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.

翻译：摘要：在估计函数型ARMA、GARCH以及可逆线性过程的参数时，会出现定义在笛卡尔积空间上过程的协方差算子和时滞互协方差算子。近年来，这些算子要么在较低通用性条件下，要么在强假设条件下得到了一致估计。本文基于温和的弱依赖条件$L^p$-$m$可逼近性，推导了定义在一般笛卡尔积希尔伯特空间中过程的时滞协方差算子和时滞互协方差算子的估计误差显式上界，从而拓展了现有文献。该上界以时滞、笛卡尔幂次及样本容量形式给出，其中时滞互协方差算子所涉及的两个过程可取值于不同空间。本文还简要提及了所得结果的推广意义。