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The discrepancy between min–max statistics of Gaussian and Gaussian-subordinated matrices

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  • Peccati, Giovanni
  • Turchi, Nicola

Abstract

We compute quantitative bounds for measuring the discrepancy between the distribution of two min–max statistics involving either one pair of Gaussian random matrices, or one Gaussian and one Gaussian-subordinated random matrix. In the fully Gaussian setup, our approach allows us to recover quantitative versions of well-known inequalities by Gordon (1985, 1987, 1992), thus generalizing the quantitative version of the Sudakov–Fernique inequality deduced in Chatterjee (2005). On the other hand, the Gaussian-subordinated case yields generalizations of estimates by Chernozhukov et al. (2015) and Koike (2019). As an application, we establish fourth moment bounds for matrices of multiple stochastic Wiener–Itô integrals, that we illustrate with an example having a statistical flavor.

Suggested Citation

  • Peccati, Giovanni & Turchi, Nicola, 2023. "The discrepancy between min–max statistics of Gaussian and Gaussian-subordinated matrices," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 315-341.
    • Handle: RePEc:eee:spapps:v:158:y:2023:i:c:p:315-341
    DOI: 10.1016/j.spa.2023.01.006
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    References listed on IDEAS

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    1. Chernozhukov, Victor & Chetverikov, Denis & Kato, Kengo, 2016. "Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3632-3651.
    2. Nourdin, Ivan & Peccati, Giovanni & Viens, Frederi G., 2014. "Comparison inequalities on Wiener space," Stochastic Processes and their Applications, Elsevier, vol. 124(4), pages 1566-1581.
    3. Victor Chernozhukov & Denis Chetverikov & Kengo Kato & Yuta Koike, 2019. "Improved Central Limit Theorem and bootstrap approximations in high dimensions," Papers 1912.10529, arXiv.org, revised May 2022.
    4. Yaowu Liu & Jun Xie, 2019. "Accurate and Efficient P-value Calculation Via Gaussian Approximation: A Novel Monte-Carlo Method," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(525), pages 384-392, January.
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