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Convolution mixtures of infinitely divisible distributions

Published online by Cambridge University Press:  24 October 2008

John T. Kent
Affiliation:
University of Leeds

Abstract

A great deal is known about infinitely divisible distributions on [0, ∞). In this paper, a simple mapping which can be used to extend this information to more general convolution algebras is examined. Some examples are given where this approach has proved fruitful.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Barndorff-Nielsen, O. and Halgreen, C.Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Z. Wahrscheinlichkeitstheorie verw. Geb. 38 (1977), 309312.CrossRefGoogle Scholar
(2)Barndorff-Nielsen, O., Kent, J. T. and Sorensen, M.Normal variance-mean mixtures and z-distributions. (Research Report no. 64, University cf Aarhus, 1980).Google Scholar
(3)Benedetto, J. J.Spectral Synthesis (Teubner, Stuttgart, 1975).CrossRefGoogle Scholar
(4)Beurling, A.Un théorème sur les fonctions bornées et uniformément continues sur l'axe réel. Acta Math. 77 (1940), 127136.CrossRefGoogle Scholar
(5)Billingsley, P.Convergence of Probability Measures. (Wiley, New York, 1968).Google Scholar
(6)Bingham, N. H.Factorization theory and domains of attraction for generalized convolution algebras. Proc. London Math. Soc. (3) 23 (1971), 1630.CrossRefGoogle Scholar
(7)Bingham, N. H.Random walks on spheres. Z. Wahrscheinliclikeitstheorie verw. Geb. 22 (1972). 169172.CrossRefGoogle Scholar
(8)Bondesson, L.On generalized gamma and generalized negative binomial convolutions, Part I and II. Scand. Actuarial J. (1979), pp. 125166.CrossRefGoogle Scholar
(9)Bondesson, L. Classes of infinitely divisible distributions and densities. Z. Wahrscheinlichkeitstheorie verw. Geb., to appear.Google Scholar
(10)Bourbaki, N.Eléments de Mathémntique., Intégration, Chapitre 9, Mesures sur les espaces topologiques séparés, première édition (Hermann, Paris, 1969).Google Scholar
(11)Embrechts, P.On a theorem of E. Lukacs. Proc. Amer. Math. Soc. 68 (1978), 292–4, erratum 75 (1979), 375.CrossRefGoogle Scholar
(12)Feller, W.An Introduction to Probability Theory and its Applications, vol. II (Wiley, New York, 1966).Google Scholar
(13)Fremlin, D. H., Garling, D. J. H. and Haydon, R. G.Bounded measures on topological spaces. Proc. London Math. Soc. (3) 25 (1972), 115–36.CrossRefGoogle Scholar
(14)Gänssler, P.Compactness and sequential compactness in spaces of measures. Z. Wahrscheinlichkeitstheorie verw. Geb. 17 (1971), 124146.CrossRefGoogle Scholar
(15)Grosswald, E.The Student t-distribution of any degree of freedom is infinitely divisible. Z. Wahrscheinlichkeitstheorie verw. Geb. 36 (1976), 103109.CrossRefGoogle Scholar
(16)Hartman, P.Completely monotone families of solutions of n'th order linear differential equations and infinitely divisible distributions. Ann. Scuola Norm. Sup. Pisa (IV), 3 (1976), 267287.Google Scholar
(17)Haydon, R. G.On compactness in spaces of measures and measure-compact spaces. Proc. London Math. Soc. (3) 29 (1974), 116.CrossRefGoogle Scholar
(18)Heyer, H.Probability Measures on Locally Compact Groups (Ergebnisse Math. 94, Springer, Berlin, 1977).CrossRefGoogle Scholar
(19)Hille, E. and Phillips, R. S.Functional Analysis and Semi-groups (American Mathematical Society, Providence, R.I., 1957).Google Scholar
(20)Kallenberg, O.Random Measures (Academic Press, London, 1979).Google Scholar
(21)Keilson, J. and Steutel, F. W.Mixtures of distributions, moment inequalities and measures of exponentiality and normality. Ann. Probability 2 (1974), 112130.CrossRefGoogle Scholar
(22)Kelker, D.Infinite divisibility and variance mixtures of the normal distributions. Ann. Math. Statist. 42 (1971), 802808.CrossRefGoogle Scholar
(23)Kendall, D. G. and Lamperti, J.A remark on topologies for characteristic functions. Proc. Cambridge Philos. Soc. 68 (1970), 703705.CrossRefGoogle Scholar
(24)Kent, J. T.The infinite divisibility of the von Mises-Fisher distribution for all values of the parameter in all dimensions. Proc. London Math. Soc. (3) 35 (1977), 359384.CrossRefGoogle Scholar
(25)Kent, J. T. The spectral decomposition of a diffusion hitting time. Ann. Probability, to appear.Google Scholar
(26)Kingman, J. F. C.Random walks with spherical symmetry. Acta Math. 109 (1963), 1153.CrossRefGoogle Scholar
(27)Lukacs, E.An essential property of the Fourier transforms of distribution functions. Proc. Amer. Math. Soc. 3 (1952), 508510.CrossRefGoogle Scholar
(28)Mase, S.Random closed sets which are infinitely divisible with respect to Minkowski addition. Advances Appl. Prob. 11 (1979), 834850.CrossRefGoogle Scholar
(29)Matheron, G.Random Sets and Integral Geometry (Wiley, London, 1975).Google Scholar
(30)Parthasarathy, K. R.Probability Measures on Metric Spaces (Academic Press, London, 1967).CrossRefGoogle Scholar
(31)Rao, R. RangaRelations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33 (1962), 659–80.CrossRefGoogle Scholar
(32)Schwartz, A.Generalized convolutions and positive definite functions associated with general orthogonal series. Pacific J. Math. 55 (1974), 565582.CrossRefGoogle Scholar
(33)Sentilles, F. D.Bounded continuous functions on a completely regular space. Trans. Amer. Math. Soc. 168 (1972), 311–36.CrossRefGoogle Scholar
(34)Steutel, F. W.Some recent results in infinite divisibility. Stochastic Processes Appl. 1 (1973), 125143.CrossRefGoogle Scholar
(35)Steutel, F. W.Infinite divisibility in theory and practice. Scand. J. Statist. 6 (1979), 5764.Google Scholar
(36)Thorin, O.On the infinite divisibility of the Pareto distribution. Scand. Actuarial J. (1977), pp. 3140.CrossRefGoogle Scholar
(37)Urbanik, K.Generalized convolution algebras. Studia Math. 23 (1964), 217245.CrossRefGoogle Scholar
(38)Varadarajan, V. S.Measures on topological spaces. Mat. Sb. (97) 55 (1961), 35100. (English translation 1965. Providence: American Mathematical Society Translations, series 2, 48, 161228.)Google Scholar