Principes d’invariance par moyennisation logarithmique pour les processus de Markov. (Invariance principles with logarithmic averaging for Markov processes). (French) Zbl 1019.60020

The author studies the almost sure central limit theorem for functionals of a recurrent Markov chain \((X_t)_{t\in T}\) with index set either \(T= \mathbb{N}_0\) or \(T= \mathbb{R}_+\) and with general state space \(S\). More specifically, additive martingale functionals are considered, i.e. processes \((M_t)_{t\in T}\) satisfying \(M_{t+s}= M_t+ M_s\circ \theta_t\) as well as \(E_x(M_t)= 0\), where \(\theta_t\) denotes the shift operator on the canonical probability space \(S^T\). In the present paper two main results are established, namely (i) an almost sure version of the functional central limit theorem for \((M_t)_{t\in T}\) and (ii) a central limit theorem and a law of the iterated logarithm extension of the strong law implicit in the almost sure CLT. The latter result only holds in the case of a positive recurrent Markov chain.


60F05 Central limit and other weak theorems
60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
60J55 Local time and additive functionals
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