Limit theorems for additive functionals of a Markov chain

Abstract

Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with α-tails with respect to π, α ∈; (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N1/α ΣnN Φ(X n) to an α-stable law. A "martingale approximation" approach and a "coupling" approach give two different sets of conditions. We extend these results to continuous time Markov jump processes Xt, whose skeleton chain satisfies our assumptions. If waiting times between jumps have finite expectation, we prove convergence of N -1/α ∞0Nt V(XS)ds to a stable process. The result is applied to show that an appropriately scaled limit of solutions of a linear Boltzman equation is a solution of the fractional diffusion equation. © Institute of Mathematical Statistics, 2009.