Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. (Limit theorems for a class of Markov chains and applications to Anosov diffeomorphisms). (French. English summary) Zbl 0649.60041

Central limit theorems, local limit theorems, and generalized renewal theorems are established for sums of functions of jumps of Markov chains. The class of Markov chains considered is that of chains whose kernels satisfy a quasi-compactness condition. Results are obtained by exploiting a theorem of Ionescu-Tulcea and Marinescu. These results are then applied to dynamical systems which can be coded as mixing sub-shifts.
Note that the proof of the majorization on page 81 is not complete as it stands. The first author has shown me how to modify the proof: the necessary (and minor) modifications are indicated below for the interested reader:
Multiply \(C(1-\epsilon)^ n\) by \(| \lambda /\sqrt{n}|\) on lines 4, 7 and 8 of that page. This is justified by an argument expressing r n(i\(\lambda)\) as a contour integral of \(z^ n R(z,\lambda)\), where \(R(z,\lambda)\) is the resolvant of \(P_{i\lambda}\).
Reviewer: W.S.Kendall


60F05 Central limit and other weak theorems
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
37A25 Ergodicity, mixing, rates of mixing
37A50 Dynamical systems and their relations with probability theory and stochastic processes
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