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

When estimating the parameters in functional ARMA, GARCH and invertible, linear processes, covariance and lagged cross-covariance operators of processes in Cartesian product spaces appear. Such operators have been consistenly estimated in recent years, either less generally or under a strong condition. This article extends the existing literature by deriving explicit upper bounds for estimation errors for lagged covariance and lagged cross-covariance operators of processes in general Cartesian product Hilbert spaces, based on the mild weak dependence condition $L^p$-$m$-approximability. The upper bounds are stated for each lag, Cartesian power(s) and sample size, where the two processes in the context of lagged cross-covariance operators can take values in different spaces. General consequences of our results are also mentioned.