Invertible processes are central to functional time series analysis, making the estimation of their defining operators a key problem. While asymptotic error bounds have been established for specific ARMA models on $L^2[0,1]$, a general theoretical framework has not yet been considered. This paper fills in this gap by deriving consistent estimators for the operators characterizing the invertible representation of a functional time series with white noise innovations in a general separable Hilbert space. Under mild conditions covering a broad class of functional time series, we establish explicit asymptotic error bounds, with rates determined by operator smoothness and eigenvalue decay. These results further provide consistency-rate estimates for operators in Hilbert space-valued causal linear processes, including functional MA, AR, and ARMA models of arbitrary order.

翻译：可逆过程是函数型时间序列分析的核心，其定义算子的估计因此成为关键问题。尽管在$L^2[0,1]$上针对特定ARMA模型已建立了渐近误差界，但通用的理论框架尚未得到考虑。本文填补了这一空白，针对一般可分希尔伯特空间中具有白噪声新息的函数型时间序列，推导了刻画其可逆表示算子的相合估计量。在涵盖广泛函数型时间序列类的温和条件下，我们建立了显式的渐近误差界，其收敛速率由算子光滑性与特征值衰减决定。这些结果进一步为希尔伯特空间值因果线性过程（包括任意阶的函数型MA、AR及ARMA模型）中的算子提供了相合性速率估计。