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.Comment: Accepted for the publication in Annals of Applied Probabilit
We consider a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4.
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