Open Access
November 2019 Rademacher complexity for Markov chains: Applications to kernel smoothing and Metropolis–Hastings
Patrice Bertail, François Portier
Bernoulli 25(4B): 3912-3938 (November 2019). DOI: 10.3150/19-BEJ1115


The concept of Rademacher complexity for independent sequences of random variables is extended to Markov chains. The proposed notion of “regenerative block Rademacher complexity” (of a class of functions) follows from renewal theory and allows to control the expected values of suprema (over the class of functions) of empirical processes based on Harris Markov chains as well as the excess probability. For classes of Vapnik–Chervonenkis type, bounds on the “regenerative block Rademacher complexity” are established. These bounds depend essentially on the sample size and the probability tails of the regeneration times. The proposed approach is employed to obtain convergence rates for the kernel density estimator of the stationary measure and to derive concentration inequalities for the Metropolis–Hastings algorithm.


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Patrice Bertail. François Portier. "Rademacher complexity for Markov chains: Applications to kernel smoothing and Metropolis–Hastings." Bernoulli 25 (4B) 3912 - 3938, November 2019.


Received: 1 June 2018; Revised: 1 December 2018; Published: November 2019
First available in Project Euclid: 25 September 2019

zbMATH: 07110160
MathSciNet: MR4010977
Digital Object Identifier: 10.3150/19-BEJ1115

Keywords: Concentration inequalities , kernel smoothing , Markov chains , Metropolis Hastings , Rademacher complexity

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4B • November 2019
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