Some Probability and Entropy Estimates for Gaussian Measures

  • V. Goodman
Part of the Progress in Probability book series (PRPR, volume 20)


We compare two estimates for the measure of Banach neighborhoods of Hilbert balls in the reproducing kernel space. Borell’s estimate [1] is quite general and is known to be sharp for certain cases which involve small probabilities. However, Talagrand [7] and Goodman [4] use the openness of certain sets to obtain alternative estimates for cases in which the probability is near one.


Gaussian Measure Reproduce Kernel Hilbert Space Alternative Estimate Separable Banach Space Tail Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Borell, The Brunn-Minkowski inequality in Gauss space, Inventiones Math. 30 (1975), 205–216.MathSciNetCrossRefGoogle Scholar
  2. [2]
    R.M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. of Functional Anal. 1 (1967), 290–330.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    V. Goodman, J. Kuelbs and J. Zinn, Some results on the LIE in Banach space with applications to weighted empirical processes, Ann. Prob. 9 (1981), 713–752.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    V. Goodman, Characteristics of normal samples, Ann. Prob. 16 (1988), 1281–1290.MATHCrossRefGoogle Scholar
  5. [5]
    J. Hoffman-Jorgensen, L.A. Shepp and R.M. Dudley, On the lower tail of Gaussian seminorms, Ann. Prob. 7 (1979), 319–342.CrossRefGoogle Scholar
  6. [6]
    A. Pajor and N. Tomczak-Jaegermann, Remarques sur les nombres d’entropie d’un opérateur et de son transposé, C.R. Acad. Sci. Paris Sér. 1 Math. 301 No. 15 (1985), 743–746.MathSciNetMATHGoogle Scholar
  7. [7]
    M. Talagrand, Sur l’intégrabilité des vecteurs gaussiens, Z. Wahrsch. verw. Gebiete 68 (1984), 1–8.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • V. Goodman
    • 1
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

Personalised recommendations