Open Access
February 2018 Rate of convergence for Hilbert space valued processes
Moritz Jirak
Bernoulli 24(1): 202-230 (February 2018). DOI: 10.3150/16-BEJ870


Consider a stationary, linear Hilbert space valued process. We establish Berry–Esseen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of non-linear Bernoulli-shift sequences is also considered. If the sequence is $m$-dependent, the optimal rate $(n/m)^{1/2}$ is reached. If the sequence is weakly geometrically dependent, the rate $(n/\log n)^{1/2}$ is obtained.


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Moritz Jirak. "Rate of convergence for Hilbert space valued processes." Bernoulli 24 (1) 202 - 230, February 2018.


Received: 1 October 2014; Revised: 1 May 2016; Published: February 2018
First available in Project Euclid: 27 July 2017

zbMATH: 1383.60008
MathSciNet: MR3706754
Digital Object Identifier: 10.3150/16-BEJ870

Keywords: Berry–Esseen , Hilbert space , linear process , Weak dependence

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 1 • February 2018
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