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

Andreas E. Kyprianou, Victor Rivero

TL;DR

This paper addresses the strong law of large numbers and a functional central limit theorem for continuous-time Markov additive processes (MAPs) with general Harris recurrent modulators, including null-recurrent cases. It develops a semimartingale framework and ergodic-theoretic approach based on additive functionals to derive SLLN results and a Lindenberg-condition–driven functional CLT, allowing for infinite invariant measures and Mittag–Leffler time changes. The central contribution is showing that, after compensating by an additive functional $A_t$, the scaled fluctuations $(\xi_{nt}-A_{nt})/\sqrt{h(n)}$ converge to a time-changed Gaussian process driven by a Mittag–Leffler clock $W^{(\alpha)}$, with explicit constants $\nu_{\langle\xi\rangle}$ and $J$, thus extending MAP results to general recurrent modulators. This unified approach reveals how recurrence epochs of the modulator induce anomalous diffusion in MAPs and provides a rigorous path to functional limit theorems beyond finite-state modulators.

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.

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

TL;DR

This paper addresses the strong law of large numbers and a functional central limit theorem for continuous-time Markov additive processes (MAPs) with general Harris recurrent modulators, including null-recurrent cases. It develops a semimartingale framework and ergodic-theoretic approach based on additive functionals to derive SLLN results and a Lindenberg-condition–driven functional CLT, allowing for infinite invariant measures and Mittag–Leffler time changes. The central contribution is showing that, after compensating by an additive functional , the scaled fluctuations converge to a time-changed Gaussian process driven by a Mittag–Leffler clock , with explicit constants and , thus extending MAP results to general recurrent modulators. This unified approach reveals how recurrence epochs of the modulator induce anomalous diffusion in MAPs and provides a rigorous path to functional limit theorems beyond finite-state modulators.

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.
Paper Structure (5 sections, 3 theorems, 84 equations)

This paper contains 5 sections, 3 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. MAP as semi-martingales
  3. Main results
  4. Proof of Theorem \ref{['SLLNMAP']}
  5. Proof of Theorem \ref{['CLTMAPS']}

Key Result

Lemma 1

Under the assumptions (H1-5), $\xi-A$ is a martingale. If moreover (H8) is satisfied with $p=1,$ we have that $\xi-A$ is a $\mathcal{F}$-square integrable martingale under $\mathbf{P}$, and its quadratic variation is the additive functional of $\Theta$

Theorems & Definitions (4)

  • Definition 1
  • Lemma 1
  • Theorem 1: Strong law of large numbers for MAP
  • Theorem 2