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

Abstract: Let $(X_{k}){k \in \mathbb Z }$ be a linear process with values in a separable Hilbert space $\mathbb{H}$ given by $X{k} =\sum_{j=0}^{\infty} (j+1)^{-N}\varepsilon_{k-j}$ for each $k \in \mathbb Z$, where $N:\mathbb{H} \to \mathbb{H}$ is a bounded, linear normal operator and $(\varepsilon_{k}){ k \in \mathbb Z }$ is a sequence of independent, identically distributed $\mathbb{H}$-valued random variables with $E\varepsilon{0}=0$ and $E| \varepsilon_{0} |^2<\infty$. We investigate the central and the functional central limit theorem for $(X_{k}){k \in \mathbb Z }$ when the series of operator norms $\sum{j=0}^{\infty} |(j+1)^{-N}|_{op}$ diverges. Furthermore we show that the limit process in case of the functional central limit theorem generates an operator self-similar process.