Advertisement

Inheritance in Multivariate Subordination

  • Alfonso Rocha-Arteaga
  • Ken-iti Sato
Chapter
Part of the SpringerBriefs in Probability and Mathematical Statistics book series (SBPMS )

Abstract

We now study inheritance of Lm property or strict stability from subordinator to subordinated in multivariate subordination. In order to observe this inheritance, we have to assume strict stability of the distribution at each s ∈ K of a K-parameter subordinand {Xs: s ∈ K}. Section 5.1 gives results and examples. Section 5.2 discusses some generalization where the defining condition of selfdecomposability or stability for distributions on \(\mathbb {R}^d\) involves a d × d matrix Q. This is called operator generalization.

References

  1. 5.
    Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics, 24, 1–13.MathSciNetCrossRefGoogle Scholar
  2. 6.
    Barndorff-Nielsen, O. E. & Halgreen, C. (1977). Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 38, 309–311.Google Scholar
  3. 8.
    Barndorff-Nielsen, O. E., Pedersen, J., & Sato, K. (2001). Multivariate subordination, selfdecomposability and stability. Advances in Applied Probability, 33, 160–187.Google Scholar
  4. 14.
    Bochner, S. (1955). Harmonic analysis and the theory of probability. Berkeley: University of California Press.Google Scholar
  5. 16.
    Bondesson, L. (1992). Generalized gamma convolutions and related classes of distribution densities. Lecture notes in statistics (Vol. 76). New York: Springer.Google Scholar
  6. 30.
    Halgreen, C. (1979). Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 47, 13–17.Google Scholar
  7. 35.
    Hudson, W. N., & Mason, J. D. (1981). Operator-stable distribution on R 2 with multiple exponents. Annals of Probability, 9, 482–489.Google Scholar
  8. 36.
    Ismail, M. E. H., & Kelker, D. H. (1979). Special functions, Stieltjes transforms and infinite divisibility. SIAM Journal on Mathematical Analysis, 10, 884–901.Google Scholar
  9. 39.
    Jurek, Z. J. (1983a). Limit distributions and one-parameter groups of linear operators on Banach spaces. Journal of Multivariate Analysis, 13, 578–604.MathSciNetCrossRefGoogle Scholar
  10. 44.
    Jurek, Z. J., & Mason, J. D. (1993). Operator-limit distributions in probability theory. New York: Wiley.Google Scholar
  11. 71.
    Pedersen, J., & Sato, K. (2003). Cone-parameter convolution semigroups and their subordination. Tokyo Journal of Mathematics, 26, 503–525.Google Scholar
  12. 79.
    Pillai, R. N. (1990). On Mittag–Leffler functions and related distributions. Annals of the Institute of Statistical Mathematics, 42, 157–161.MathSciNetCrossRefGoogle Scholar
  13. 80.
    Ramachandran, B. (1997). On geometric-stable laws, a related property of stable processes, and stable densities of exponent one. Annals of the Institute of Statistical Mathematics, 49, 299–313.MathSciNetCrossRefGoogle Scholar
  14. 86.
    Sato, K. (1985). Lectures on multivariate infinitely divisible distributions and operator-stable processes. Technical Report Series, Laboratory for Research in Statistics and Probability, Carleton University and University of Ottawa, No. 54, Ottawa.Google Scholar
  15. 87.
    Sato, K. (1987). Strictly operator-stable distributions. Journal of Multivariate Analysis, 22, 278–295.MathSciNetCrossRefGoogle Scholar
  16. 93.
    Sato, K. (1999). Lévy processes and infinitely divisible distributions. Cambridge: Cambridge University Press.Google Scholar
  17. 94.
    Sato, K. (2001a). Subordination and selfdecomposability. Statistics & Probability Letters, 54, 317–324.Google Scholar
  18. 100.
    Sato, K. (2009). Selfdecomposability and semi-selfdecomposability in subordination of cone-parameter convolution semigroups. Tokyo Journal of Mathematics, 32, 81–90.MathSciNetCrossRefGoogle Scholar
  19. 104.
    Sato, K., Watanabe, T., Yamamuro, K., & Yamazato, M. (1996). Multidimensional process of Ornstein–Uhlenbeck type with nondiagonalizable matrix in linear drift terms. Nagoya Mathematical Journal, 141, 45–78.Google Scholar
  20. 111.
    Sato, K., & Yamazato, M. (1985). Completely operator-selfdecomposable distributions and operator-stable distributions. Nagoya Mathematical Journal, 97, 71–94.Google Scholar
  21. 112.
    Shanbhag, D. N., & Sreehari, M. (1979). An extension of Goldie’s result and further results in infinite divisibility. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 47, 19–25.Google Scholar
  22. 113.
    Sharpe, M. (1969). Operator-stable probability distributions on vector groups. Transactions of the American Mathematical Society, 136, 51–65.MathSciNetCrossRefGoogle Scholar
  23. 119.
    Takano, K. (1989/1990). On mixtures of the normal distribution by the generalized gamma convolutions. Bulletin of the Faculty of Science, Ibaraki University. Series A, 21, 29–41. Correction and addendum, 22, 49–52.MathSciNetCrossRefGoogle Scholar
  24. 128.
    Urbanik, K. (1972a). Lévy’s probability measures on Euclidean spaces. Studia Mathematica, 44, 119–148.MathSciNetCrossRefGoogle Scholar
  25. 133.
    Watanabe, T. (1998). Sato’s conjecture on recurrence conditions for multidimensional processes of Ornstein–Uhlenbeck type. Journal of the Mathematical Society of Japan, 50, 155–168.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alfonso Rocha-Arteaga
    • 1
  • Ken-iti Sato
    • 2
  1. 1.Facultad de Ciencias Físico-MatemáticasUniversidad Autónoma de SinaloaCuliacánMexico
  2. 2.Hachiman-yama 1101-5-103Tenpaku-kuJapan

Personalised recommendations