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

TL;DR

This paper establishes limit theorems for Hilbert space-valued linear processes exhibiting long-range dependence, demonstrating convergence to operator self-similar processes under divergence conditions of the series of operator norms.

Contribution

It extends classical limit theorems to infinite-dimensional Hilbert space processes with long-range dependence, identifying conditions for convergence to operator self-similar processes.

Findings

Proves central limit theorem for long-range dependent Hilbert space processes.

Shows functional limit theorem leads to operator self-similar processes.

Identifies divergence of operator norm series as key condition.

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.