Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

Version 1 : Received: 12 May 2021 / Approved: 13 May 2021 / Online: 13 May 2021 (08:47:17 CEST)
Version 2 : Received: 9 September 2021 / Approved: 10 September 2021 / Online: 10 September 2021 (11:05:11 CEST)

How to cite: Kühnert, S. Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces. Preprints 2021, 2021050277 (doi: 10.20944/preprints202105.0277.v2). Kühnert, S. Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces. Preprints 2021, 2021050277 (doi: 10.20944/preprints202105.0277.v2).

Abstract

A major task in Functional Time Series Analysis is measuring the dependence within and between processes, for which lagged covariance and cross-covariance operators have proven to be a practical tool in well-established spaces. This article deduces estimators and asymptotic upper bounds of the estimation errors for lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces for fixed and increasing lag and Cartesian powers. We allow the processes to be non-centered, and to have values in different spaces when investigating the dependence between processes. Also, we discuss features of estimators for the principle components of our covariance operators.

Keywords

Estimation; functional time series; lagged covariance operator; lagged cross covariance operator; principle components

Comments (1)

Comment 1
Received: 10 September 2021
Commenter: Sebastian Kühnert
Commenter's Conflict of Interests: Author
Comment: - I significantly improved the readibility (especially by shortening chapter 1 and definitions in chapter 2)
- I also slightly restated some assertions in a more readable way
- I changed/got rid of some notations
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