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Decomposition of probability measures on locally compact abelian groups

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Analytical Methods in Probability Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 861))

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Abstract

Let G be a locally compact, separable, Abelian metric group. Let B be the σ-field of Borel subsets of G and let P be the class of all probability measures on B. Let I ⊂ P be the class of all infinitely divisible probability measures. Let I0 ⊂ I be the class of all measures which have no indecomposable or idempotent factors. One of the fundamental problems in analytic probability theory is to obtain a precise description of the class I0. This problem is very difficult and has net yet been solved even for the case G = ℝ. It is therefore important to determine conditions under which a measure P∈I does or does not belong to I0. This paper surveys the recent work on this subject.

Work done while the author was visiting the Department of Statistics, The Ohio State University, Columbus, Ohio.

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References

  1. Cramér, H., Uber eine Eigenshaft der normalen Vertielungsfunktion. Math. Z., 41(1936), 405–414.

    Article  MathSciNet  MATH  Google Scholar 

  2. Cuppens, R., Décomposition des fonctions caractéristiques des vecteurs aléatoires. Publ. Inst. Statist. Univ. Paris, 16 (1967), 61–153.

    MathSciNet  MATH  Google Scholar 

  3. Cuppens, R., Decomposition of Multivariate Probability. Academic Press, New York 1975.

    MATH  Google Scholar 

  4. Cuppens, R., The arithmetic of distribution functions, in Contributions to Probability. Ed. by J. Gani and V. K. Rohatgi, Academic Press, New York 1981.

    Google Scholar 

  5. Feldman, G. M., On the generalized Poisson distribution on groups. Theor. Probability Appl., 20(1975), 641–644.

    Article  MathSciNet  Google Scholar 

  6. Feldman, G. M., On the decomposition of Gaussian distributions on groups. Theory of Prob. and Applications, 22(1977), 133–140.

    Article  Google Scholar 

  7. Feldman, G. M., and Fryntov, A. E., On the decomposition of the convolution of the Gaussian and Poisson distributions on locally compact Abelian groups. Functional Analysis and its Applications (Russian), 13(1979), 93–94.

    MathSciNet  Google Scholar 

  8. Feldman, G. M., and Fryntov, A. E., On the deomposition of the convolution of the Gaussian and Poisson distribution on locally compact Abelian groups. (Russian preprint) (to appear), J. Mult. Analysis.

    Google Scholar 

  9. Lévy, P., L’arithmétique des lois de probabilité et les produits finis et lois de Poisson. Actualites Sci. Indust., #736 (1938), 25–59.

    MATH  Google Scholar 

  10. Linnik, Ju. V., On the decomposition of the convolution of Gauss and Poissonian laws. Theor. Probability Appl., 2(1957), 34–59.

    MathSciNet  MATH  Google Scholar 

  11. Linnik, Ju. V., and Ostrovskii, I. V., Decomposition of Random Variables and Vectors. American Math. Soc., Providence 1971.

    MATH  Google Scholar 

  12. Livsič, L. Z., and Ostrovskii, I. V., On multivariate infinitely divisible components. Tr. FTINT AN USSR, Matem. Fizika i. Funktsion Analiz. vyp., 2(1971),, 61–75 (Russian).

    Google Scholar 

  13. Livsič, L. Z., Ostrovskii, I. V., and Čistjakov, G. P., The arithmetic of probability laws. Teor. Verojatnost. Mat. Statist., Teoret. Kibernet. VINIT, 12(1975), 5–42 (Russian).

    MathSciNet  Google Scholar 

  14. Marcinkiewicz, J., Sur les fonctions independantes, III. Fund. Math., 31(1938), 86–102.

    MATH  Google Scholar 

  15. Ostrovskii, I. V., A multidimensional analogue of Ju. V. Linnik’s theorem on decomposition of a Gauss and a Poisson law. Theor. Probability Appl., 10(1965), 673–677.

    Article  MathSciNet  Google Scholar 

  16. Ostrovskii, I. V., On certain classes of infinitely divisible laws. Izv. ANNSSR, Ser. Matem., 34(1970), 923–944 (Russian).

    MathSciNet  Google Scholar 

  17. Ostrovskii, I. V., The arithmetic of probability distributions. J. Mult. Anal., 7(1977), 475–490.

    Article  MathSciNet  MATH  Google Scholar 

  18. Parthasarathy, K. R., Probability Measures on Metric Spaces. Academic Press, New York 1967.

    Book  MATH  Google Scholar 

  19. Raikov, D. A., Decomposition of Gauss and Poisson laws. Izv. Akad. Nauk SSSR Ser. Mat., 2(1938), 91–124 (Russian).

    MATH  Google Scholar 

  20. Ruhin, A. L., Some statistical and probabilistic problems on groups. Trudy Matem. In-ta. im. Steklov Institute USSR, 69(1970), 52–108 (Russian).

    MathSciNet  Google Scholar 

  21. Ruhin, A. L., On Poisson law on groups. Mat. Sbornik, Akad. Nauk SSR, 10(1970), 537–543.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Bowling Green State University, 43403, Bowling Green, Ohio

    R. G. Laha & V. K. Rohatgi

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  1. R. G. Laha
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  2. V. K. Rohatgi
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Daniel Dugué Eugene Lukacs Vijay K. Rohatgi

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© 1981 Springer-Verlag

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Laha, R.G., Rohatgi, V.K. (1981). Decomposition of probability measures on locally compact abelian groups. In: Dugué, D., Lukacs, E., Rohatgi, V.K. (eds) Analytical Methods in Probability Theory. Lecture Notes in Mathematics, vol 861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097317

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  • DOI: https://doi.org/10.1007/BFb0097317

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10823-8

  • Online ISBN: 978-3-540-36785-7

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