Geometrical and Statistical Aspects of Probability in Banach by Christer Borell (auth.), Xavier Fernique, Bernard Heinkel,

By Christer Borell (auth.), Xavier Fernique, Bernard Heinkel, Paul-André Meyer, Michael B. Marcus (eds.)

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Extra resources for Geometrical and Statistical Aspects of Probability in Banach Spaces: Actes des Journées SMF de Calcul des Probabilités dans les Espaces de Banach, organisées à Strasbourg les 19 et 20 juin 1985

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J. [2J BERKES I. P. : Almost exchangeable sequences of random variables. To appear in Zeitschrift fUr Wahrscheinlichkeitstheorie verw. Gebiete. [3J BRUNEL A. and SUCHESTON L. : On B-convex Banach spaces. Math. Systems theory, t. 7 n 0 4, 1973. [ 4J DACUNHA-CASTELLE D. : Variables aleatoires echangeables et espaces d'Orlicz. Seminaire Maurey-Schwartz, Ecole Poly technique, 1974/75, exposes 10 et 11 • [5J DACUNHA-CASTELLE D. L. , 26 (1977), 320-351. [6J GUERRE S. Types et suites symetriques dans LP , 1,; p<+ A paraitre dans Israel Journal of Math.

Let E be a Banach space. •• + X • A random variable n is said to satisfy the central limit theorem (CLT Ill) n (sn / r" if the sequence from some probability space E X in short) converges weakly to a Gaussian Radon probability E. In his remarkable work on the Glivenko-Cantelli problem, observed the following characterization of the CLT M. ) Although it seems rather difficult to verify these conditions on small balls, the preceding property is intriguing since it reduces a central limit property in Banach spaces to some kind of weak convergence on the line by taking norm.

In all the sequel we will denote by (B, II II) a real separable Banach space which is p-uniformly smooth (1 < P " 2); this means that its modulus of smoothness 'If t > U', P (t) = sup( 1/2 ( Ilx+tYII + IIx-tyll) - 1 , Ilxll = IIYII = 1 ) , p: 30 satisfies: p (t) "c tP C being a positive constant. It is well known that the norm II II is differentiable away from the origin let's de- note by D the derivative of II II. If now one as sociates to D the following fonction F p l 'l xt-O F(x) :: Ilxll - D(xl Ilxll) and: B ~ B' F( 0):: :) one can check that F has the following two properties [ 19J ( i) IIF(x)II B , :: Ilxlltl ~ C > 0: 'l (x.

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