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August 2014 Approximating class approach for empirical processes of dependent sequences indexed by functions
Herold Dehling, Olivier Durieu, Marco Tusche
Bernoulli 20(3): 1372-1403 (August 2014). DOI: 10.3150/13-BEJ525

Abstract

We study weak convergence of empirical processes of dependent data $(X_{i})_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class $\mathcal{F}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_{i}))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index class $\mathcal{F}$ which fits this setting. Our results apply to the empirical process of data $(X_{i})_{i\geq0}$ satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron–Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, for example, to ergodic torus automorphisms.

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Herold Dehling. Olivier Durieu. Marco Tusche. "Approximating class approach for empirical processes of dependent sequences indexed by functions." Bernoulli 20 (3) 1372 - 1403, August 2014. https://doi.org/10.3150/13-BEJ525

Information

Published: August 2014
First available in Project Euclid: 11 June 2014

zbMATH: 1307.60027
MathSciNet: MR3217447
Digital Object Identifier: 10.3150/13-BEJ525

Keywords: dependent data , dynamical systems , empirical processes indexed by classes of functions , ergodic torus automorphism , Markov chains , weak convergence

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability

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Vol.20 • No. 3 • August 2014
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