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Limit theorems for some functionals with heavy tails of a discrete time Markov chain. (English) Zbl 1328.60093

Summary: Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain \((X_{n},n\geq 0)\) with invariant distribution \(\mu\). We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional \(S_{n}=\sum_{i=1}^{n}f(X_{i})\) for a possibly non square integrable function \(f\). To this end, we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence to stable distributions for stationary mixing sequences. Contrary to the usual \(L^2\) framework studied by Cattiaux et al., where weak forms of ergodicity are sufficient to ensure the validity of the central limit theorem, we will need here strong ergodic properties: the existence of a spectral gap, hyperboundedness (or hypercontractivity). These properties are also discussed. Finally, we give explicit examples.

MSC:

60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60J05 Discrete-time Markov processes on general state spaces
60J55 Local time and additive functionals
60E07 Infinitely divisible distributions; stable distributions
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