Weak convergence for the covariance operators of a Hilbertian linear process.
type: Item
instance of: scholarly article
title: Weak convergence for the covariance operators of a Hilbertian linear process.
zbMATH Open document ID: 1059.60026
DOI: 10.1016/S0304-4149(02)00087-X
author: André Mas
published in: Stochastic Processes and their Applications
publication date: 2005-02-25T00:00:00Z
review text: Let \(H\) be a separable Hilbert space, \(a_k \in L(H,H)\), and let \(\varepsilon _k\) be i.i.d.\ centred random variables with values in \(H\). The author considers the stochastic process \(X_i= \sum _{k=-\infty }^{\infty } a_k\varepsilon _{i-k}\). Under the assumptions \({\mathbf E}\| \varepsilon _0\| ^4 <\infty \) and \(\sum _{k}\| a_k\| <\infty \) he obtains the following central limit theorem for the vector of empirical covariance operators of \((X_i)\): \[ \sqrt {n}(V_{n,0},\dots ,V_{n,k})@>w>> G,\qquad n\to \infty , \] where \(V_{n,i} = \frac {1}{n} \sum _{j=1}^n \langle X_{i+j},\cdot \rangle X_j - {\mathbf E}(\langle X_i,\cdot \rangle X_0)\) are Hilbert-Schmidt operators and \(G\) is a Gaussian centred random \((n+1)\)-dimensional vector having its components in the space of Hilbert-Schmidt operators. Also, a connection to the principal component analysis of the process \((X_i)\) is mentioned.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 2138965
zbMATH Keywords: linear operators on Hilbert space
zbMATH Keywords: covariance operators
zbMATH Keywords: weak convergence of random elements
describes a project that uses: fda (R)
MaRDI profile type: Publication
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full work available at URL: s0304-4149(02)00087-x
OpenAlex ID: W2081273940
description: scientific article; zbMATH DE number 2138965