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December 2009 Limit theorems for additive functionals of a Markov chain
Milton Jara, Tomasz Komorowski, Stefano Olla
Ann. Appl. Probab. 19(6): 2270-2300 (December 2009). DOI: 10.1214/09-AAP610

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/αnNΨ(Xn) 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/α0NtV(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.

Citation

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Milton Jara. Tomasz Komorowski. Stefano Olla. "Limit theorems for additive functionals of a Markov chain." Ann. Appl. Probab. 19 (6) 2270 - 2300, December 2009. https://doi.org/10.1214/09-AAP610

Information

Published: December 2009
First available in Project Euclid: 25 November 2009

zbMATH: 1232.60018
MathSciNet: MR2588245
Digital Object Identifier: 10.1214/09-AAP610

Subjects:
Primary: 60F05, 60F17
Secondary: 76P05

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.19 • No. 6 • December 2009
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