Journal of Mathematical Sciences

, Volume 78, Issue 1, pp 102–108 | Cite as

Multivariate cauchy distributions as locally gaussian distributions

  • S. Ya. Shatskikh
Article

Abstract

In the present paper, we propose a definition of locally Gaussian probability distributions of random vectors based on the linearization of their conditional quantiles. We prove that the Cauchy distribution inRn is locally Gaussian and give explicit formulas for the vectors of expectations and covariance matrices of locally Gaussian approximations. We show that locally Gaussian approximations with different dimensionalities are in some sense compatible: all of them have equal corresponding correlation coefficients. For the Cauchy distribution in a Hilbert space we prove a limit theorem on the convergence of squared finite-dimensional conditional quantiles to the stable Lévy distribution.

Keywords

Covariance Gaussian Distribution Probability Distribution Hilbert Space Limit Theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Cramér,Mathematical Methods of Statistics, Princeton Univ. Press, Princeton (1974).Google Scholar
  2. 2.
    I. V. Proskuryakov,A Collection of Problems in Linear Algebra [in Russian], Nauka, Moscow (1970).Google Scholar
  3. 3.
    C. R. Rao,Linear Statistical Inference and Its Applications, Wiley, New York (1965).Google Scholar
  4. 4.
    L. N. Bol'shev and N. V. Smirnov,Tables of Mathematical Statistics [in Russian], Nauka, Moscow (1983).Google Scholar
  5. 5.
    V. M. Zolotarev,One-Dimensional Stable Distributions, American Mathematical Society, Providence (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • S. Ya. Shatskikh
    • 1
  1. 1.Samara State UniversitySamaraRussia

Personalised recommendations