Renewal Theory for a System with Internal States

¹ Institut für Physik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
² Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
³ Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel

^⋆ Corresponding author. E-mail: Eli.Barkai@biu.ac.il

Abstract

We investigate a stochastic signal described by a renewal process for a system with N states. Each state has an associated joint distribution for the signal’s intensity and its holding time. We calculate multi-point distributions, correlation functions, and the power-spectrum of the signal. Focusing on fat tailed power-law distributed sojourn times in the states of the system, we investigate 1/f noise in this widely applicable model. When the mean waiting time is infinite, the averaged sample spectrum depends both on the age of the process, i.e. the time elapsing from start of the process and the start of observation, and on the total time of observation. Fluctuations of the periodogram estimator of the power-spectrum are investigated for aged systems and are found to be determined by the distribution of the number of renewals in the observation time window. These reduce to the Mittag-Leffler distribution when the start of observation is also the start of the process. When the average waiting time is finite we find a time independent Wienerian spectrum computed from the stationary correlation function of the signal.