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

Doukhan, P.; Leon, J.; Portal, F.

Probab. Math. Stat. 6, 19-27 (1985).

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

Keywords:

speed of convergence; central limit theorem; Dudley and Lévy’s metrics; stationary and mixing random variables; strong mixing hypotheses