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Weak convergence of the integral of semi-Markov processes

Andrea Pedicone, Fabrizio Cinque

TL;DR

This work establishes functional weak convergence results for the integral of semi-Markov velocity processes, showing that, under mild moment and mixing conditions, the normalized integral $X_\lambda(t)=\lambda^{-1/2}(X(\lambda t)-\lambda\theta t)$ converges to a scaled Brownian motion $\mu^{-1/2}\gamma W$ in $C[0,\infty)$, where $\theta$ is the mean drift and $\gamma^2$ is determined by the stationary Markov renewal structure. The authors build a unified framework using regenerative and Markov renewal theory, proving ergodic, mixing, and renewal-type limit theorems for related functionals, and derive a precise expression for the limiting variance in terms of cycle sums and cross-correlations. They treat both the general semi-Markov case and the alternating renewal (periodic) specialization, addressing the challenges of non-ergodicity by delayed regenerative analysis and providing a consistent Brownian limit with explicit drift and diffusion parameters. As an application, they specialize to the generalized telegraph process, showing convergence to Brownian motion with drift and giving connections to the telegraph equation and its Kac-type diffusion limit to the heat equation. These results extend the weak-limit theory of telegraph-type motions and offer rigorous tools for analyzing finite-velocity particle models driven by semi-Markov dynamics.

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.

Weak convergence of the integral of semi-Markov processes

TL;DR

This work establishes functional weak convergence results for the integral of semi-Markov velocity processes, showing that, under mild moment and mixing conditions, the normalized integral converges to a scaled Brownian motion in , where is the mean drift and is determined by the stationary Markov renewal structure. The authors build a unified framework using regenerative and Markov renewal theory, proving ergodic, mixing, and renewal-type limit theorems for related functionals, and derive a precise expression for the limiting variance in terms of cycle sums and cross-correlations. They treat both the general semi-Markov case and the alternating renewal (periodic) specialization, addressing the challenges of non-ergodicity by delayed regenerative analysis and providing a consistent Brownian limit with explicit drift and diffusion parameters. As an application, they specialize to the generalized telegraph process, showing convergence to Brownian motion with drift and giving connections to the telegraph equation and its Kac-type diffusion limit to the heat equation. These results extend the weak-limit theory of telegraph-type motions and offer rigorous tools for analyzing finite-velocity particle models driven by semi-Markov dynamics.

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.
Paper Structure (6 sections, 17 theorems, 153 equations)

This paper contains 6 sections, 17 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Limit theorems for regenerative processes
  3. Limit theorems for Markov renewal processes
  4. Weak convergence of the integral of semi-Markov process
  5. Weak convergence of the integral of an alternating renewal process
  6. Application to the generalized telegraph process

Key Result

Lemma 2.1

Let $\{\eta_k\}_{k\geq 1}$ be a sequence of random variables with $\mathsf{E}[\eta_k] = 0$, $\mathsf{E}[\eta_k^2] = \sigma^2 <+\infty$ for every $k \in \mathbb{N}$. Define the processes where $\theta$ is a positive constant, $\{a_n\}$ is a positive divergent sequence and $(\nu_{t})_{t \geq 0}$ is a family of integer random variables. If it holds that then $X_n \Rightarrow W$ on $D[0,1]$, implies

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 3
  • ...and 27 more