Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 013, 18 pages      arXiv:1409.4213      https://doi.org/10.3842/SIGMA.2015.013

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

Margit Rösler a and Michael Voit b
a) Institut für Mathematik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany
b) Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany

Received October 14, 2014, in final form February 03, 2015; Published online February 10, 2015

Abstract
We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a sharp Mehler-Heine formula, that is an approximation of the Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of $G/K$, which are constructed by successive decompositions of tensor powers of spherical representations of $G$. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.

Key words: Mehler-Heine formula; Heckman-Opdam polynomials; Grassmann manifolds; spherical functions; central limit theorem; asymptotic representation theory.

pdf (427 kb)   tex (27 kb)

References

  1. Aslaksen H., Quaternionic determinants, Math. Intelligencer 18 (1996), 57-65.
  2. Ben Saïd S., Ørsted B., Analysis on flat symmetric spaces, J. Math. Pures Appl. 84 (2005), 1393-1426.
  3. Bloom W.R., Heyer H., Harmonic analysis of probability measures on hypergroups, de Gruyter Studies in Mathematics, Vol. 20, Walter de Gruyter & Co., Berlin, 1995.
  4. Clerc J.L., Une formule asymptotique du type Mehler-Heine pour les zonoles d'un espace riemannien symétrique, Studia Math. 57 (1976), 27-32.
  5. Clerc J.L., Roynette B., Un théorème central-limite, in Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1976-1978), II, Lecture Notes in Math., Vol. 739, Springer, Berlin, 1979, 122-131.
  6. de Jeu M., Paley-Wiener theorems for the Dunkl transform, Trans. Amer. Math. Soc. 358 (2006), 4225-4250, math.CA/0404439.
  7. Dunkl C.F., The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc. 179 (1973), 331-348.
  8. Eymard P., Roynette B., Marches aléatoires sur le dual de ${\rm SU}(2)$, in Analyse harmonique sur les groupes de Lie, Lecture Notes in Math., Vol. 497, Editors P. Eymard, J. Faraut, G. Schiffmann, R. Takahashi, Springer-Verlag, Berlin - New York, 1975, 108-152.
  9. Faraut J., Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, in Analyse harmonique (Université de Nancy I, 1980), Editors J.L. Clerc, P. Eymard, J. Faraut, M. Raïs, R. Takahasi, Les Cours du C.I.M.P.A., Nice, France, 1983, 315-446.
  10. Faraut J., Korányi A., Analysis on symmetric cones, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.
  11. Gallardo L., Comportement asymptotique des marches aléatoires associées aux polynômes de Gegenbauer et applications, Adv. in Appl. Probab. 16 (1984), 293-323.
  12. Heckman G., Dunkl operators, Astérisque (1997), exp. No. 828, 223-246.
  13. Heckman G., Schlichtkrull H., Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics, Vol. 16, Academic Press, Inc., San Diego, CA, 1994.
  14. Helgason S., Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs, Vol. 83, Amer. Math. Soc., Providence, RI, 2000.
  15. Jewett R.I., Spaces with an abstract convolution of measures, Adv. Math. 18 (1975), 1-101.
  16. Koornwinder T.H., Two-variable analogues of the classical orthogonal polynomials, in Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), Academic Press, New York, 1975, 435-495.
  17. Koornwinder T.H., Harmonics and spherical functions on Grassmann manifolds of rank two and two-variable analogues of Jacobi polynomials, in Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976), Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977, 141-154.
  18. Koornwinder T.H., Positive convolution structures associated with quantum groups, in Probability Measures on Groups, X (Oberwolfach, 1990), Plenum, New York, 1991, 249-268.
  19. Macdonald I.G., Orthogonal polynomials associated with root systems, Sém. Lothar. Combin. 45 (2000), Art. B45a, 40 pages, math.QA/0011046.
  20. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W., NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.
  21. Opdam E.M., Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), 333-373.
  22. Opdam E.M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121.
  23. Remling H., Rösler M., Convolution algebras for Heckman-Opdam polynomials derived from compact Grassmannians, J. Approx. Theory, to appear.
  24. Rösler M., Bessel convolutions on matrix cones, Compos. Math. 143 (2007), 749-779, math.CA/0512474.
  25. Rösler M., Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type $BC$, J. Funct. Anal. 258 (2010), 2779-2800, arXiv:0907.2447.
  26. Rösler M., Voit M., Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type $BC$, Trans. Amer. Math. Soc., to appear, arXiv:1402.5793.
  27. Szegö G., Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, R.I., 1959.
  28. van Diejen J.F., Properties of some families of hypergeometric orthogonal polynomials in several variables, Trans. Amer. Math. Soc. 351 (1999), 233-270, q-alg/9604004.
  29. Voit M., Central limit theorems for a class of polynomial hypergroups, Adv. in Appl. Probab. 22 (1990), 68-87.
  30. Voit M., Bessel convolutions on matrix cones: algebraic properties and random walks, J. Theoret. Probab. 22 (2009), 741-771, math.CA/0603017.
  31. Voit M., Central limit theorems for hyperbolic spaces and Jacobi processes on $[0,\infty [$, Monatsh. Math. 169 (2013), 441-468, arXiv:1201.3490.
  32. Zeuner H., Moment functions and laws of large numbers on hypergroups, Math. Z. 211 (1992), 369-407.

Previous article  Next article   Contents of Volume 11 (2015)