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2022 A sharp lower-tail bound for Gaussian maxima with application to bootstrap methods in high dimensions
Miles E. Lopes, Junwen Yao
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Electron. J. Statist. 16(1): 58-83 (2022). DOI: 10.1214/21-EJS1961

Abstract

Although there is an extensive literature on the maxima of Gaussian processes, there are relatively few non-asymptotic bounds on their lower-tail probabilities. The aim of this paper is to develop such a bound, while also allowing for many types of dependence. Let (ξ1,,ξN) be a centered Gaussian vector with standardized entries, whose correlation matrix R satisfies maxijRijρ0 for some constant ρ0(0,1). Then, for any ϵ0(0,1ρ0), we establish an upper bound on the probability P(max1jNξjϵ02log(N)) in terms of (ρ0,ϵ0,N). The bound is also sharp, in the sense that it is attained up to a constant, independent of N. Next, we apply this result in the context of high-dimensional statistics, where we simplify and weaken conditions that have recently been used to establish near-parametric rates of bootstrap approximation. Lastly, an interesting aspect of this application is that it makes use of recent refinements of Bourgain and Tzafriri’s “restricted invertibility principle”.

Funding Statement

Miles E. Lopes was supported in part by NSF grant DMS 1613218 and 1915786

Citation

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Miles E. Lopes. Junwen Yao. "A sharp lower-tail bound for Gaussian maxima with application to bootstrap methods in high dimensions." Electron. J. Statist. 16 (1) 58 - 83, 2022. https://doi.org/10.1214/21-EJS1961

Information

Received: 1 February 2021; Published: 2022
First available in Project Euclid: 6 January 2022

Digital Object Identifier: 10.1214/21-EJS1961

Subjects:
Primary: 60E15 , 60G15
Secondary: 62G09 , 62G32

Keywords: bootstrap , Gaussian processes , High-dimensional statistics , tail bounds

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Vol.16 • No. 1 • 2022
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