Journal of Theoretical Probability

, Volume 8, Issue 2, pp 221–259

Random continued fractions and inverse Gaussian distribution on a symmetric cone

  • Evelyne Bernadac
Article
  • 108 Downloads

Abstract

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesHn+(ℝ) and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onHn+(ℝ) andC by means of real Lagrangians forHn+(ℝ) and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onHn+(ℝ) andC.

Key Words

Continued fraction inverse Gaussian distribution Jordan algebra symmetric cone Wishart distribution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barndorff-Nielsen, O., and Halgreen, C. (1977). Infinite divisibility of the hyperbolic and generalized inverse Gaussian distribution,Z. Wahrsch. Verw. Gebiete 38, 309–311.Google Scholar
  2. 2.
    Bernadac, E. (1992). Fractions continues sur les matrices symétriques réelles et la loi gaussienne inverse,C. R. Acad. Sci. Paris, t. 315, Série I, pp. 329–332.Google Scholar
  3. 3.
    Bernadac, E. (1993). Fractions continues aléatoires sur un cône symétrique,C. R. Acad. Sci. Paris, t. 316, Série I, pp. 859–864.Google Scholar
  4. 4.
    Bonnefoy-Casalis, M. (1990). Familles exponentielles naturelles invariantes par un groupe, Thèse, Université Paul Sabatier, Toulouse.Google Scholar
  5. 5.
    Chhikara, R. S., and Folks, J. L. (1989).The Inverse Gaussian Distribution, M. Dekker, New York.Google Scholar
  6. 6.
    Dunau, J. L., and Sénateur, H. (1988). A characterization of the type of the Cauchy-Hua measure on real symmetric matrices,J. Theoret. Probab. 1 (3), 263–270.Google Scholar
  7. 7.
    Faraut, J. (1988). Algèbres de Jordan et cônes symétriques, Notes d'un cours de l'Ecole d'Eté, CIMPA, Université de Poitiers.Google Scholar
  8. 8.
    Faraut, J., and Koranyi, A. (1994). Analysis on symmetric cones, Oxford University Press.Google Scholar
  9. 9.
    Faraut, J., and Travaglini, G. (1987). Bessel functions associated with representations of formally real Jordan algebras,J. Funct. Anal. 71, 123–141.Google Scholar
  10. 10.
    Good, I. J. (1953). The population frequencies of spaces and the estimation of population parameters,Biometrika 40, 237–264.Google Scholar
  11. 11.
    Hallin, M. (1984). Spectral factorization of non-stationary moving average processes,Ann. Statist. 12, 172–192.Google Scholar
  12. 12.
    Hallin, M. (1986). Non-stationaryq-dependent processes and time-varying moving-average models: invertibility properties and the forecasting problem,Adv. Appl. Prob. 18, 170–210.Google Scholar
  13. 13.
    Herz, C. S. (1955). Bessel functions of matrix argument,Ann. of Math. 61 (3), 474–523.Google Scholar
  14. 14.
    Jacobson, N. (1968). Structure and representations of Jordan algebras,Amer. Math. Soc., Providence, R.I.Google Scholar
  15. 15.
    Jorgensen, B. (1982). Statistical properties of the generalized inverse Gaussian distribution,Lectures Notes in Statistics, No. 9, Springer, New York.Google Scholar
  16. 16.
    Letac, G., and Seshadri, V. (1983). A characterization of the generalized inverse Gaussian distribution by continued fractions,Z. Wahrsch. Verw. Gebiete 62, 485–489.Google Scholar
  17. 17.
    Morlat, G. (1956). Les lois de probabilité de Halphen,Revue de Statist. Appl. 4 (3), 21–46.Google Scholar
  18. 18.
    Muirhead, R. J. (1982).Aspects of Multivariate Statistical Theory Wiley, New York.Google Scholar
  19. 19.
    Seshadri, V. Inverse Gaussian distribution, to appear.Google Scholar
  20. 20.
    Vallois, P. (1989). Sur le passage de certaines marches aléatoires planes au-dessus d'une hyperbole équilatère,Ann. Inst. Henri Poincaré 25(4), 443–456.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Evelyne Bernadac
    • 1
  1. 1.Laboratoire de Mathématiques AppliquéesUniversité de Pau et des Pays de l'Adour, CNRS, U.R.A. 1204, IPRAPauFrance

Personalised recommendations