Estimating lagged (cross‐)covariance operators of Lp‐m‐approximable processes in Cartesian product Hilbert spaces

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$$ {L}^p $$‐ m$$ m $$‐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 eigen elements and parameters in functional AR(MA) models are also discussed.