Limit theorems for additive functionals of a Markov chain

TL;DR

This paper establishes conditions under which additive functionals of Markov chains and jump processes converge in distribution to stable laws or processes, extending classical limit theorems to heavy-tailed and continuous-time settings.

Contribution

It provides new sufficient conditions for stable law convergence of additive functionals of Markov chains and jump processes, including cases with infinite mean waiting times.

Findings

Convergence to $ ext{alpha}$-stable laws for discrete-time Markov chains.

Extension of results to continuous-time Markov jump processes.

Identification of convergence to Mittag-Leffler processes for infinite mean waiting times.

Abstract

Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $\pi$. Let $\Psi$ a function on the state space of the chain, with $\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. ``Martingale approximation'' approach and ``coupling'' approach give two different sets of conditions. We extend these results to continuous time Markov jump processes $X_t$, whose skeleton chain satisfies our assumptions. If waiting time between jumps has finite expectation, we prove convergence of $N^{-1/\alpha}\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.