A multidimensional functional central limit theorem for an empirical estimator of a continuous-time semi-Markov kernel

Abstract

In this article, we consider the empirical estimator of the kernel of a semi-Markov process on continuous time with finite state space. We obtain a functional central limit theorem for this estimator in multidimensional form. Next, we present the corresponding theorem for the empirical estimator of the conditional sojourn-time distribution function. The proofs of our results are based on semi-martingales.

Keywords:

• semi-Markov process
• semi-Markov kernel
• empirical estimator
• functional central limit theorem
• semi-martingales

AMS Subject Classifications :

• 60F05
• 60F17
• 60K15
• 62G05
• 62G20

Acknowledgements

The authors are grateful to the referees for their useful comments on the manuscript.