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Limit theorems for additive functionals of a markov chain. arXiv, math.PR

By M. Jara, T. Komorowski and S. Olla


Abstract. Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ 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 N 1/α ∑ N n Ψ(Xn) to a α-stable law. “Martingale approximation ” approach and “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 time between jumps has finite expectation, we prove convergence of N −1/α ∫ Nt 0 V (Xs)ds to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process. 1

Year: 2008
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