Weak convergence of the integral of semi-Markov processes

TL;DR

This paper investigates the weak convergence properties of integrals of semi-Markov processes, demonstrating under mild conditions their convergence to scaled Brownian motion, including applications to the generalized telegraph process.

Contribution

It provides a theoretical framework for the weak convergence of semi-Markov process integrals, extending to the generalized telegraph process.

Findings

Weak convergence to scaled Brownian motion under mild conditions

Establishment of ergodic and renewal properties for Markov renewal processes

Application to the classical generalized telegraph process

Abstract

We study the asymptotic properties, in the weak sense, of regenerative processes and Markov renewal processes. For the latter, we derive both renewal-type results, also concerning the related counting process, and ergodic-type ones, including the so-called phi-mixing property. This theoretical framework permits us to study the weak limit of the integral of a semi-Markov process, which can be interpret as the position of a particle moving with finite velocities taken for a random time according to the Markov renewal process underlying the semi-Markov one. Under mild conditions, we obtain the weak convergence to scaled Brownian motion. As a particular case, this result establishes the weak convergence of the classical generalized telegraph process.