Operator self-similar processes and functional central limit theorems

TL;DR

This paper investigates the asymptotic behavior of operator self-similar processes derived from linear Hilbert space-valued processes, establishing conditions for functional central limit theorems in the presence of diverging operator norm series.

Contribution

It provides new sufficient conditions for the functional central limit theorem for operator self-similar processes with diverging operator norm series.

Findings

Established conditions for the functional central limit theorem in Hilbert space setting.

Showed the limit process is an operator self-similar process.

Extended understanding of long-range dependence in infinite-dimensional processes.

Abstract

Let $\{X_k:k\ge1\}$ be a linear process with values in the separable Hilbert space $L_2(\mu)$ given by $X_k=\sum_{j=0}^\infty(j+1)^{-D}\varepsilon_{k-j}$ for each $k\ge1$, where $D$ is defined by $Df=\{d(s)f(s):s\in\mathbb S\}$ for each $f\in L_2(\mu)$ with $d:\mathbb S\to\mathbb R$ and $\{\varepsilon_k:k\in\mathbb Z\}$ are independent and identically distributed $L_2(\mu)$-valued random elements with $\operatorname E\varepsilon_0=0$ and $\operatorname E\|\varepsilon_0\|^2<\infty$. We establish sufficient conditions for the functional central limit theorem for $\{X_k:k\ge1\}$ when the series of operator norms $\sum_{j=0}^\infty\|(j+1)^{-D}\|$ diverges and show that the limit process generates an operator self-similar process.