We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and obtain several asymptotic results as well as finite sample error bounds. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition.
翻译:我们考虑波兰空间固定式随机过程的自动操作者,该过程已嵌入一个复制核心Hilbert空间的波兰空间。我们调查这些操作者的经验估计如何在各种条件下实现该过程的同时汇集在一起。特别是,我们检查随机和强烈混合过程,并获得若干无症状结果以及有限的抽样误差界限。我们运用我们的理论,对内核五氯苯甲醚的一致性结果提供依赖数据和有条件的过渡概率平均嵌入。最后,我们利用我们的方法,审查Markov过渡操作者的非参数估计,并突出我们的理论如何为包括内核动态模式分解在内的大型光谱分析方法提供一致性分析。