Limit theorems for additive functionals of a Markov chain

Abstract

Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $π$. Let $Ψ$ a function on the state space of the chain, with $α$-tails with respect to $π$, $α\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/α}\sum_n^N Ψ(X_n)$ to a $α$-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 $N^{-1/α}\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.