numerical extraction of the decay constant θ for different autocorrelation functions.

Good agreement with simulations for most cases except certain power-law correlations.

Abstract

The persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval. Although persistence is a simple concept to understand, it is in general hard to calculate. Here we consider zero mean Gaussian stationary processes in discrete time $n$. Few results are known for the persistence $P_0(n)$ in discrete time, except the large time behavior which is characterized by the nontrivial constant $\theta$ through $P_0(n)\sim \theta^n$. Using a modified version of the Independent Interval Approximation (IIA) that we developed before, we are able to calculate $P_0(n)$ analytically in $z$-transform space in terms of the autocorrelation function $A(n)$. If $A(n)\to0$ as $n\to\infty$, we extract $\theta$ numerically, while if $A(n)=0$, for finite $n>N$, we find $\theta$ exactly (within the IIA). We apply our results to three special cases: the nearest neighbor-correlated "first order moving average process" where $A(n)=0$ for $ n>1$, the double exponential-correlated "second order autoregressive process" where $A(n)=c_1\lambda_1^n+c_2\lambda_2^n$, and power law-correlated variables where $A(n)\sim n^{-\mu}$. Apart from the power-law case when $\mu<5$, we find excellent agreement with simulations.