Exponential integrability in Gauss space
Exponential integrability in Gauss space
复制标题
高斯空间中的指数可积性
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Ryan Russell
中科院分区：
文献类型：
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作者：
P. Ivanisvili;Ryan Russell
Talagrand观察到$mathbb{E}， E ^{frac{1}{2}|的有限性
Talagrand observed that finiteness of $mathbb{E}, e^{frac{1}{2}|
abla f(X)|^{2}}$ 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}|
abla f|^{2}} (1+|
abla f|)^{-1}$ implies finiteness of $mathbb{E}, e^{, f(X)}$, and we also obtain quantitative bounds
egin{align*} log, mathbb{E}, e^{, f} leq 10, mathbb{E}, e^{frac{1}{2}|
abla f|^{2}} (1+|
abla f|)^{-1}.
end{align*} Moreover, the extra factor $(1+|
abla f|)^{-1}$ is the best possible in the sense that there is smooth $f$ with $mathbb{E}, e^{,f} =infty$ but $mathbb{E}, e^{frac{1}{2}|
abla f|^{2}} (1+|
abla f|)^{-c} 1$.