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.

翻译：估计函数型ARMA、GARCH及可逆过程的参数需要估计笛卡尔积希尔伯特空间值过程的滞后协方差算子与滞后交叉协方差算子。近年来已有渐近结果被推导出来，但要么普适性不足，要么依赖于严格条件。本文基于对每个滞后阶数、笛卡尔幂次及样本量均成立的温和条件——Lp-m-可逼近性，推导了此类算子估计误差的上界；在滞后交叉协方差算子的情形下，两个过程可取值于不同空间。本文还讨论了我们的结果对特征元、函数型AR(MA)模型参数及其他一般情形的影响。