We study the optimal linear prediction of a random function, assuming it takes values in an infinite dimensional Hilbert space. We begin by characterizing the mean square prediction error (MSPE) associated with a linear predictor and discussing the minimal achievable MSPE. This analysis reveals that, in general, there are multiple non-unique linear predictors that minimize the MSPE, and even if a unique solution exists, consistently estimating it from finite samples is generally impossible. Nevertheless, we can define asymptotically optimal linear operators whose empirical MSPEs approach the minimal achievable level as the sample size increases. We show that, interestingly, standard post-dimension reduction estimators, which have been widely used in the literature, attain such asymptotic optimality under minimal conditions.

翻译：我们研究随机函数的最优线性预测问题，假设该随机函数取值于无穷维希尔伯特空间。首先，我们刻画与线性预测器相关的均方预测误差（MSPE），并讨论其最小可达水平。分析表明，一般情况下存在多个非唯一的线性预测器可实现MSPE最小化；即使存在唯一解，通过有限样本对其进行一致估计通常也是不可能的。然而，我们仍可定义渐近最优线性算子，其经验MSPE随样本量增加趋近于最小可达水平。有趣的是，文献中广泛使用的标准后降维估计量在极弱条件下即可达到这种渐近最优性。