Universal behaviors of the multi-time correlation functions of random processes with renewal: the step noise case (the random velocity of a L\'evy walk)

Abstract

Stochastic processes with renewal properties are powerful tools for modeling systems where memory effects and long-time correlations play a significant role. In this work, we study a broad class of renewal processes where a variable's value changes according to a prescribed Probability Density Function (PDF), $p(\xi)$, after random waiting times $\theta$. This model is relevant across many fields, including classical chaos, nonlinear hydrodynamics, quantum dots, cold atom dynamics, biological motion, foraging, and finance. We derive a general analytical expression for the $n$-time correlation function by averaging over process realizations. Our analysis identifies the conditions for stationarity, aging, and long-range correlations based on the waiting time and jump distributions. Among the many consequences of our analysis, two new key results emerge. First, for Poissonian waiting times, the correlation function quickly approaches that of telegraphic noise. Second, for power-law waiting times with $\mu>2$, , \emph{any $n$-time correlation function asymptotically reduces to the two-time correlation evaluated at the earliest and latest time points}. This second result reveals a universal long-time behavior where the system's full statistical structure becomes effectively two-time reducible. Furthermore, if the jump PDF $p(\xi)$ has fat tails, this convergence becomes independent of the waiting time PDF and is significantly accelerated, requiring only modest increases in either the number of realizations or the trajectory lengths. Building upon earlier work that established the universality of the two-point correlation function (i.e., a unique formal expression depending solely on the variance of $\xi$ and on the waiting-time PDF), the present study extends that universality to the full statistical description of a broad class of renewal-type stochastic processes.