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Limit theorems for additive functionals of a Markov chain

Olla, Stefano; Komorowski, Tomasz; Jara, Milton (2009), Limit theorems for additive functionals of a Markov chain, The Annals of Applied Probability, 19, 6, p. 2270-2300. http://dx.doi.org/10.1214/09-AAP610

Type
Article accepté pour publication ou publié
External document link
http://projecteuclid.org/euclid.aoap/1259158772
Date
2009
Journal name
The Annals of Applied Probability
Volume
19
Number
6
Publisher
Institute of Mathematical Statistics
Pages
2270-2300
Publication identifier
http://dx.doi.org/10.1214/09-AAP610
Metadata
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Author(s)
Olla, Stefano cc
Komorowski, Tomasz
Jara, Milton
Abstract (EN)
Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $\pi$. Let $\Psi$ a function on the state space of the chain, with $\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. ``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 time between jumps has finite expectation, we prove convergence of $N^{-1/\alpha}\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.
Subjects / Keywords
martingale approximations; fractional heat equation; Boltzmann phonon equation; Stable processes

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