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

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.

💡 Research Summary

This paper revisits two fundamental asymptotic results – the strong law of large numbers (SLLN) and a functional central limit theorem (FCLT) – for continuous‑time Markov additive processes (MAPs). A MAP consists of an “ordinate” ξ and a “modulator” Θ; conditional on the whole past of Θ, the future increments of ξ are stationary and independent. Historically, most MAP literature assumes that Θ is a finite‑state, ergodic Markov chain, which makes the analysis relatively straightforward. Recent work has begun to treat the case where Θ is a general Markov process, but the results are still limited.

The authors adopt a different route: they combine an ergodic theorem for additive functionals of Harris‑recurrent Markov processes with the semimartingale representation of ξ. This allows them to treat both positive‑recurrent (π finite) and null‑recurrent (π σ‑finite but infinite) modulators, and to obtain a CLT‑type result under the same framework.

Model and assumptions.

• (H1) The bivariate process ((ξₜ,Θₜ),t≥0) is a Hunt process.
• (H2) Θ is Harris recurrent with invariant σ‑finite measure π; π finite corresponds to positive recurrence, π infinite to null recurrence.
• (H3) ξ is a semimartingale (this follows from the MAP structure).
• (H4) The Lévy system’s time‑change functional Hₜ equals t∧ζ, which can always be arranged by a deterministic time change.
• (H5)–(H8) impose integrability on the drift, Lévy jump kernel, and their Revuz measures, ensuring that the predictable compensator of ξ has a finite mean m₁ and that a higher moment (p>0) exists for the jump part.

Using C̆inlar’s decomposition, ξ can be written as
ξₜ = χₜ + ξᶜₜ + ξᶠₜ + ξᵈₜ,
where ξᶜ is a continuous Gaussian MAP, ξᶠ is a pure‑jump MAP whose jump times are Θ‑measurable, ξᵈ is a conditionally independent‑increments MAP, and χ is a possibly irregular drift functional of Θ. The Lévy system (H,Π) yields the conditional characteristic function (5) and identifies the predictable drift Bₜ and quadratic variation Cₜ.

Define the additive functional
Aₜ := Bᶜₜ + ∫₀ᵗ μ_d(Θ_s) ds,
where μ_d(θ) aggregates the jump intensity of Π. Under (H5)–(H7) the process ξₜ−Aₜ is a zero‑mean martingale (Lemma 1). Moreover, the Revuz measures of Bᶜ and C are finite, which guarantees that the mean of ξ over one unit of time under the invariant law π is
m₁ = ‖ν_{Bᶜ}‖ + ∫S μ_d(θ) π(dθ) = E{0,π}