Comparison and anti-concentration bounds for maxima of Gaussian random vectors

2013/01/21 by Victor Chernozhukov, Denis Chetverikov, Chernozhukov, Victor +3 · 9 citations
Mathematics · #60E15 #60G15 #62E20 #Applied mathematics #Combinatorics #Covariance #Covariance function #Dimension (graph theory) #FOS: Mathematics #Gaussian #Gaussian process #Gaussian random field #Law of total covariance #Mathematical analysis #Mathematics #Maxima #Multivariate random variable #Physics #Point processes and geometric inequalities #Probability (math.PR) #Random field #Random variable #Statistical Methods and Inference #Statistical physics #Statistics #Statistics Theory (math.ST) #Upper and lower bounds #math.PR #math.ST #msc:60E15 #msc:60G15 #msc:62E20 #stat.TH

paper · pdf · doi:10.48550/arxiv.1301.4807

published in arXiv (Cornell University) (Cornell University) · 22 pages; discussions and references updated

openalex publication_date 2013/01/21 · arxiv created 2014/04/13 · arxiv updated 2014/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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.

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