Abstract
There are introduced moments on polynomial hypergroups. These moments are used to prove strong laws of large number (SSLLNs) for random walks on the nonnegative integers that are homogeneous with respect to a polynomial hypergroup where SLLNs of different kind appear for polynomial hypergroups thth different properties. Furthermore, we discuss polynomial hypergroups that are associated with some discrete semigroups in a canonical way, and, using SLLNs for polynomial hypergroups, we get SLLNs for isotropic random walks on some discrete semigroups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Askey, R., and Wilson, J. A. (1985). Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.Memoirs Am. Math. Soc. 54(319).
Feller, W. (1971).An Introduction to Probability Theory and its Applications, Vol. II, 2nd ed. Wiley, New York.
Gallardo, L. (1984). Comportement asymptotique des marches aléatoires associées aux polynômes de Gegenbauer.Adv. Appl. Prob. 16, 293–323.
Gallardo, L., and Ries, V. (1979). La loi de grandes nombres pour les marches aléatoires sur le dual deSU(2),Studia Math. LXVI, 93–105.
Gasper, G. (1975). Positivity and special functions. InTheory and Applications of Special Functions, Askey, R. (Ed.), pp. 375–434. Academic Press, New York.
Heyer, H. (1984). Probability theory on hypergroups: A survey. InProc. Conf., Oberwolfach, 1983, pp. 481–550. Lecture Notes in Mathematics, Vol. 1064. Springer-Verlag, Berlin.
Jewett, R. I. (1975). Spaces with an abstract convolution of measures,Adv. Math. 18, 1–101.
Lasser, R. (1983). Orthogonal polynomials and hypergroups.Rend. Math. Appl. 2, 185–209.
Lasser, R. (1984). On the Levy-Hincin formula for commutative hypergroups. InProc. Conf., Oberwolfach, 1983, pp. 298–308. Lecture Notes in Mathematics, Vol. 1064. Springer-Verlag, Berlin.
Lasser, R. (1987). Convolution semigroups on hypergroups.Pacific J. Math. 127, 353–371.
Sawyer, S. (1978). Isotropic random walks in a tree.Z. Wahrsch. verw. Geb. 42, 279–292.
Soardi, P. M. (1989). Limit theorems for random walks on discrete semigroups related to nonhomogeneous trees and Chebyshef polynomials.Math. Z. 200, 313–327.
Stout, W. F. (1974).Almost Sure Convergence. Academic Press, New York.
Voit, M. (1988). Positive characters on commutative hypergroups and some applications.Math. Z. 198, 405–421.
Voit, M. (1990). Central limit theorems for a class of polynomial hypergroups. To appear inAdv. Appl. Prob. 1990.
Zeuner, H. M. (1989). Laws of large numbers for hypergroups on ℝ n .Math. Ann. 283, 657–678.
Zeuner, H. M. (1989). The central limit theorem for Chebli-Trimeche hypergroups.J. Theoret. Prob. 2, 51–63.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Voit, M. Laws of large numbers for polynomial hypergroups and some applications. J Theor Probab 3, 245–266 (1990). https://doi.org/10.1007/BF01045161
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01045161