Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T23:57:11.232Z Has data issue: false hasContentIssue false
  • English
  • Français

Factorization of probability measures on symmetric hypergroups

Part of: Distribution theory - Probability Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  09 April 2009

Michael Voit
Michael Voit
Affiliation:
Mathematisches Institut Technische Universität MünchenArcisstr. 21 D-8000 München 2, Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Generalizing known results for special examples, we derive a Khintchine type decomposition of probability measures on symmetric hypergroups. This result is based on a triangular central limit theorem and a discussion of conditions ensuring that the set of all factors of a probability measure is weakly compact. By our main result, a probability measure satisfying certain restrictions can be written as a product of indecomposable factors and a factor in I0(K), the set of all measures having decomposable factors only. Some contributions to the classification of I0(K) are given for general symmetric hypergroups and applied to several families of examples like finite symmetric hypergroups and hypergroup joins. Furthermore, all results are discussed in detail for a class of discrete symmetric hypergroups which are generated by infinitely many joins, for a class of countable compact hypergroups, for Sturm-Liouville hypergroups on [0, ∞[ and, finally, for polynomial hypergroups.

Keywords

Khintchine decompositioninfinitely divisible measuresindecomposable measuressymmetric hypergroupspolynomial hypergroupsSturm-Liouville hypergroupshypergroup joins

