Abstract EN:

We study here asymptomatic properties of mixing processes and their statistical applications. First we give sufficient conditions for mixing of classical processes like non-linear autoregressive ones. After that we give fundamental moment inequalities for sums and for cumulants sums; they allow us to obtain good versions of central limit theorem giving rates with respect to Dudley, Levy of Prohorov metrics. With these tools we give a weak invariance principle for the empirical multidimensional repartition function with arithmetic rate of convergence. We also give rates of convergence in the weak invariance principle for empirical measure in Sobolev spaces and for kernel estimates of the density and of the regression of mixing sequences. From another hand we give asymptotically Gaussian results for quadratic deviation of non parametric estimates from a various kind. Finally we give invariance principles and functional law of the iterated logarithm in the cases of the local time of a Markov recurrent process and of the empirical spectral density of a stationary mixing random process.