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Submartingale Condition for Weak Convergence for Semi-Markov Processes

Vitaliy Golomoziy

TL;DR

This paper extends the classical Strook–Varadhan submartingale condition for weak convergence to semi-Markov processes by embedding the process in a Markov renewal framework. It shows that a straightforward discrete-time restatement is insufficient and introduces an additional condition requiring uniformly vanishing expected waiting times after short horizons, yielding tightness in the Skorokhod space $D[0,\infty)$. A counterexample demonstrates the necessity of the extra condition, and the authors develop a space–time scaling approach to verify the condition in practice, including a criterion expressed via holding-time tails. The results advance the theory of weak convergence for semi-Markov processes and inform practical approximation schemes via space–time scaling in applications.

Abstract

In this paper, we consider a modified version of a well-known submartingale condition fortheweak convergence of probabilitymeasures, adapted to the semi-Markov case. In this setting, it is convenient to work with an embedded Markov chain and the filtration generated by jump times. We demonstrate that a straightforward restatement of the classical result is not valid, and that an additional condition is required.

Submartingale Condition for Weak Convergence for Semi-Markov Processes

TL;DR

This paper extends the classical Strook–Varadhan submartingale condition for weak convergence to semi-Markov processes by embedding the process in a Markov renewal framework. It shows that a straightforward discrete-time restatement is insufficient and introduces an additional condition requiring uniformly vanishing expected waiting times after short horizons, yielding tightness in the Skorokhod space . A counterexample demonstrates the necessity of the extra condition, and the authors develop a space–time scaling approach to verify the condition in practice, including a criterion expressed via holding-time tails. The results advance the theory of weak convergence for semi-Markov processes and inform practical approximation schemes via space–time scaling in applications.

Abstract

In this paper, we consider a modified version of a well-known submartingale condition fortheweak convergence of probabilitymeasures, adapted to the semi-Markov case. In this setting, it is convenient to work with an embedded Markov chain and the filtration generated by jump times. We demonstrate that a straightforward restatement of the classical result is not valid, and that an additional condition is required.
Paper Structure (4 sections, 3 theorems, 69 equations)

This paper contains 4 sections, 3 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Discussion and examples
  4. Space-time scaled semi-Markov processes

Key Result

Theorem 1

[D. Strook, S. Varadhan] Let $\Omega = C([0,\infty); \mathbb{R}^d)$ be the space of continuous functions on $[0,\infty)$ with values in $\mathbb{R}^d$, and let $(\EuScript{M}_t)_{t\ge 0}$ be the corresponding canonical filtration. Let $\EuScript{P}$ be a family of probability measures on $(\Omega, ( is a non-negative $P$-submartingale. Assume also that $A_f$ can be selected in such a way that it w

Theorems & Definitions (12)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 2
  • Remark 1
  • proof
  • Remark 2
  • Theorem 3
  • ...and 2 more