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Weak convergence of the integral of semi-Markov processes

Andrea Pedicone & Fabrizio Cinque

Journal of Applied Probability2026https://doi.org/10.1017/jpr.2026.10088preprint
AJG 2ABDC A
Weight
0.50

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 results, including the so-called $\varphi$ -mixing property. This theoretical framework permits us to study the weak limit of the integral of a semi-Markov process, which can be interpreted 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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https://doi.org/https://doi.org/10.1017/jpr.2026.10088

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@article{andrea2026,
  title        = {{Weak convergence of the integral of semi-Markov processes}},
  author       = {Andrea Pedicone & Fabrizio Cinque},
  journal      = {Journal of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1017/jpr.2026.10088},
}

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