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June 2019 Effective Berry–Esseen and concentration bounds for Markov chains with a spectral gap
Benoît Kloeckner
Ann. Appl. Probab. 29(3): 1778-1807 (June 2019). DOI: 10.1214/18-AAP1438

Abstract

Applying quantitative perturbation theory for linear operators, we prove nonasymptotic bounds for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions $\mathscr{X}$. The main results are concentration inequalities and Berry–Esseen bounds, obtained assuming neither reversibility nor “warm start” hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform $\mathscr{X}$-ergodicity hypothesis, and when $\mathscr{X}$ consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the flexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.

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Benoît Kloeckner. "Effective Berry–Esseen and concentration bounds for Markov chains with a spectral gap." Ann. Appl. Probab. 29 (3) 1778 - 1807, June 2019. https://doi.org/10.1214/18-AAP1438

Information

Received: 1 November 2017; Revised: 1 July 2018; Published: June 2019
First available in Project Euclid: 19 February 2019

zbMATH: 07057466
MathSciNet: MR3914556
Digital Object Identifier: 10.1214/18-AAP1438

Subjects:
Primary: 65C05
Secondary: 60J22, 62E17

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 3 • June 2019
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