Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-29T13:34:29.522Z Has data issue: false hasContentIssue false
  • English
  • Français

Central limit theorems for a class of polynomial hypergroups

Published online by Cambridge University Press:  01 July 2016

Michael Voit
Michael Voit*
Affiliation:
Technische Universität Munchen
*
∗Postal address: Institut für Mathematik, Technische Universität München, Arcisstr. 21, D-8000 München 2, W. Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

Central limit theorems are proved for random walks on the non-negative integers where the transition probabilities are homogeneous with respect to a sequence of orthogonal polynomials. Assuming some restrictions concerning the three-term recursion formula of these polynomials, one gets a Rayleigh distribution as limit distribution where bounds of the order of convergence can be computed explicitly. These central limit theorems are applied to generalized birth and death random walks and random walks on polynomial hypergroups. Finally some examples of polynomial hypergroups are discussed in view of the limit theorems above.

Keywords

BIRTH AND DEATH RANDOM WALKCENTRAL LIMIT THEOREMS WITH ERROR BOUNDSRAYLEIGH DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 68 - 87
Copyright
Copyright © Applied Probability Trust 1990 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Askey, R. (1970) Orthogonal polynomials and positivity. In Studies in Applied Mathematics 6, Wave Propagation and Special Functions , ed. Ludwig, D. and Olver, F. W. J., SIAM, Philadelphia, 64–85.Google Scholar
[2] Askey, R. and Wilson, J. A. (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem Amer. Math. Soc. 54, Number 319.Google Scholar
[3] Bressoud, D. M. (1981) On partitions, orthogonal polynomials and the expansion of certain infinite products. Proc. London Math. Soc. 42, 473–500.Google Scholar
[4] Bressoud, D. M. (1981) Linearization and related formulas for q-ultraspherical polynomials. SIAM J. Math. Anal. 12, 161–168.Google Scholar
[5] Chichara, T. S. (1978) An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
[6] Eymard, P. and Roynette, B. (1975) Marches aléatoires sur le dual de SU(2). In Analyse harmonique sur les groupes de Lie, Lecture Notes in Mathematics 497, Springer-Verlag, Berlin.Google Scholar
[7] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. Wiley, New York.Google Scholar
[8] Finck, U. (1986) Beiträge zur Wahrscheinlichkeitstheorie auf einer Kingman Struktur. Dissertation, Tübingen.Google Scholar
[9] Gallardo, L. (1984) Comportement asymptotique des marches aléatoires associées aux polynomes de Gegenbauer. Adv. Appl. Prob. 16, 293–323.Google Scholar
[10] Gallardo, L. and Ries, V. (1979) La loi de grandes nombres pour les marches aléatoires sur le dual de SU(2). Studia Math. LXVI, 93–105.CrossRefGoogle Scholar
[11] Gasper, G. (1975) Positivity and special functions. In Theory and Applications of Special Functions, ed. Askey, R., Academic Press, New York, 375–434.Google Scholar
[12] Geronimus, Ja. L. (1977) Orthogonal Polynomials. English translation: Am. Math. Soc. Trans. (2) 108, 37–130 (1977).Google Scholar
[13] Heyer, H. (1984) Probability theory on hypergroups: A survey. Proc. Conf., Oberwolfach, 1983, Lecture Notes in Mathematics 1064, Springer-Verlag, Berlin, 481–550.Google Scholar
[14] Kingman, J. F. C. (1963) Random walks with spherical symmetry. Acta Math. 109, 11–53.Google Scholar
[15] Lamperti, J. (1962) A new class of probability limit theorems. J. Math. Mech. 11, 749–772.Google Scholar
[16] Lasser, R. (1983) Orthogonal polynomials and hypergroups. Rend. Math. Appl. 2, 185–209.Google Scholar
[17] Lasser, R. (1983) Linearization of the product of associated Legendre polynomials. SIAM J. Math. Anal. 14, 403–408.Google Scholar
[18] Lasser, R. (1984) On the Levy-Hincin formula for commutative hypergroups. Proc. Conf., Oberwolfach, 1983, Lecture Notes in Mathematics 1064, Springer-Verlag, Berlin, 298–308.Google Scholar
[19] Lasser, R. (1983) Bochner theorems for hypergroups and their application to orthogonal polynomial expansions. J. Approx. Theory 37, 311–327.Google Scholar
[20] Soardi, P. M. (1989) Limit theorems for random walks on discrete semigroups related to nonhomogeneous trees and Chebyshef polynomials. Math Z. 200, 313–329.Google Scholar
[21] Szegö, G. (1959) Orthogonal Polynomials. Am. Math. Soc. Coll. Publ. 23, Amer. Math. Soc. Providence, R.I. Google Scholar
[22] Voit, M. (1988) Positive characters on commutative hypergroups and some applications. Math. Z. 198, 405–421.CrossRefGoogle Scholar
[23] Voit, M. (1989) Laws of large numbers for polynomial hypergroups and some applications. J. Theoret. Prob. To appear.Google Scholar
[24] Zeuner, H. M. (1989) One-dimensional hypergroups. Adv. Math. 76, 1–18.Google Scholar
[25] Zeuner, H. M. (1989) Laws of large numbers for hypergroups on ℝ+ . Math. Ann. 283, 657–678.Google Scholar
[26] Zeuner, H. M. (1989) The central limit theorem for Chebli-Trimeche hypergroups. J. Theoretical Prob. 2, 51–63.Google Scholar