6533b86ffe1ef96bd12cdbcf

RESEARCH PRODUCT

Exponential inequalities and estimation of conditional probabilities

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This paper deals with the problems of typicality and conditional typicality of “empirical probabilities” for stochastic process and the estimation of potential functions for Gibbs measures and dynamical systems. The questions of typicality have been studied in [FKT88] for independent sequences, in [BRY98, Ris89] for Markov chains. In order to prove the consistency of estimators of transition probability for Markov chains of unknown order, results on typicality and conditional typicality for some (Ψ)-mixing process where obtained in [CsS, Csi02]. Unfortunately, lots of natural mixing process do not satisfy this Ψ -mixing condition (see [DP05]). We consider a class of mixing process inspired from [DP05]. For this class, we prove strong typicality and strong conditional typicality. In the particular case of Gibbs measures (or complete connexions chains) and for certain dynamical systems, from the typicality results we derive an estimation of the potential as well as a procedure to test the nullity of the asymptotic variance of the process. More formally, we consider X0, ...., Xn, ... a stochastic process taking values on an complete set Σ and a sequence of countable partitions of Σ, (Pk)k∈N such that if P ∈ Pk then there exists a unique P ∈ Pk−1 such that almost surely, Xj ∈ P implies Xj−1 ∈ P . Our aim is to obtain empirical estimates of the probabilities: P(Xj ∈ P ), P ∈ Pk ,