The asymptotics of spherical functions and the central limit theorem on symmetric cones

Genkai Zhang

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 2, page 565-575
  • ISSN: 0373-0956

Abstract

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We prove a central limit theorem for certain invariant random variables on the symmetric cone in a formally real Jordan algebra. This extends form the previous results of Richards and Terras on the cone of real positive definite n × n matrices.

How to cite

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Zhang, Genkai. "The asymptotics of spherical functions and the central limit theorem on symmetric cones." Annales de l'institut Fourier 45.2 (1995): 565-575. <http://eudml.org/doc/75129>.

@article{Zhang1995,
abstract = {We prove a central limit theorem for certain invariant random variables on the symmetric cone in a formally real Jordan algebra. This extends form the previous results of Richards and Terras on the cone of real positive definite $n\times n$ matrices.},
author = {Zhang, Genkai},
journal = {Annales de l'institut Fourier},
keywords = {symmetric cones; formally real Jordan algebras; spherical functions; central limit theorems; symmetric spaces},
language = {eng},
number = {2},
pages = {565-575},
publisher = {Association des Annales de l'Institut Fourier},
title = {The asymptotics of spherical functions and the central limit theorem on symmetric cones},
url = {http://eudml.org/doc/75129},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Zhang, Genkai
TI - The asymptotics of spherical functions and the central limit theorem on symmetric cones
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 2
SP - 565
EP - 575
AB - We prove a central limit theorem for certain invariant random variables on the symmetric cone in a formally real Jordan algebra. This extends form the previous results of Richards and Terras on the cone of real positive definite $n\times n$ matrices.
LA - eng
KW - symmetric cones; formally real Jordan algebras; spherical functions; central limit theorems; symmetric spaces
UR - http://eudml.org/doc/75129
ER -

References

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  3. [FK] J. FARAUT and A. KORANYI, Function spaces and reproducing kernels on bounded symmetric domains, J. Func. Anal., 89 (1990), 64-89. Zbl0718.32026MR90m:32049
  4. [G1] P. GRACZYK, A central limit theorem on the space of positive definite symmetric matrices, Ann. Inst. Fourier, 42-1,2 (1992), 857-874. Zbl0736.60025MR93m:60023
  5. [G2] P. GRACZYK, Dispersions and a central limit theorem, preprint. Zbl0829.43002
  6. [H1] S. HELGASON, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, London, 1978. Zbl0451.53038
  7. [H2] S. HELGASON, Groups and geometric analysis, Academic Press, London, 1984. Zbl0543.58001
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  9. [Ku] H. B. KUSHNER, The linearization of the product of two zonal polynomials, SIAM J. Math. Anal., 19 (1988), 686-717. Zbl0642.33024MR89e:33026
  10. [L] O. LOOS, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977. Zbl0228.32012
  11. [M] R. J. MURIHEAD, Aspects of multivariate statistical theory, John Wiley & Sons, New-York, 1982. Zbl0556.62028
  12. [R] D. St. P. RICHARDS, The central limit theorem on spaces of positive definite matrices, J. Multivariate Anal., 23 (1987), 13-36. 
  13. [St] R. P. STANLEY, Some Combinatorial Properties of Jack Symmetric Functions, Adv. Math., 77 (1989), 76-115. Zbl0743.05072MR90g:05020
  14. [T1] A. TERRAS, Harmonic analysis on symmetric spaces and applications, II, Springer-Verlag, New-York, 1985. Zbl0574.10029MR87f:22010
  15. [T2] A. TERRAS, Asymptotics of special functions and the central limit theorem on the space Pn of positive n x n matrices, J. Multivariate Anal., 23 (1987) 13-36. Zbl0627.43009MR88j:43006
  16. [UU] A. UNTERBERGER and H. UPMEIER, Berezin transform and invariant differential operators, preprint. Zbl0843.32019
  17. [U1] H. UPMEIER, Jordan algebras in analysis, operator theory, and quantum mechanics, Regional conference series in mathematics, n° 67, Amer. Math. Soc., 1987. Zbl0608.17013
  18. [U2] H. UPMEIER, Toeplitz operators on bounded symmetric domains, Tran. Amer. Math. Soc., 280 (1983), 221-237. Zbl0527.47019MR85g:47042
  19. [Vr] L. VRETARE, Elementary spherical functions on symmetric spaces, Math. Scand., 39 (1976), 343-358. Zbl0387.43009MR56 #6289
  20. [Z] G. ZHANG, Some recurrence formula for spherical polynomials on tube domains, Trans. Amer. Math. Soc., to appear. Zbl0839.22019

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