Weak convergence of the integral of semi-Markov processes

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.