Statistics of the occupation time of renewal processes

TL;DR

This paper systematically analyzes the occupation time statistics in renewal processes with various interval distributions, revealing different scaling behaviors and providing methods applicable to more complex stochastic processes.

Contribution

It introduces a comprehensive analysis of occupation time statistics in renewal processes with different interval distributions, including narrow and broad cases, and offers methods for studying more complex processes.

Findings

Different scaling laws for occupation time depending on interval distribution

Analytical methods for narrow and broad interval distributions

Framework applicable to complex stochastic processes

Abstract

We present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time. According to the nature of the distribution of time intervals, the probability density functions of these random variables have very different scalings in time. We analyze successively the cases where this distribution is narrow, where it is broad with index $\theta <1$, and finally where it is broad with index $1<\theta <2$. The methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes (see joint paper by G. De Smedt, C. Godr\`{e}che, and J.M. Luck).