Advertisement

Sufficient Conditions for the Existence of Conditional Moments of Stable Random Variables

  • Renata Cioczek-Georges
  • Murad S. Taqqu
Part of the Trends in Mathematics book series (TM)

Abstract

Conditional moments E[X 2|p|X 1 = x] of an α-stable random vector (X 1,X 2) may exist even if p ⩾ α. The precise conditions are stated in Cioczek-Georges and Taqqu [4]_and Samorodnitsky and Taqqu [12]. This paper provides the proof for the most delicate cases, namely 1 < α< 2 and p < 2α+ 1, which is the maximal range of possible p’s when the vector (X 1, X 2) is nondegenerate.

Keywords

Characteristic Function Triangle Inequality Conditional Variance Integral Sign Conditional Moment 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Cambanis and S. B. Fotopoulos. Conditional variance for stable random vectors. Probability and Mathematical Statistics, 15:195–214, 1995.MathSciNetzbMATHGoogle Scholar
  2. [2]
    S. Cambanis, S. B. Fotopoulos, and L. He. On the conditional variance for scale mixtures of normal distributions. Preprint, 1996.Google Scholar
  3. [3]
    S. Cambanis and W. Wu. Multiple regression on stable vectors. Journal of Multivariate Analysis, 41:243–272, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    R. Cioczek-Georges and M. S. Taqqu. How do conditional moments of stable vectors depend on the spectral measure? Stochastic Processes and their Applications, 54:95–111, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    R. Cioczek-Georges and M. S. Taqqu. Form of the conditional variance for symmetric stable random variables. Statistica Sinica, 5:351–361, 1995.MathSciNetzbMATHGoogle Scholar
  6. [6]
    R. Cioczek-Georges and M. S. Taqqu. Necessary conditions for the existence of conditional moments of stable random variables. Stochastic Processes and their Applications, 56:233–246, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    S. B. Fotopoulos and L. He. Form of the conditional variance-covariance matrix for α-stable scale mixtures of normal distributions. Preprint, 1996.Google Scholar
  8. [8]
    C. D. Hardin Jr., G. Samorodnitsky, and M. S. Taqqu. Non-linear regression of stable random variables. The Annals of Applied Probability, 1:582–612, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    B. Ramachandran. On characteristic functions and moments. Sankhya, 31(Series A):1–12, 1969.MathSciNetzbMATHGoogle Scholar
  10. [10]
    H. L. Royden. Real Analysis. Macmillan, third edition, 1988.Google Scholar
  11. [11]
    G. Samorodnitsky and M. S. Taqqu. Conditional moments and linear regression for stable random variables. Stochastic Processes and their Applications, 39:183–199, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York, London, 1994.zbMATHGoogle Scholar
  13. [13]
    W. Wu and S. Cambanis. Conditional variance of symmetric stable variables. In S. Cambanis, G. Samorodnitsky, and M. S. Taqqu, editors, Stable Processes and Related Topics, volume 25 of Progress in Probability, pp. 85–99, Birkhäuser, Boston, 199Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Renata Cioczek-Georges
    • 1
  • Murad S. Taqqu
    • 2
  1. 1.Department of StatisticsUniversity of North Carolina at Chapel HillChapel HillUSA
  2. 2.Department of MathematicsBoston UniversityBostonUSA

Personalised recommendations