Persistence of non-Markovian Gaussian stationary processes in discrete time

摘要

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) ～ θn. 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_1λ_1~n+ c_2λ_2~n, and power-law-correlated variables, where A(n) ～ n~(-μ). Apart from the power-law case whenμ < 5, we find excellent agreement with simulations.