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 $X_k={\sum }_{j=0}^{\infty }(j+1{)}^{-N}{\epsilon }_{k-j}$ for each $k\in \mathbb{Z}$, where $N:\mathbb{H}\to \mathbb{H}$ is a bounded, linear normal operator and $({\epsilon }_k{)}_{k\in \mathbb{Z}}$ is a sequence of independent, identically distributed $\mathbb{H}$-valued random variables with $E{\epsilon }_0=0$ and $E\Vert {\epsilon }_0{\Vert }^2<\infty$. We investigate the central and the functional central limit theorem for $(X_k{)}_{k\in \mathbb{Z}}$ when the series of operator norms ${\sum }_{j=0}^{\infty }\Vert (j+1{)}^{-N}{\Vert }_{op}$ diverges. Furthermore we show that the limit process in case of the functional central limit theorem generates an operator self-similar process.

References in corpus (1)

• The central limit theorem for a sequence of random processes with space varying long memory