Optimal constants in concentration inequalities on the sphere and in the Gauss space
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| Format: | Preprint |
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2024
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| author | Aubrun, Guillaume Jenkinson, Justin Szarek, Stanislaw J. |
| author_facet | Aubrun, Guillaume Jenkinson, Justin Szarek, Stanislaw J. |
| contents | 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 $n\geq 3$, $f : S^{n-1} \to \mathbb{R}$ is $1$-Lipschitz, $M$ is the median of $f$, and $t >0$, then $μ\big(f \geq M +t\big) \leq \frac 12 e^{-nt^2/2}$. If $M$ is the mean of $f$, we have a two-sided bound $μ\big(|f - M| \geq t\big) \leq e^{-nt^2/2}$. Consequently, if $γ$ is the standard Gaussian measure on $\mathbb{R}^n$ and $f : \mathbb{R}^{n} \to \mathbb{R}$ (again, $1$-Lipschitz, with the mean equal to $M$), then $γ\big(|f - M| \geq t\big) \leq e^{-t^2/2}$. These bounds are slightly better and arguably more elegant than those available elsewhere in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13581 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal constants in concentration inequalities on the sphere and in the Gauss space Aubrun, Guillaume Jenkinson, Justin Szarek, Stanislaw J. Probability Functional Analysis 46B, 60B, 60E 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 $n\geq 3$, $f : S^{n-1} \to \mathbb{R}$ is $1$-Lipschitz, $M$ is the median of $f$, and $t >0$, then $μ\big(f \geq M +t\big) \leq \frac 12 e^{-nt^2/2}$. If $M$ is the mean of $f$, we have a two-sided bound $μ\big(|f - M| \geq t\big) \leq e^{-nt^2/2}$. Consequently, if $γ$ is the standard Gaussian measure on $\mathbb{R}^n$ and $f : \mathbb{R}^{n} \to \mathbb{R}$ (again, $1$-Lipschitz, with the mean equal to $M$), then $γ\big(|f - M| \geq t\big) \leq e^{-t^2/2}$. These bounds are slightly better and arguably more elegant than those available elsewhere in the literature. |
| title | Optimal constants in concentration inequalities on the sphere and in the Gauss space |
| topic | Probability Functional Analysis 46B, 60B, 60E |
| url | https://arxiv.org/abs/2406.13581 |