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Théorème de convergence vers des lois stables pour une classe de chaines de Markov. (A theorem of convergence to a stable law for a class of Markov chains). (French) Zbl 0597.60066

The author considers a Markov chain \((x_ k)^{\infty}\), on a compact metric space X, which has an invariant distribution \(\pi\). He studies the behaviour of the sequence \(S_ n/n^{1/\alpha}\) where \(S_ n=f(x_ 1)+...+f(x_ n)\), and f is a real-valued function on X having a (centered) distribution (with respect to \(\pi)\) in the domain of attraction of a stable law with exponent \(\alpha\). It is shown that under fairly general conditions the distribution of \(S_ n/n^{1/\alpha}\) tends to the stable law.
The proof uses Fourier transforms and the behaviour of the dominant eigenvalue of a Fourier-type linear operator. The theorem generalizes a central limit result by Y. Guivarc’h [Lect. Notes Math. 1096, 301- 332 (1984; Zbl 0562.60074)], and a result by A. H. Hoekstra and the reviewer [Linear Algebra Appl. 60, 65-77 (1984; Zbl 0548.60068)].
Reviewer: F.W.Steutel

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
60B05 Probability measures on topological spaces
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