MSC classification

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 3 , June 1991 , pp. 417 - 467
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Askey, R., ‘Orthogonal polynomials and positivity’, in Studies in Applied Mathematics 6, Wave Propagation and Special Functions, Ludwig, D. and Olver, F. W. J., eds., 64–85, Philadelphia, SIAM 1970.Google Scholar
[2]Askey, R. and Wilson, J. A., ‘Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials’, Memoirs Amer. Math. Soc. 319 (1985).Google Scholar
[3]Berg, C. and Forst, G., Potential Theory on Locally Compact Abelian Groups, Springer, Berlin, Heidelberg, New York, 1975.Google Scholar
[4]Bingham, N. H., ‘Positive definite functions on spheres’, Proc. Cambridge Philos. Soc. 73 (1973), 145–156.CrossRefGoogle Scholar
[5]Bingham, N. H., ‘Factorization theory and domains of attraction for generalized convolution algebras’, Proc. London Math. Soc. 23 (1971), 16–30.Google Scholar
[6]Bloom, W., ‘Idempotent measures on commutative hypergroups’, Probability Measures on Groups VIII, Proc. Conf., Oberwolfach, 1985, 13–23. Lecture Notes in Math. 1210, Springer, Berlin, Heidelberg, New York, 1986.Google Scholar
[7]Bloom, W. and Heyer, H., ‘The Fourier transform of probability measures on hypergroups’, Rend. Mat. 2 (1982), 315–334.Google Scholar
[8]Bloom, W. and Heyer, H., ‘Convolution semigroups and resolvent families of measures on hypergroups’, Math. Z. 188 (1985), 449–474.CrossRefGoogle Scholar
[9]Bloom, W. and Heyer, H., ‘Continuity of convolution semigroups on hypergroups’, J. Theoret. Prob. 1 (1988), 271–286.Google Scholar
[10]Bressoud, D. M., ‘Linearization and related formulas for q-ultraspherical polynomials’, SIAM J. Math. Anal. 12 (1981), 161–168.Google Scholar
[11]Chebli, H., ‘Opérateurs de translation généralisee et semigroupes de convolution’. Théorie du Potentiel et Analyse Harmonique, Proc. Conf. Strasbourg, 1973, PP. 33–59. Springer, Lecture Notes in Math. 404, Berlin, Heidelberg, New York; 1974.Google Scholar
[12]Chebli, H., ‘Sur un theoreme de Paley-Wiener associe a la decomposition spectrale d'un operateur de Sturm-Liouville sur ]0, ∞[’, J. Funct. Anal. 17 (1974), 447–461.CrossRefGoogle Scholar
[13]Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.Google Scholar
[14]Connett, W. C. and Schwartz, A. L., ‘Analysis of a class of probability preserving measure algebras on compact intervals’, Trans. Amer. Math. Soc., 317 (1990), 371–393.CrossRefGoogle Scholar
[15]Dunkl, C. F., ‘The measure algebra of a locally compact hypergroup’, Trans. Amer. Math. Soc. 179 (1973), 331–348.Google Scholar
[16]Dunkl, C. F. and Ramirez, D. E., ‘A family of countable compact P*-hypergroups’, Trans. Amer. Math. Soc. 202 (1975), 339–356.Google Scholar
[17]Fel'dman, G. M., ‘On generalized Poisson distributions on groups’, Theory Probab. and its Appl. 20 (1975), 641–644.Google Scholar
[18]Finkh, U., Beiträge zur Wahrscheinlichkeitstheorie auf einer Kingman Struktur, Dissertation. Tübingen, 1986.Google Scholar
[19]Gallardo, L. and Gebuhrer, O., ‘Lois de probabiite infiniment divisibles sur les hypergroupes commutatifs, discrets, denomerables’, Probability Measures on Groups VII, Proc. Conf., Oberwolfach, 1983, 116–130. Springer, Lecture Notes in Math. 1064, Berlin, Heideberg, New York, 1984.Google Scholar
[20]Gasper, G., ‘Positivity and special functions’, in Theory and Applications of Special Functions, ed. Askey, R., pp. 375–434, Academic Press, New York, 1975.CrossRefGoogle Scholar
[21]Heyer, H., Probability Measures on Locally Compact Groups, Springer, Berlin, Heidelberg, New York, 1977.Google Scholar
[22]Heyer, H., ‘Probability theory on hypergroups: a survey’, Probability Measures on Groups VII, Proc. Conf., Oberwolfach, 1983, 481–550, Springer, Lecture Notes in Math. 1064, Berlin, Heidelberg, New York, 1984.Google Scholar
[23]Heyer, H., ‘Convolution semigroups and potential kernels on a commutative hypergroup’, in Hofmann, K. H. et al. (eds.), The Analytical and Topological Theory of Semigroups, pp. 273–312, de Gruyter Berlin, New York, 1990.Google Scholar
[24]Holland, A. S. B., Introduction to the Theory of Entire Functions, Academic Press, New York, London, 1973.Google Scholar
[25]Jewett, R. I., ‘Spaces with an abstract convolution of measures’, Adv. Math. 18 (1975),1–101.CrossRefGoogle Scholar
