Estimating Lagged (Cross-)Covariance Operators of $L^p$-$m$-approximable Processes in Cartesian Product Hilbert Spaces

Sebastian Kühnert

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arXiv · 2402.08110

Estimating Lagged (Cross-)Covariance Operators of $L^p$-$m$-approximable Processes in Cartesian Product Hilbert Spaces

Sebastian Kühnert

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Abstract

Estimating parameters of functional ARMA, GARCH and invertible processes requires estimating lagged covariance and cross-covariance operators of Cartesian product Hilbert space-valued processes. Asymptotic results have been derived in recent years, either less generally or under a strict condition. This article derives upper bounds of the estimation errors for such operators based on the mild condition Lp-m-approximability for each lag, Cartesian power(s) and sample size, where the two processes can take values in different spaces in the context of lagged cross-covariance operators. Implications of our results on eigenelements, parameters in functional AR(MA) models and other general situations are also discussed.

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BibTeX RIS

Sebastian Kühnert. 2024-02-12. Estimating Lagged (Cross-)Covariance Operators of $L^p$-$m$-approximable Processes in Cartesian Product Hilbert Spaces. https://arxiv.org/abs/2402.08110

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