Weak convergence of the integral of semi-Markov processes

Andrea Pedicone; Fabrizio Cinque

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arXiv · 2505.13274

Weak convergence of the integral of semi-Markov processes

Andrea Pedicone·Fabrizio Cinque

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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.

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Andrea Pedicone, Fabrizio Cinque. 2025-05-19. Weak convergence of the integral of semi-Markov processes. https://arxiv.org/abs/2505.13274

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