Estimating lagged (cross-)covariance operators of \(L^{p}\)-\(\it m\)-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 \(L^{p}\)-\(\it m\)-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.