2 00 9 Limit Theorems for Additive Functionals of a Markov Chain

Consider a Markov chain {X n } 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 N 1/α N n Ψ(X n) to an α-stable law. A " 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 times between jumps have finite expectation, we prove convergence of N −1/α N t 0 V (X s)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.