Some deviation inequalities.(English)Zbl 0756.60018

The author gives short proofs of a deviation inequality of Talagrand and of some concentration results for Gaussian measures as e.g. Pisier’s inequality, $$E\exp{\lambda \over \sqrt{2}}(\varphi(X)-\varphi(Y))\leq e^{\lambda^ 2/2}$$ for all real $$\lambda$$ and independent Gaussian vectors with independent components and Lipschitzian’s $$\varphi$$. The proofs are based on stability properties of the “property $$(\tau)$$” of a pair $$(\mu,w)$$ defined by the class of inequalities $$(\int e^{\varphi\square w}d\mu)(\int e^{-\varphi}d\mu)\leq 1$$ for all bounded $$\varphi$$; $$\varphi\square w$$ denoting the infimal convolution on $$\mathbb{R}^ n$$.

MSC:

 60E15 Inequalities; stochastic orderings 60B05 Probability measures on topological spaces
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References:

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