Persistence of Non-Markovian Gaussian Stationary Processes in Discrete Time

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 $ntoinfty$, 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 $A(n)=c_1lambda_1^n+c_2lambda_2^n$, 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.