The Annals of Applied Probability

Limit theorems for additive functionals of a Markov chain

Milton Jara, Tomasz Komorowski, and Stefano Olla

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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.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 6 (2009), 2270-2300.

Dates
First available in Project Euclid: 25 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1259158772

Digital Object Identifier
doi:10.1214/09-AAP610

Mathematical Reviews number (MathSciNet)
MR2588245

Zentralblatt MATH identifier
1232.60018

Subjects
Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Secondary: 76P05: Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05]

Keywords
Stable laws self-similar Lévy processes limit theorems linear Boltzmann equation fractional heat equation anomalous heat transport

Citation

Jara, Milton; Komorowski, Tomasz; Olla, Stefano. Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19 (2009), no. 6, 2270--2300. doi:10.1214/09-AAP610. https://projecteuclid.org/euclid.aoap/1259158772


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