Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### On recursive estimation for locally stationary time varying autoregressive processes

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Résumé: This paper focuses on recursive estimation of locally stationary autoregressive processes. The stability of the model is revisited and uniform results are provided when the time-varying autoregression parameters belong to appropiate smoothness classes. An adequate normalization for the correction term used in the recursive estimation procedure allows for very mild assumptions on the innovations distributions. The rate of convergence of the pointwise estimates are shown to be minimax in $\beta$-Lipschitz classes for $0 < \beta \leq 1$. For $1 < \beta \leq 2$, this property does no longer hold; a bias reduction method is proposed for recovering the minimax rate. Finally, an asymptotic expansion of the estimation error is given, allowing both for an explicit asymptotic expansion of the mean-square error and for a central limit theorem.