Lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces

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 wellestablished spaces. This article focuses on estimating these operators of processes in Cartesian products of abstract Hilbert spaces. We derive precise asymptotic results for the estimation errors for fixed and increasing lag and Cartesian powers under very mild conditions, presumably even under the mildest that can be assumed, establish estimators for the principal components, and conduct a simulation study.

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

“…Theorem 3.1 states explicit upper bounds for the estimation errors for lagged covariance operators of processes in Cartesian products of arbitrary separable Hilbert spaces for any lag h, Cartesian power m and sample size N under L 4 -m-approximability, and thus adds to the existing literature: Under L 4 -m-approximability, Hörmann and Kokoszka (2010) stated upper bounds for covariance operators of processes in L 2 [0, 1], where their upper bound matches ours except for their additional factor √ 2 in the series term; Aue and Klepsch (2017) derived upper bounds for any m, N for covariance operators of processes in Cartesian products defined via linear processes in L 2 [0, 1]; and Kuenzer (2024) mentions asymptotic upper bounds for covariance and specific lagged covariance operators. Moreover, Kühnert (2022) only deduced precise limits for the estimation errors of lagged covariance operators in Cartesian product under a stricter condition.…”

“…We can allow k also to depend on the sample size N. This is needed when estimating operators involving projecting onto a k-dimensional subspace and allowing the number of projections k to go to infinity, see Bosq (2000), Kühnert (2022), Kühnert et al (2024), Kuenzer (2024), to name a few.…”

“…Furthermore, also under L 4 -m-approximibility, Aue and Klepsch (2017) gave upper bounds for covariance operators of processes in Cartesian product spaces originating from L 2 [0, 1]-valued linear processes, Hörmann and Kidziński (2015) for covariance and crosscovariance operators, and in Kuenzer (2024) asymptotic upper bounds for covariance and certain lagged covariance operators can be found. Moreover, Kühnert (2022) established exact limits for the estimation errors for lagged covariance and lagged cross-covariance operators of quite general, not necessarily centered processes in Cartesian product spaces under a rather implicit weak dependence condition. Finally, we would like to refer to Rice and Shum (2019) who studied lagged cross-covariance operators of L 2 [0, 1]-valued processes in detail.…”