This habilitation manuscript presents my research work on statistics for weakly dependent processes. Asymptotical results for the Quasi Maximum Likelihood Estimator in general affine models are given in the first part. To detect stationarity breaks, we suggest to penalize the Quasi Likelihood criteria by the number of breaks. For some volatility models such as EGARCH model, the procedure is not stable and we suggest to constrain the criteria on the continuously invertible domain. Then we consider the one-step prediction of weakly dependent processes, establishing new oracle inequalities. Such non asymptotical results need to assert the gaussian concentration properties of weakly dependent measures. To this aim, we propose a notion of weak transport and new conditional transport inequalities. Finally, we introduce the cluster index to characterize the extremal behavior of regularly varying partial sums. We obtain limit properties such as the $\alpha$-stable limits in the Central Limit Theorem or the large deviations in presence of dependent extremes. Solutions of linear stochastic recurrent equation and the GARCH model are examples of heavy tailed processes. We apply our results to characterize asymptotically the estimation errors of heavy tailed processes autocovariances.
