Limit theorems for additive functionals of a Markov chain

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.

Accepted for the publication in Annals of Applied Probability

References in corpus (4)

• Fractional diffusion limit for collisional kinetic equations
• Limit theorems for additive functionals of a Markov chain
• Kinetic Limit for Wave Propagation in a Random Medium
• Energy transport in stochastically perturbed lattice dynamics

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