Moments of the Cramér transform of log-concave probability measures

Abstract

Let $\mu$ be a centered log-concave probability measure on ${\mathbb{R}}^n$ and let ${\Lambda }_{\mu }^{\ast }$ denote the Cramér transform of $\mu$, i.e. ${\Lambda }_{\mu }^{\ast }(x)=\sup \{⟨x,\xi ⟩-{\Lambda }_{\mu }(\xi ):\xi \in {\mathbb{R}}^n\}$ where ${\Lambda }_{\mu }$ is the logarithmic Laplace transform of $\mu$. We show that ${\mathbb{E}}_{\mu }\left[\exp \left(\frac{c_1}{n}{\Lambda }_{\mu }^{\ast }\right)\right]<\infty$ where $c_1>0$ is an absolute constant. In, particular, ${\Lambda }_{\mu }^{\ast }$ has finite moments of all orders. The proof, which is based on the comparison of certain families of convex bodies associated with $\mu$, implies that $\Vert {\Lambda }_{\mu }^{\ast }{\Vert }_{L^2(\mu )}⩽c_{2}n\ln n$. 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.

24 pages, Journal of Functional Analysis (to appear)