Title: Moments of the Cramér transform of log-concave probability measures Show Chinese title

Title: 对数凹概率测度的Cramér变换矩

Abstract: Let $\mu$ be a centered log-concave probability measure on ${\mathbb R}^n$ and let $\Lambda_{\mu}^{\ast}$ denote the Cram\'{e}r transform of $\mu$, i.e. $\Lambda_{\mu}^{\ast}(x)=\sup\{\langle x,\xi\rangle-\Lambda_{\mu}(\xi):\xi\in\mathbb{R}^n\}$ 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 $\|\Lambda_{\mu}^{\ast}\|_{L^2(\mu)}\leqslant c_2n\ln n$. 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. Show Chinese abstract

Abstract: 设 $\mu$ 是 ${\mathbb R}^n$ 上的一个中心对数凹概率测度，并令 $\Lambda_{\mu}^{\ast}$ 表示 $\mu$ 的Cramér变换，即 \[$\Lambda_{\mu}^{\ast}(x)=\sup\{\langle x,\xi\rangle-\Lambda_{\mu}(\xi):\xi\in\mathbb{R}^n\}$\] 其中 $\Lambda_{\mu}$ 是 $\mu$ 的对数Laplace变换。 我们证明了 $\mathbb{E}_{\mu}\left[\exp\left(\frac{c_1}{n}\Lambda_{\mu}^{\ast }\right)\right]<\infty $，其中 $c_1>0$是一个绝对常数。 特别地， $\Lambda_{\mu}^{\ast}$的所有阶矩都是有限的。 该证明基于与 $\mu$相关的某些凸体族的比较，意味着 $\|\Lambda_{\mu}^{\ast}\|_{L^2(\mu)}\leqslant c_2n\ln n$。 欧几里得球上均匀测度的例子表明，当维度 $n$趋向于无穷大时，这个估计对于 $n$是最优的。