Weak dependence beyond mixing and asymptotics for nonparametric regression

Abstract

We consider a new concept of weak dependence, introduced by Doukhan and Louhichi [Stochastic Process. Appl. 84 (1999) 313–342], which is more general than the classical frameworks of mixing or associated sequences. The new notion is broad enough to include many interesting examples such as very general Bernoulli shifts, Markovian models or time series bootstrap processes with discrete innovations.

Under such a weak dependence assumption, we investigate nonparametric regression which represents one (among many) important statistical estimation problems. We justify in this more general setting the “whitening by windowing principle” for nonparametric regression, saying that asymptotic properties remain in first order the same as for independent samples. The proofs borrow previously used strategies, but precise arguments are developed under the new aspect of general weak dependence.