Anti-Concentration Inequalities for the Difference of Maxima of Gaussian Random Vectors
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917757473259520 |
|---|---|
| author | Belloni, Alexandre Fang, Ethan X. Shen, Shuting |
| author_facet | Belloni, Alexandre Fang, Ethan X. Shen, Shuting |
| contents | We derive novel anti-concentration bounds for the difference between the maximal values of two Gaussian random vectors across various settings. Our bounds are dimension-free, scaling with the dimension of the Gaussian vectors only through the smaller expected maximum of the Gaussian subvectors. In addition, our bounds hold under the degenerate covariance structures, which previous results do not cover. In addition, we show that our conditions are sharp under the homogeneous component-wise variance setting, while we only impose some mild assumptions on the covariance structures under the heterogeneous variance setting. We apply the new anti-concentration bounds to derive the central limit theorem for the maximizers of discrete empirical processes. Finally, we back up our theoretical findings with comprehensive numerical studies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13348 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anti-Concentration Inequalities for the Difference of Maxima of Gaussian Random Vectors Belloni, Alexandre Fang, Ethan X. Shen, Shuting Statistics Theory Probability We derive novel anti-concentration bounds for the difference between the maximal values of two Gaussian random vectors across various settings. Our bounds are dimension-free, scaling with the dimension of the Gaussian vectors only through the smaller expected maximum of the Gaussian subvectors. In addition, our bounds hold under the degenerate covariance structures, which previous results do not cover. In addition, we show that our conditions are sharp under the homogeneous component-wise variance setting, while we only impose some mild assumptions on the covariance structures under the heterogeneous variance setting. We apply the new anti-concentration bounds to derive the central limit theorem for the maximizers of discrete empirical processes. Finally, we back up our theoretical findings with comprehensive numerical studies. |
| title | Anti-Concentration Inequalities for the Difference of Maxima of Gaussian Random Vectors |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2408.13348 |