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

Abstract In this note, we revisit 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, cf. (Asmussen in Applied probability and queues, Springer-Verlag, New York, 2003; Asmussen and Albrecher in Ruin probabilities, Hackensack, 2010), not to mention the classical contributions of Pacheco and Prabhu (Markov additive processes of arrivals, CRC, Boca Raton, 1995), Prabhu (Stochastic storage processes, Springer-Verlag, New York, 1998). Recent works have addressed the strong law of large numbers when the underlying modulating process is a general Markov processes; cf. as reported (Kyprianou et al. Entrance laws at the origin of self-similar Markov processes in high dimensions, 2019; Yaran and Çağlar in ALEA Lat Am J Probab Math Stat 22:991–1010, 2025) . We add to the latter with a different approach based on an ergodic theorem for additive functionals and on the semimartingale 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.