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

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  • Victor Chernozhukov
  • Denis Chetverikov
  • Kengo Kato

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 the 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 maxima of Gaussian random vectors, which derives a useful upper bound on the Lévy concentration function for the maximum of (not necessarily independent) Gaussian random variables. The bound is universal 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.

Suggested Citation

  • Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Comparison and anti-concentration bounds for maxima of Gaussian random vectors," CeMMAP working papers 71/13, Institute for Fiscal Studies.
  • Handle: RePEc:azt:cemmap:71/13
    DOI: 10.1920/wp.cem.2013.7113
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    References listed on IDEAS

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    1. Liqing Yan, 2009. "Comparison Inequalities for One Sided Normal Probabilities," Journal of Theoretical Probability, Springer, vol. 22(4), pages 827-836, December.
    2. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Anti-concentration and honest, adaptive confidence bands," CeMMAP working papers CWP69/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
    4. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Central limit theorems and multiplier bootstrap when p is much larger than n," CeMMAP working papers CWP45/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
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    1. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Testing Many Moment Inequalities," CeMMAP working papers 65/13, Institute for Fiscal Studies.
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    3. Andrews, Donald W.K. & Shi, Xiaoxia, 2017. "Inference based on many conditional moment inequalities," Journal of Econometrics, Elsevier, vol. 196(2), pages 275-287.
    4. Demian Pouzo, 2014. "Bootstrap Consistency for Quadratic Forms of Sample Averages with Increasing Dimension," Papers 1411.2701, arXiv.org, revised Aug 2015.

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