Inequalities for absolutely regular sequences: application to density estimation

This paper investigates the problem of density estimation for absolutely regular observations. In a first part, we state two important results: a new variance inequality and a Rosenthal type inequality. This allows us to study the ? _p -integrated risk, p≧ 2, of a large class of density estimators including kernel or projection estimators. Under the summability condition on the mixing coefficients ∑ _{k≧ 0} (k+1) ^{p− 2} β _k <∞, the rates obtained are those known to be optimal in the independent setting. Received: 17 October 1995 / In revised form: 26 October 1996