[26]Kendall, D., ‘Delphic semigroups, infinitely divisible regenerative phenomena, and the arithmetic of p-functions’, Z. Wahrscheinlichkeitsth. Verw. Geb. 9 (1968), 163–195.Google Scholar
[27]Kingman, J. F. C., ‘Random walks with spherical symmetry’, Acta Math. 109 (1963),11–53.CrossRefGoogle Scholar
[28]Koornwinder, T., ‘Jacobi functions and analysis of noncompact semisimple Lie groups’, in Askey, R. A. et al. (eds.), Special Functions: Group Theoretical Aspects and Applications, pp. 1–85, Reidel Dordrecht, Boston, Lancaster, 1984.Google Scholar
[29]Lamperti, J., ‘The arithmetic of certain semi-groups of positive operators’, Proc. Cambridge Philos. Soc. 64 (1968), 161–166.CrossRefGoogle Scholar
[30]Lesser, R., ‘Orthogonal polynomials and hypergroups’, Rend. Math. Appl. 2 (1983), 185–209.Google Scholar
[31]Lasser, R., ‘Bochner theorems for hypergroups and their application to orthogonal polynomial expansions’, J. Approx. Theory 37 (1983), 311–327.Google Scholar
[32]Lasser, R., ‘Convolution semigroups on hypergroups’, Pacific J. Math. 127 (1987), 353–371.CrossRefGoogle Scholar
[33]Lasser, R., ‘Linearization of the product of associated Legendre polynomials’, SIAM J. Math. Anal. 14 (1983), 403–408.CrossRefGoogle Scholar
[34]Linnik, Yu. V., Decomposition of Probability Distributions, Oliver and Boyd, Edinburgh, London, 1964.Google Scholar
[35]Nevai, P., ‘Orthogonal polynomials’, Mem. Amer. Math. Soc. 213 (1979).Google Scholar
[36]Ostrovskii, I. V., ‘Description of the I0 class in a special semigroup of probability measures’, Soviet Math. Dokl. 14 (1973), 525–529.Google Scholar
[37]Ostrovskii, I. V. and Truhina, I. R., ‘The arithmetic of Schoenberg-Kennedy semigroups’. in Questions of mathematical physics and functional analysis (Proc. Res. Sem. Inst. Low Temp. Phys. Engrg., Kharkov), pp. 11–19, 171, “Naukova Dunka”, Kiev, 1976.Google Scholar
[38]Parthasarathy, K. R., Probability Measures on Metric Spaces, Academic Press, New York, London, 1967.CrossRefGoogle Scholar
[39]Rudin, W., Fourier analysis on Groups, Interscience, New York, 1962.Google Scholar
[40]Rusza, I. Z. and Szekely, G. J., Algebraic Probability Theory, Wiley, Chichester, New York, 1988.Google Scholar
[41]Schwartz, A. L., ‘l1-convolution algebras: representation and factorization’, Z. Wahrscheinlichkeitsth. Verw. Geb. 41 (1977), 161–176.CrossRefGoogle Scholar
[42]Schwartz, A. L., ‘Classification of one-dimensional hypergroups’, Proc. Amer. Math. Soc. 239 (1988), 1073–1081.CrossRefGoogle Scholar
[43]Soardi, P. M., ‘Limit theorems for random walks on discrete semigroups related to nonhomogeneous trees and Chebyshef polynomials’, Math. Z. 200 (1989), 313–327.Google Scholar
[44]Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ 23, Providence, R.I., 1959.Google Scholar
[45]Truhina, I. R., ‘A problem related to the arithmetic of probability measures on the sphere’, J. Soviet. Math. 17 (1981), 2321–2333.CrossRefGoogle Scholar
[46]Voit, M., ‘Positive characters on commutative hypergroups and some applications’, Math. Z. 198 (1988), 405–421.Google Scholar
[47]Voit, M., ‘Positive and negative definite functions on the dual space of a commutative hypergroup’, Analysis 9 (1989), 371–387.Google Scholar
[48]Voit, M., ‘Laws of large numbers for polynomial hypergroups and some applications’, J. Theoret. Prob. 3 (1990), 245–266.Google Scholar
[49]Voit, M., ‘Central limit theorems for a class of polynomial hypergroups’, Adv. Appl. Prob. 22 (1990), 66–87.Google Scholar
[50]Voit, M., ‘Negative definite functions on commutative hypergroups’, Probability Measures on Groups IX, Proc. Conf., Oberwolfach, 1988, 376–388, Springer, Lecture Notes in Math. 1379, Berlin, Heidelberg, New York, 1989.Google Scholar
[51]Voit, M., ‘Central limit theorems for random ralks on N0 that are associated with a sequence of orthogonal polynomials’, J. Multivariate Anal. 34 (1990), 290–322.CrossRefGoogle Scholar
[52]Vrem, C., ‘Hypergroup joins and their dual objects’, Pacific J. Math. 111 (1984), 483–495.CrossRefGoogle Scholar
[53]Zeuner, Hm., ‘One-dimensional hypergroups’, Adv. Math. 76 (1989), 1–18.Google Scholar
[54]Zeuner, Hm., ‘Properties of the cosh hypergroup’, Probability Measures on Groups IX, Proc. Conf., Oberwolfach, 1988, 425–434, Springer Lecture Notes in Math. 1379, Berlin, Heidelberg, New York, 1989.Google Scholar