Weak convergence of the integral of semi-Markov processes

2025/05/19 by Pedicone, Andrea, Cinque, Fabrizio
#60F05 #60F17 #60K15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2505.13274

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.

Citations

• On the distribution of the telegraph meander and its properties
• Multidimensional random motions with a natural number of finite velocities
• A Note on the Conditional Probabilities of the Telegraph Process
• On the Exact Distributions of the Maximum of the Asymmetric Telegraph\n Process
• Cyclic random motions with orthogonal directions
• On a directionally reinforced random walk
• Stochastic velocity motions and processes with random time

Related

• Weak approximation for Gaussian processes from renewal processes
• Strong limit of processes constructed from a renewal proces
• Submartingale Condition for Weak Convergence for Semi-Markov Processes
• From persistent random walks to the telegraph noise
• Markov Renewal Processes with Finitely Many States