Universal behavior of the two-times correlation functions of random processes with renewal

Abstract

Stochastic processes with renewal properties, or semi-Markovian processes, have emerged as powerful tools for modeling phenomena where the assumption of complete independence between temporally spaced events is unrealistic. These processes find applications across diverse disciplines, including biology, neuroscience, health sciences, social sciences, ecology, climatology, geophysics, oceanography, chemistry, physics, and finance. Investigating their statistical properties is crucial for understanding complex systems. Here we obtain a simple exact expression for the two-times correlation function, a key descriptor of renewal processes, as it determines the power spectrum and impacts the diffusion properties of systems influenced by such processes. Although results for the two-times correlation function have been derived, the exact expression has been evaluated only for some specific cases, as for systems with $N$ states notably the simplest is the dichotomous scenario. By averaging over trajectory realizations, we obtain a universal result for the two-times correlation function, independent of the jump statistics, provided the variance is finite. Under the standard assumption for reaching asymptotic stationarity, where waiting times decay as $t^{-\mu }$ with $\mu >2$, we show that stationarity depends solely on the first time $t_1$, i.e., the time distance from the preparation time, while the time difference $t_2-t_1$ is inconsequential. For systems where stationarity is unattainable ($1<\mu <2$), we provide a universal asymptotic form of the correlation function for large $t_1$, extending previous results limited to specific time difference regimes. We examine two interpretations of renewal processes: shot noise and step noise—, relevant to physical systems such as general Continuous Time Random Walks and Lévy walks with random velocities. While this study focuses on two-times correlations, the simple methodology is generalizable to $n$-times correlations, offering a pathway for future research into the statistical mechanics of renewal processes.