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 θ through P_{0}(n)∼θ^{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)→0 as n→∞, we extract θ numerically, while if A(n)=0, for finite n>N, we find θ 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}λ_{1}^{n}+c_{2}λ_{2}^{n}, and power-law-correlated variables, where A(n)∼n^{-μ}. Apart from the power-law case when μ<5, we find excellent agreement with simulations.