How to overcome the curse of long-memory errors

Long-memory errors dramatically slow down the convergence of minimax risks in a fixed design nonparametric regression. The problem becomes even more complicated for the case of adaptive estimation. This defines the curse of long-memory errors. I show that using a random design, instead of a fixed one, allows one to overcome this curse and make familiar data-driven estimators robust. Moreover, the result holds for a wide class of nonstationary errors with bounded moments (including bounded deterministic errors). Possible extensions are discussed.