The strong law of large numbers and a functional central limit theorem for general Markov additive processes

TL;DR

This paper establishes strong law of large numbers and a functional central limit theorem for continuous-time Markov additive processes, extending classical results to more general modulating Markov processes using ergodic theorems and semi-martingale techniques.

Contribution

It introduces a new approach based on ergodic theorems and semi-martingale structure to prove LLN and CLT for MAPs with general Markov modulating processes.

Findings

Proved strong law of large numbers for general MAPs.

Established a functional central limit theorem for MAPs.

Extended results to null recurrent modulating processes.

Abstract

In this note we re-visit the fundamental question of the strong law of large numbers and central limit theorem for processes in continuous time with conditional stationary and independent increments. For convenience we refer to them as Markov additive processes, or MAPs for short. Historically used in the setting of queuing theory, MAPs have often been written about when the underlying modulating process is an ergodic Markov chain on a finite state space. Recent works have addressed the strong law of large numbers when the underlying modulating process is a general Markov processes. We add to the latter with a different approach based on an ergodic theorem for additive functionals and on the semi-martingale structure of the additive part. This approach also allows us to deal with the setting that the modulator of the MAP is either positive or null recurrent. The methodology additionally inspires a CLT-type result.