Comparison and anti-concentration bounds for maxima of Gaussian random vectors
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Author(s)
Chetverikov, Denis
Kato, Kengo
Chernozhukov, Victor V
Date Issued
May 2014
Journal
Probability Theory and Related Fields
Publisher
Springer Berlin Heidelberg
Citation
Chernozhukov, Victor, Denis Chetverikov, and Kengo Kato. “Comparison and Anti-Concentration Bounds for Maxima of Gaussian Random Vectors.” Probab. Theory Relat. Fields 162, no. 1-2 (May 9, 2014): 47–70.
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Author's final manuscript
Abstract
Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Lévy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.
MIT Department
Massachusetts Institute of Technology. Department of Economics
Massachusetts Institute of Technology. Operations Research Center
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DOI of Published Version
https://doi.org/10.1007/s00440-014-0565-9
