Anti-concentration of Suprema of Gaussian Processes and Gaussian Order Statistics

2023/10/18 by Alexander Giessing, Giessing, Alexander · 1 citation
Environmental Science · #60G15 (Primary) #60G60 (Secondary) #Analysis of environmental and stochastic processes #FOS: Mathematics #Hydrology and Drought Analysis #Probability (math.PR) #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2310.12119

openalex publication_date 2023/10/18 · openalex created_date 2023/10/21 · openalex updated_date 2026/07/28

Abstract

We derive, up to a constant factor, matching lower and upper bounds on the concentration functions of suprema of separable centered Gaussian processes and order statistics of Gaussian random fields. These bounds reveal that suprema of separable centered Gaussian processes \X_u : u ∈ U\ exhibit the same anti-concentration properties as a single Gaussian random variable with mean zero and variance Var(sup_{u ∈ U} X_u). To apply these results to high-dimensional statistical problems, it is therefore essential to understand the asymptotic behavior of Var(sup_{u ∈ U} X_u) as the dimension or metric entropy of the index set U increases. Consequently, we also derive lower and upper bounds on this quantity.

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