## Calcul de la vitesse de convergence dans le théorème central limite vis a vis des distances de Prohorov, Dudley et Levy dans le cas de variables aléatoires depéndantes. (Speed of convergence in the central limit theorem with respect to Prokhorov, Dudley and Levy distances in the case of dependent random variables).(French)Zbl 0607.60019

The paper gives a general framework to estimate Dudley and Lévy’s metrics for Hilbert space valued random variables and Prohorov’s one for the k-dimensional distributions of an $${\mathbb{R}}^ d$$-valued process, in the case of central limit theorem for stationary and mixing random variables. The speeds of convergence obtained here are approximately $$n^{-1/4}$$, $$n^{-1/12}$$ and $$k^{5/8}n^{-1/12}$$, where n is the length of the observed sample and with quite strong mixing hypotheses.

### MSC:

 60F05 Central limit and other weak theorems 60G10 Stationary stochastic processes