Limit theorems for additive functionals of a Markov chain

Jara, Milton ; Komorowski, Tomasz ; Olla, Stefano

Ann. Appl. Probab., Tome 19 (2009) no. 1, p. 2270-2300 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2009-12-15
Classification:  Stable laws,  self-similar Lévy processes,  limit theorems,  linear Boltzmann equation,  fractional heat equation,  anomalous heat transport,  60F05,  60F17,  76P05