Universal behavior of the two-times correlation functions of random processes with renewal
Marco Bianucci,
Mauro Bologna,
Daniele Lagomarsino-Oneto and
Riccardo Mannella
Chaos, Solitons & Fractals, 2025, vol. 196, issue C
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−μ with μ>2, we show that stationarity depends solely on the first time t1, i.e., the time distance from the preparation time, while the time difference t2−t1 is inconsequential. For systems where stationarity is unattainable (1<μ<2), we provide a universal asymptotic form of the correlation function for large t1, 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.
Keywords: Renewal processes; Correlation function; CTRW; Lévy walk; Aging; 1/f-noise; Weak ergodicity breaking (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:
Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0960077925003649
Full text for ScienceDirect subscribers only
Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.
Export reference: BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text
Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:196:y:2025:i:c:s0960077925003649
DOI: 10.1016/j.chaos.2025.116351
Access Statistics for this article
Chaos, Solitons & Fractals is currently edited by Stefano Boccaletti and Stelios Bekiros
More articles in Chaos, Solitons & Fractals from Elsevier
Bibliographic data for series maintained by Thayer, Thomas R. ().