Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence

Abstract

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.

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• Normal approximation and concentration of spectral projectors of sample covariance
• A Rigorous Theory of Conditional Mean Embeddings
• Nonparametric estimation of the stationary density and the transition density of a Markov chain
• A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings
• Hilbert Space Embeddings of POMDPs
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• High-probability bounds for the reconstruction error of PCA

Cited by in corpus (2)

• Nonparametric approximation of conditional expectation operators
• Sobolev Norm Learning Rates for Conditional Mean Embeddings