Rate of convergence for Hilbert space valued processes

Abstract

Consider a stationary, linear Hilbert space valued process. We establish Berry-Esseen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of non-linear Bernoulli-shift sequences is also considered. If the sequence is m-dependent, the optimal rate (n/m)1/2 is reached. If the sequence is weakly geometrically dependent, the rate (n/log n)1/2 is obtained.