估计笛卡尔积Hilbert空间中$L^p$-$m$-可逼近过程的滞后（交叉）协方差算子
Estimating Lagged (Cross-)Covariance Operators of $L^p$-$m$-approximable Processes in Cartesian Product Hilbert Spaces
AI总结：
本文基于温和条件$L^p$-$m$-可逼近性，推导了笛卡尔积Hilbert空间值过程的滞后协方差与交叉协方差算子估计误差的上界，并探讨了其对函数型时间序列模型参数估计等推论。
摘要：
估计函数型ARMA、GARCH及可逆过程的参数，需要估计笛卡尔积Hilbert空间值过程的滞后协方差和交叉协方差算子。近年虽已推导出渐近结果，但要么不够一般，要么条件严格。本文基于温和条件——即针对每个滞后、笛卡尔幂和样本量的$L^p$-$m$-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情形下，两个过程可取值于不同空间。本文还讨论了这些结果对本征元素、函数型AR(MA)模型中的参数及其他一般情况的推论。
英文摘要：
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.
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