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

Guivarc’h, Yves; Hardy, J.

Ann. Inst. Henri Poincaré, Probab. Stat. 24, No. 1, 73-98 (1988).

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

MSC:

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

Keywords:

Anosov diffeomorphisms; Central limit theorems; local limit theorems; generalized renewal theorems; Markov chains; mixing sub-shifts