论文标题
固定过程的内核自动加盟操作员：估计和收敛
Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence
论文作者
论文摘要
我们考虑在波兰空间上的固定随机过程的自动助行操作员，该过程嵌入了繁殖的内核希尔伯特空间中。我们研究了这些运营商的经验估计如何在各种条件下沿该过程的实现。特别是，我们检查了偏僻的和强烈的混合过程，并获得了几个渐近结果以及有限的样本误差界。我们以依赖数据和过渡概率的条件平均嵌入为内核PCA的一致性结果提供了理论的应用。最后，我们使用我们的方法来检查马尔可夫过渡算子的非参数估计，并强调我们的理论如何为包括基于内核的动态模式分解的大型光谱分析方法提供一致性分析。
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.
