Limit theorems for additive functionals of a Markov chain

Abstract

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^NΨ(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 X_t, whose skeleton chain satisfies our assumptions. If waiting times between jumps have finite expectation, we prove convergence of N^−1/α∫₀^NtV(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.