François ROUEFF, Telecom Paris Tech

We introduce locally stationary time series through the  local approximation of the non-stationary covariance structure by a  stationary one.  This allows us to define autoregression coefficients in a  non-stationary context, which, in the particular case of a locally stationary  Time Varying Autoregressive (TVAR) process, coincide with the generating
coefficients. We provide and study an estimator of the time varying  autoregression coefficients in a general setting. The proposed estimator of  these coefficients enjoys an optimal minimax convergence rate under limited  smoothness conditions. In a second step, using a bias reduction technique, we  derive a minimax-rate estimator for arbitrarily smooth time-evolving  coefficients, which outperforms the previous one for large data sets. In  turn, for TVAR processes, the predictor derived from the estimator  exhibits an optimal minimax prediction rate.