Berry-Esseen Estimates in Hilbert Space and an Application to the Law of the Iterated Logarithm

Abstract

We establish Berry-Esseen type estimates for random variables with values in a real separable Hilbert space $H$. These estimates are then used to prove the law of the iterated logarithm for sequences of $H$-valued random variables and also to prove a functional form of the law of the iterated logarithm for $H$-valued partial sums as given by Strassen. We also prove a $\log \log$ result for $H$-valued symmetric stable random variables.