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Article Dans Une Revue The Annals of Applied Probability Année : 2009

Limit theorems for additive functionals of a Markov chain

Résumé

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.
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Dates et versions

hal-00315784 , version 1 (31-08-2008)
hal-00315784 , version 2 (18-09-2008)
hal-00315784 , version 3 (28-10-2008)
hal-00315784 , version 4 (25-04-2009)
hal-00315784 , version 5 (15-12-2009)

Identifiants

Citer

Milton Jara, Tomasz Komorowski, Stefano Olla. Limit theorems for additive functionals of a Markov chain. The Annals of Applied Probability, 2009, 19 (6), pp.2270-2300. ⟨10.1214/09-AAP610⟩. ⟨hal-00315784v5⟩
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