On Large Deviations of Gaussian Measures in Banach Spaces

  • Marek Slaby
Part of the Progress in Probability book series (PRPR, volume 30)


Throughout this paper E will denote a separable Banach space and γ will be a full centered Gaussian measure on E. Define an operator S: E’E by S f = ∫ x f(x)γ(dx), and a scalar product <, >γ on SE’ by
$$ <Sf,Sg>_\gamma =\int f(x)g(x)\gamma(dx) $$
The completion of S E’ with respect to the norm ∥x;∥γ = √<x,x >γ is denoted by Hγ and called the reproducing kernel Hilbert space of γ. Since
$$||Sf||\leq _{||g||E^{*}\leq1}^<Superscript>p</Superscript>[\int g^{2}(x)\gamma(dx)]^{1/2}\cdot ||Sf||_{\gamma} $$
can and will be viewed as a subset of E. For details on the construction and properties of the reproducing kernel Hilbert space we refer the reader to [2] or [5].


Banach Space Random Vector Moderate Deviation Gaussian Measure Reproduce Kernel Hilbert Space 
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  1. [1]
    de Acosta, A. Moderate deviations and associated Laplace approximations for sums of independent random vectors. Preprint.Google Scholar
  2. [2]
    Azencott, R.(1980). Grandes déviations et applications. Lecture Notes in Math. No.774. Springer-Verlag, Berlin, Heidelberg and New York.Google Scholar
  3. [3]
    Badrikian, A. and Chevet, S. (1974). Measures cylindriques, espaces de Wiener et fonctions aléatoires Gaussiennes. Lecture Notes in Math. No. 379. Springer-Verlag, Berlin, Heidelberg and New York.Google Scholar
  4. [4]
    Borell, C. (1976). Gaussian Radon measures on locally convex spaces. Math. Scand. 38, 265–284.MathSciNetMATHGoogle Scholar
  5. [5]
    Goodman, V., Kuelbs, J. and Zinn, J. (1981). Some results on the LIL in Banach space with applications to weighted empirical processes. Ann. Prob. 9, 713–752.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Ledoux, M. (1992). Sur les deviations moderees des sommes de variables aléatoires vectorielles independantes de meme loi. Anm. Inst. H. Poincare, to appear.Google Scholar
  7. [7]
    Slaby, M. (1988). On the upper bound for large deviations of sums of i.i.d. random vectors. Ann. Probab. 16, No.3, 978–990.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Slaby, M. Singularities in large deviations of sums of i.i.d. random vectors in R2. Preprint.Google Scholar
  9. [9]
    Xia, C. (1991). Probabilities of moderate deviations for independent random vectors in a Banach space. Chinese J. Appl. Prob. Statist. 7, 24–32.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Marek Slaby
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of Nebraska - LincolnLincolnUSA

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