Convergence rates for density estimators of weakly dependent time series

Abstract

Assuming that $(X_t)_{t\in\Z}$ is a vector valued time series with a common marginal distribution admitting a density $f$, our aim is to provide a wide range of consistent estimators of $f$. We consider different methods of estimation of the density as kernel, projection or wavelets ones. Various cases of weakly dependent series are investigated including the Doukhan & Louhichi (1999)'s $\eta$-weak dependence condition, and the $\tilde \phi$-dependence of Dedecker & Prieur (2005). We thus obtain results for Markov chains, dynamical systems, bilinear models, non causal Moving Average... From a moment inequality of Doukhan & Louhichi (1999), we provide convergence rates of the term of error for the estimation with the $\L^q$ loss or almost surely, uniformly on compact subsets.