论文标题
对数符号测量的指数可集成性
Exponential integrability for log-concave measures
论文作者
论文摘要
talagrand观察到，$ \ mathbb {e} \，e^{\ frac {\ frac {1} {2} {2} | \ nabla f（x）|^{2}} $的有限性暗示$ \ m athbbb {e} $ \ mathbb {r}^{n} $和$ f $是平均平均值的平滑函数。但是，在本文中，我们表明$ \ Mathbb {e} \，E^{\ frac {1} {2} {2} | \ nabla f |^{2}}（1+ | \ nabla f |）获得定量界限 \ begin {Align*} \ log \，\ Mathbb {e} \，e^{\，f} \ leq 10 \，\ Mathbb {e} \，e} \，e^{\ frac {1} {1} {2} {2} {2} {2} | \ nabla f |^{2} {2} {2}}}（1+ | | | 此外，在某种意义上说，带有$ \ Mathbb {e} \，e} \，e^{\ f} = \ f} = \ infty $ butty $ \ narbb，$ f $的意义上说，额外的因子$（1+ | \ nabla f |）^{ - 1} $是最好的可能性。 e^{\ frac {1} {2} | \ nabla f |^{2}}（1+ | \ nabla f |）作为应用程序，我们显示了离散时间二元组及其二次变化的相应双重不平等。
Talagrand observed that finiteness of $\mathbb{E}\, e^{\frac{1}{2}|\nabla f(X)|^{2}}$ implies finiteness of $\mathbb{E}\, e^{\, f(X)}$ where $X$ is the standard Gaussian vector in $\mathbb{R}^{n}$ and $f$ is a smooth function with zero average. However, in this paper we show that finiteness of $ \mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-1}$ implies finiteness of $\mathbb{E}\, e^{\, f(X)}$, and we also obtain quantitative bounds \begin{align*} \log\, \mathbb{E}\, e^{\, f} \leq 10\, \mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-1}. \end{align*} Moreover, the extra factor $(1+|\nabla f|)^{-1}$ is the best possible in the sense that there is smooth $f$ with $\mathbb{E}\, e^{\,f} =\infty$ but $\mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-c}<\infty$ for all $c>1$. As an application we show corresponding dual inequalities for the discrete time dyadic martingales and its quadratic variations.
