Limit theorems for Hilbert space-valued linear processes under long range dependence

Let (Xk)k∈Z be a linear process with values in a separable Hilbert space H given by Xk=∑j=0∞(j+1)−Nεk−j for each k∈Z, where N:H→H is a bounded, linear normal operator and (εk)k∈Z is a sequence of independent, identically distributed H-valued random variables with Eε0=0 and E‖ε0‖2<∞. We investigate the central and the functional central limit theorem for (Xk)k∈Z when the series of operator norms ∑j=0∞‖(j+1)−N‖op diverges. Furthermore, we show that the limit process in case of the functional central limit theorem generates an operator self-similar process.