Given the joint distribution of two random variables $X,Y$ on some second countable locally compact Hausdorff space, we investigate the statistical approximation of the $L^2$-operator defined by $[Pf](x) := \mathbb{E}[ f(Y) \mid X = x ]$ under minimal assumptions. By modifying its domain, we prove that $P$ can be arbitrarily well approximated in operator norm by Hilbert-Schmidt operators acting on a reproducing kernel Hilbert space. This fact allows to estimate $P$ uniformly by finite-rank operators over a dense subspace even when $P$ is not compact. In terms of modes of convergence, we thereby obtain the superiority of kernel-based techniques over classically used parametric projection approaches such as Galerkin methods. This also provides a novel perspective on which limiting object the nonparametric estimate of $P$ converges to. As an application, we show that these results are particularly important for a large family of spectral analysis techniques for Markov transition operators. Our investigation also gives a new asymptotic perspective on the so-called kernel conditional mean embedding, which is the theoretical foundation of a wide variety of techniques in kernel-based nonparametric inference.

翻译：给定定义在第二可数局部紧Hausdorff空间上的两个随机变量$X,Y$的联合分布，我们在最小假设下研究$L^2$算子$[Pf](x) := \mathbb{E}[ f(Y) \mid X = x ]$的统计逼近。通过修改其定义域，我们证明$P$可以在算子范数下被作用在再生核希尔伯特空间上的希尔伯特-施密特算子任意好地逼近。这一事实允许我们在$P$非紧的情况下，也能在稠密子空间上通过有限秩算子一致估计$P$。从收敛模式的角度，我们由此获得了基于核的技术相对于经典参数投影方法（如Galerkin方法）的优越性。这还为$P$的非参数估计收敛到的极限对象提供了新视角。作为应用，我们证明这些结果对于一大类马尔可夫转移算子的谱分析技术尤为重要。我们的研究还给出了所谓的核条件均值嵌入的新渐近视角，而这一嵌入是众多基于核的非参数推断技术的理论基础。