An Exponential Concentration Inequality for the Components of a Uniform Random Vector on the Sphere
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911100352593920 |
|---|---|
| author | Samani, Joshua |
| author_facet | Samani, Joshua |
| contents | We show that if $\vec X = (X_1, \dots, X_N)$ is a uniform random vector on the unit Euclidean sphere, the empirical CDF of the components of $\sqrt N \vec X = (\sqrt N X_1, \dots, \sqrt N X_N)$ concentrates exponentially rapidly in $N$ around the standard Gaussian CDF $Φ$. More precisely, we find explicit functions $γ$ and $g_\pm$ such that the Kolmogorov-Smirnov distance between the empirical CDF of the components of $\sqrt N \vec X$ and $Φ$ deviates by more than $ε+ γ(t)$ with probability at most $2e^{-2Nε^2} + e^{-Ng_+(t)^2} + e^{-Ng_-(t)^2}$ for $ε> 0$ and $t\in[0,1)$. A weaker but more transparent inequality replacing $γ$ and $g_\pm$ with linear functions is obtained as a corollary. All functions and constants are explicit, so our bounds offer finite-sample guarantees for statistical applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06748 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Exponential Concentration Inequality for the Components of a Uniform Random Vector on the Sphere Samani, Joshua Probability 60E15 (Primary) 60F10, 60B10, 62G30 (Secondary) We show that if $\vec X = (X_1, \dots, X_N)$ is a uniform random vector on the unit Euclidean sphere, the empirical CDF of the components of $\sqrt N \vec X = (\sqrt N X_1, \dots, \sqrt N X_N)$ concentrates exponentially rapidly in $N$ around the standard Gaussian CDF $Φ$. More precisely, we find explicit functions $γ$ and $g_\pm$ such that the Kolmogorov-Smirnov distance between the empirical CDF of the components of $\sqrt N \vec X$ and $Φ$ deviates by more than $ε+ γ(t)$ with probability at most $2e^{-2Nε^2} + e^{-Ng_+(t)^2} + e^{-Ng_-(t)^2}$ for $ε> 0$ and $t\in[0,1)$. A weaker but more transparent inequality replacing $γ$ and $g_\pm$ with linear functions is obtained as a corollary. All functions and constants are explicit, so our bounds offer finite-sample guarantees for statistical applications. |
| title | An Exponential Concentration Inequality for the Components of a Uniform Random Vector on the Sphere |
| topic | Probability 60E15 (Primary) 60F10, 60B10, 62G30 (Secondary) |
| url | https://arxiv.org/abs/2508.06748 |