Limit theorems for additive functionals of a Markov chain - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2008

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.
Fichier principal
Vignette du fichier
jkosm30_08_08.pdf (393.77 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

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. 2008. ⟨hal-00315784v1⟩
228 Consultations
271 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More