Submartingale Condition for Weak Convergence for Semi-Markov Processes

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.

Summary

This paper investigates the weak convergence of semi-Markov processes using a submartingale condition. It demonstrates that the classical submartingale condition is insufficient for semi-Markov processes and introduces an additional condition related to the frequency of jumps, which is crucial for establishing tightness, a key property for weak convergence.

Key Insights

• •The paper identifies that directly applying the classical submartingale condition (Strook and Varadhan, 2006) is insufficient for proving weak convergence in semi-Markov processes due to the random holding times between jumps.
• •The key contribution is the introduction of an additional condition (condition (iii) or (iv)) that ensures tightness, which is essential for establishing weak convergence in the semi-Markov setting.
• •The paper provides a counterexample to demonstrate the necessity of the additional condition, highlighting that the submartingale property alone does not guarantee tightness in semi-Markov processes.
• •The paper analyzes a specific class of semi-Markov processes obtained by space-time scaling and provides a verifiable condition (condition (14)) for ensuring tightness in this context, particularly relevant for diffusion and averaging approximations.

Practical Implications

• •The findings clarify the theoretical requirements for proving weak convergence in semi-Markov models, which are widely used in various fields such as queuing theory, reliability analysis, and finance.
• •The paper suggests future research directions focused on weakening the additional condition or exploring alternative conditions that guarantee tightness in the semi-Markov setting, potentially leading to more general and applicable results.
• •The analysis of space-time scaled processes provides a practical tool for verifying the conditions in specific applications, especially in the context of diffusion and averaging approximations, enabling researchers to apply these results to real-world models.

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