Admissible Sequences for Talagrand's $γ_2$-functional
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917055173754880 |
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| author | Diaconu, Simona |
| author_facet | Diaconu, Simona |
| contents | Suprema of random processes appear naturally in a plethora of disciplines, and Talagrand's majorizing theorem yields a geometric interpretation for them: for a centered Gaussian random process $(X_t)_{t \in T},$ $\mathbb{E}[\sup_{t \in T}{X_t}]$ is comparable to the $γ_2$-functional of $T,$ a quantity that depends solely on the space $(T,d),$ where $d$ denotes the pseudometric $d(u,v)=\sqrt{\mathbb{E}[(X_u-X_v)^2]}.$ Despite the explicit definition of this functional, an infimum over admissible sequences, this tool tends to be used exclusively as a means to bound the expectation of the supremum of a random process by that of another. This work considers the $γ_2$-functional as a proxy for the quantity of interest by constructing admissible sequences that are close to being optimal, and aims to provide a promising avenue towards understanding expectations of suprema of Gaussian random processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Admissible Sequences for Talagrand's $γ_2$-functional Diaconu, Simona Probability Suprema of random processes appear naturally in a plethora of disciplines, and Talagrand's majorizing theorem yields a geometric interpretation for them: for a centered Gaussian random process $(X_t)_{t \in T},$ $\mathbb{E}[\sup_{t \in T}{X_t}]$ is comparable to the $γ_2$-functional of $T,$ a quantity that depends solely on the space $(T,d),$ where $d$ denotes the pseudometric $d(u,v)=\sqrt{\mathbb{E}[(X_u-X_v)^2]}.$ Despite the explicit definition of this functional, an infimum over admissible sequences, this tool tends to be used exclusively as a means to bound the expectation of the supremum of a random process by that of another. This work considers the $γ_2$-functional as a proxy for the quantity of interest by constructing admissible sequences that are close to being optimal, and aims to provide a promising avenue towards understanding expectations of suprema of Gaussian random processes. |
| title | Admissible Sequences for Talagrand's $γ_2$-functional |
| topic | Probability |
| url | https://arxiv.org/abs/2511.00942 |