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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:

[1] [B]C. Borell, The Brunn-Minkowski inequality in Gauss space, Inventiones Math. 30 (1975), 205–216. · Zbl 0311.60007 · doi:10.1007/BF01425510
[2] [C]L. Chen, An inequality for the multivariate normal distribution, J. Multivariate Anal. 12 (1982) 306–315. · Zbl 0483.60011 · doi:10.1016/0047-259X(82)90022-7
[3] [E]A. Ehrhard, Symétrisation dans l’espace de Gauss, Math. Scand. 53 (1983) 281–301. · Zbl 0542.60003
[4] [FLM]T. Figiel, J. Lindenstrauss, V. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977) 53–94. · Zbl 0375.52002 · doi:10.1007/BF02392234
[5] [JS]W. Johnson, G. Schechtman, Remarks on Talagrand’s deviation inequality for Rademacher functions, Texas Functional Analysis Seminar 1988–89.
[6] [L]L. Leindler, On a certain converse of Hölder’s inequality, Acta Sci. Math. 33 (1972) 217–223. · Zbl 0245.26011
[7] [M]V. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Func. Anal. Appl. 5 (1971) 28–37.
[8] [MS]V. Milman, G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Springer Lecture Notes in Math. 1200 (1986). · Zbl 0606.46013
[9] [Pi1]G. Pisier, Probabilistic methods in the geometry of Banach spaces, CIME Varenna 1985, Springer Lecture Notes in Math. 1206, 167–241.
[10] [Pi2]G. Pisier, Volume of convex bodies and Banach spaces geometry, Cambridge University Press.
[11] [Pr]A. Prekopa, On logarithmically concave measures and functions, Acta Sci. Math. 34 (1973) 335–343. · Zbl 0264.90038
[12] [ST]V. Sudakov, B. Tsirelson, Extremal properties of half spaces for spherically invariant measures, Zap. Nauch. Sem. LOMI 41, (1974) 14–24 translated in J. Soviet Math. 9 (1978) 9–18.
[13] [Ta1]M. Talagrand, Seminar lecture in Paris, Spring 1990.
[14] [Ta2]M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, to appear in GAFA90, Springer LNM (1991), 94–124. · Zbl 0818.46047
[15] [Ta3]M. Talagrand, An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities, Proc. Amer. Math. Soc. 104 (1988) 905–909. · Zbl 0691.60015 · doi:10.1090/S0002-9939-1988-0964871-7
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