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 $P_0\left(n\right)$ in discrete time, except the large time behavior which is characterized by the nontrivial constant $\theta$ through $P_0\left(n\right)\sim {\theta }^n$. Using a modified version of the independent interval approximation (IIA) that we developed before, we are able to calculate $P_0\left(n\right)$ analytically in $z$-transform space in terms of the autocorrelation function $A\left(n\right)$. If $A\left(n\right)\to 0$ as $n\to \infty$, we extract $\theta$ numerically, while if $A\left(n\right)=0$, 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 $A\left(n\right)=0$ for $n>1$, the double exponential-correlated “second order autoregressive process”, where $A\left(n\right)=c_1{\lambda }_1^n+c_2{\lambda }_2^n$, and power-law-correlated variables, where $A\left(n\right)\sim n^{-\mu }$. Apart from the power-law case when $\mu <5$, we find excellent agreement with simulations.

Figure 1

Stochastic time series describing the continuous position $x\left(n\right)$ as a function of the discrete time $n$. Time intervals $j_1,j_2,...$ denote times spent above and above the origin.

Figure 2

Persistence constant $\theta \left(\alpha \right)$ as a function of $\alpha$ for the process described by the equation of motion $x\left(n\right)=\eta \left(n\right)+\alpha \eta (n-1)$. Inset displays the persistence $P_0\left(n\right)$ for the case $\alpha =1$. Simulations are averaged over ${10}^6$ realizations.

Figure 3

(a): Persistence constant $\theta$ for the two-step memory process, $x\left(n\right)={\varphi }_{1}x(n-1)+{\varphi }_{2}x(n-2)+\eta \left(n\right)$, for different values of ${\lambda }_{1,2}$ (${\varphi }_{1,2}$). Inset displays the persistence probability $P_0\left(n\right)$ for fixed ${\lambda }_2=0.35$. Simulations are averaged over ${10}^6$ realizations. (b) Heat map displaying the mean absolute error of the persistence up to $n=35$: recursive formula vs simulation. Simulation is averaged over $4\times {10}^6$ realizations. In both (a) and (b), note that the points where ${\lambda }_1={\lambda }_2$ are excluded, as they make the autocorrelator divergent.

Figure 4

Persistence probability $P_0\left(n\right)$ (on the ordinate), simulation vs recursive formula. Simulations plotted as far as they have converged, averaged over ${10}^8$ realizations.