Persistence of Non-Markovian Gaussian Stationary Processes in Discrete Time

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.