Operator self-similar processes and functional central limit theorems

Abstract

Let $\{X_k:k\ge 1\}$ be a linear process with values in the separable Hilbert space $L_2(\mu )$ given by $X_k={\sum }_{j=0}^{\infty }(j+1{)}^{-D}{\epsilon }_{k-j}$ for each $k\ge 1$, 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 $\{{\epsilon }_k:k\in \mathbb{Z}\}$ are independent and identically distributed $L_2(\mu )$-valued random elements with $E{\epsilon }_0=0$ and $E\Vert {\epsilon }_0{\Vert }^2<\infty$. We establish sufficient conditions for the functional central limit theorem for $\{X_k:k\ge 1\}$ when the series of operator norms ${\sum }_{j=0}^{\infty }\Vert (j+1{)}^{-D}\Vert$ diverges and show that the limit process generates an operator self-similar process.

22 pages

References in corpus (2)

• Integral representations and properties of operator fractional Brownian motions
• Covariance function of vector self-similar process