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 \(\mathbb{R}^ n\).


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