Moments of the Cramér transform of log-concave probability measures
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915826312937472 |
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| author | Giannopoulos, Apostolos Tziotziou, Natalia |
| author_facet | Giannopoulos, Apostolos Tziotziou, Natalia |
| contents | Let $μ$ be a centered log-concave probability measure on ${\mathbb R}^n$ and let $Λ_μ^{\ast}$ denote the Cramér transform of $μ$, i.e. $Λ_μ^{\ast}(x)=\sup\{\langle x,ξ\rangle-Λ_μ(ξ):ξ\in\mathbb{R}^n\}$ where $Λ_μ$ is the logarithmic Laplace transform of $μ$. We show that $\mathbb{E}_μ\left[\exp\left(\frac{c_1}{n}Λ_μ^{\ast }\right)\right]<\infty $ where $c_1>0$ is an absolute constant. In, particular, $Λ_μ^{\ast}$ has finite moments of all orders. The proof, which is based on the comparison of certain families of convex bodies associated with $μ$, implies that $\|Λ_μ^{\ast}\|_{L^2(μ)}\leqslant c_2n\ln n$. The example of the uniform measure on the Euclidean ball shows that this estimate is optimal with respect to $n$ as the dimension $n$ grows to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19528 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moments of the Cramér transform of log-concave probability measures Giannopoulos, Apostolos Tziotziou, Natalia Metric Geometry Functional Analysis Probability Primary 60D05, Secondary 60E15, 62H05, 52A22, 52A23 Let $μ$ be a centered log-concave probability measure on ${\mathbb R}^n$ and let $Λ_μ^{\ast}$ denote the Cramér transform of $μ$, i.e. $Λ_μ^{\ast}(x)=\sup\{\langle x,ξ\rangle-Λ_μ(ξ):ξ\in\mathbb{R}^n\}$ where $Λ_μ$ is the logarithmic Laplace transform of $μ$. We show that $\mathbb{E}_μ\left[\exp\left(\frac{c_1}{n}Λ_μ^{\ast }\right)\right]<\infty $ where $c_1>0$ is an absolute constant. In, particular, $Λ_μ^{\ast}$ has finite moments of all orders. The proof, which is based on the comparison of certain families of convex bodies associated with $μ$, implies that $\|Λ_μ^{\ast}\|_{L^2(μ)}\leqslant c_2n\ln n$. The example of the uniform measure on the Euclidean ball shows that this estimate is optimal with respect to $n$ as the dimension $n$ grows to infinity. |
| title | Moments of the Cramér transform of log-concave probability measures |
| topic | Metric Geometry Functional Analysis Probability Primary 60D05, Secondary 60E15, 62H05, 52A22, 52A23 |
| url | https://arxiv.org/abs/2503.19528 |