First return time probability in correlated stationary signals

Abstract

We study the distribution of first return times at a given level L in stationary correlated signals. Our approach makes use of the relation between the characteristic function of the first return probability density function (PDF) and the occupation probability of the state L. In this work we consider a discrete in time and space Ornstein-Uhlenbeck (OU) process with exponential decaying correlation function and then, by a subordination approach, we treat the case of a process with power-law tail correlation function and diverging correlation time. In the first case, by inverting the Laplace transforms we write down an exact analytical expression for the first return time PDF as a function of the level L, while in the second case we obtain the expressions for the first two asymptotic behaviors. In both cases no simple form of the return time statistics like stretched-exponential is obtained.