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Limit theorems for additive functionals of a Markov chain | | Milton Jara
; Tomasz Komorowski
; Stefano Olla
; | | Date: |
1 Sep 2008 | | Abstract: | Consider a Markov chain ${X_n}_{nge 0}$ with an ergodic probability
measure $pi$. Let $Psi$ a function on the state space of the chain, with
$alpha$-tails with respect to $pi$, $alphain (0,2)$. 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. | | Source: | arXiv, 0809.0177 | | Services: | Forum | Review | PDF | Favorites | |
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