Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
121 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Optimal constants in concentration inequalities on the sphere and in the Gauss space (2406.13581v2)

Published 19 Jun 2024 in math.PR and math.FA

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 $\mu$ 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 $\mu\big(f \geq M +t\big) \leq \frac 12 e{-nt2/2}$. If $M$ is the mean of $f$, we have a two-sided bound $\mu\big(|f - M| \geq t\big) \leq e{-nt2/2}$. Consequently, if $\gamma$ 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 $\gamma \big(|f - M| \geq t\big) \leq e{-t2/2}$. These bounds are slightly better and arguably more elegant than those available elsewhere in the literature.

Citations (1)

Summary

We haven't generated a summary for this paper yet.