Optimal constants in concentration inequalities on the sphere and in the Gauss space

2024/06/19 by Aubrun, Guillaume, Jenkinson, Justin, Szarek, Stanislaw J.
#46B #60B #60E #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2406.13581

Abstract

We show several variants of concentration inequalities on the sphere stated as subgaussian estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided bounds for deviation from the median as well as from the mean. For example, we show that if μ is the normalized surface measure on S^n-1 with n≥ 3, f : S^n-1 → ℝ is 1-Lipschitz, M is the median of f, and t >0, then μ(f ≥ M +t) ≤ \frac 12 e^-nt2/2. If M is the mean of f, we have a two-sided bound μ(|f - M| ≥ t) ≤ e^-nt2/2. Consequently, if γ is the standard Gaussian measure on ℝⁿ and f : ℝⁿ → ℝ (again, 1-Lipschitz, with the mean equal to M), then γ(|f - M| ≥ t) ≤ e^-t2/2. These bounds are slightly better and arguably more elegant than those available elsewhere in the literature.

Related

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• Variations and extensions of the Gaussian concentration inequality, Part II
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