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world:log-sobolev [2025/07/09 10:58] rdouc [Theorem] |
world:log-sobolev [2025/07/09 11:02] (current) rdouc [Concentration inequalities and logarithmic Sobolev inequality] |
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| - | ====== Concentration inequalities and logarithmic Sobolev inequality ====== | + | ====== Logarithmic Sobolev inequality and concentration ====== |
| $\newcommand{\Ent}{\mathrm{Ent}}$ | $\newcommand{\Ent}{\mathrm{Ent}}$ | ||
| - | ====== Theorem (Thm 7.4.1 in the book "Sur les inégalités de Sobolev logarithmiques")====== | + | ====== Theorem ====== |
| + | * Taken from Thm 7.4.1 in the book "Sur les inégalités de Sobolev logarithmiques". | ||
| + | <WRAP center round tip 80%> | ||
| Let $\mu$ be a probability measure on $\mathbb{R}^n$ satisfying the following logarithmic Sobolev inequality: | Let $\mu$ be a probability measure on $\mathbb{R}^n$ satisfying the following logarithmic Sobolev inequality: | ||
| $$ | $$ | ||
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| \mu(|F - \mathbb{E}_\mu(F)| \geq r) \leq 2 \exp\left(-\frac{r^2}{c}\right). | \mu(|F - \mathbb{E}_\mu(F)| \geq r) \leq 2 \exp\left(-\frac{r^2}{c}\right). | ||
| $$ | $$ | ||
| + | |||
| + | </WRAP> | ||
| ===== Proof. ===== | ===== Proof. ===== | ||
world/log-sobolev.1752051507.txt.gz · Last modified: 2025/07/09 10:58 by rdouc
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