Strong Law of Large Numbers and Central Limit Theorems for Functionals of Inhomogeneous Semi-Markov Processes

Abstract

Limit theorems for functionals of classical (homogeneous) Markov renewal and semi-Markov processes have been known for a long time, since the pioneering work of Pyke Schaufele (Limit theorems for Markov renewal processes, Ann. Math. Statist., 35(4):1746–1764, 1964). Since then, these processes, as well as their time-inhomogeneous generalizations, have found many applications, for example, in finance and insurance. Unfortunately, no limit theorems have been obtained for functionals of inhomogeneous Markov renewal and semi-Markov processes as of today, to the best of the authors’ knowledge. In this article, we provide strong law of large numbers and central limit theorem results for such processes. In particular, we make an important connection of our results with the theory of ergodicity of inhomogeneous Markov chains. Finally, we provide an application to risk processes used in insurance by considering a inhomogeneous semi-Markov version of the well-known continuous-time Markov chain model, widely used in the literature.