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On a functional equation generalizing the class of semistable distributions

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Abstract

With ϕ(p),p≥0 the Laplace-Stieltjes transform of some infinitely divisible probability distribution, we consider the solutions to the functional equation ϕ(p-e −pβΠ m i=1 ϕγi (c i p) for somem≥1,c i>0, γ i >0,i=1., …,m, β ε ®. We supply its complete solutions in terms of semistable distributions (the ones obtained whenm=1). We then show how to obtain these solutions as limit laws (r → ∞) of normalized Poisson sums of iid samples when the Poisson intensity λ(r) grows geometrically withr.

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References

  • Bertoin, J. (1996).Lévy Processes, Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Crump, K. and Mode, C. J. (1968a). A general age-dependent branching processes,Journal of Mathematical Analysis and Its Applications Part I,24, 497–508.

    Google Scholar 

  • Crump, K. and Mode, C. J. (1968b). A general age-dependent branching processes,Journal of Mathematical Analysis and Its Applications, Part II,25, 8–17.

    Article  MathSciNet  Google Scholar 

  • Feller, W. (1971).An Introduction to Probability Theory and Its Applications,2, Wiley, New York.

    MATH  Google Scholar 

  • Grinevich, I. V. and Khokhlov, Y. S. (1993). The domains of attraction of the semistable laws,Theory of Probability and Its Applications,37, 361–366.

    MathSciNet  Google Scholar 

  • Guivarc’h, Y. (1990). Sur une extension de la notion de loi semi-stable,Annales de l’Institut Henri Poincaré, Probabilités et Statistiques,26, 261–285.

    MathSciNet  MATH  Google Scholar 

  • Huillet, T., Porzio, A. and Ben Alaya, M. (2001). On Lévy stable and semistable distributions,Fractals,9(3), 347–365.

    Article  MathSciNet  MATH  Google Scholar 

  • Kahane, J. P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot,Advances in Mathematics,22, 131–345.

    Article  MathSciNet  MATH  Google Scholar 

  • Kruglov, V. M. (1972) On the extension of the class of stable distributions,Theory of Probability and Its Applications,17, 685–694.

    Article  MATH  Google Scholar 

  • Lévy, P. (1937).Théorie de l’Addition des Variables Aléatoires, Gauthier Villars, Paris.

    Google Scholar 

  • Liu, Q. (1997). Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten-Stigum concernant des processus de branchement,Advances in Applied Probability,29, 353–373.

    Article  MathSciNet  MATH  Google Scholar 

  • Lukacs, E. (1983).Development in Characteristic Function Theory, C. Griffin and Co., London and High Wycombe.

    Google Scholar 

  • Mandelbrot, B. B. (1974a). Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire,Comptes Rendus de l’Académie des Sciences de Paris,278, 289–292.

    MathSciNet  MATH  Google Scholar 

  • Mandelbrot, B. B. (1974b). Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire: quelques extensions,Comptes Rendus de l’Académie des Sciences de Paris,278, 355–358.

    MathSciNet  MATH  Google Scholar 

  • Pillai, R. N. (1971). Semistable laws as limit distributions,The Annals of Mathematical Statistics,42, 780–783.

    Article  MathSciNet  Google Scholar 

  • Ramachandran, B. and Rao, C. R. (1968). Some results on characteristic functions and characterizations of the normal and generalized stable laws,Sankhyā, Series A,30, 125–140.

    MathSciNet  MATH  Google Scholar 

  • Rao, C. R. and Shanbhag, D. N. (1994).Choquet-Deny Type Functional Equations with Applications to Stochastic Models, Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons, Chichester.

    MATH  Google Scholar 

  • Sato, K. (1999).Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Shimizu, R. (1970). On the domain of partial attraction of semistable distributions,The Annals of the Institute of Statistical Mathematics,22, 245–255.

    Article  MathSciNet  MATH  Google Scholar 

  • Shimizu, R. (1978) Solution to a functional equation and its applications to some characterization problems,Sankhyä, Series A 40(4), 319–332.

    MathSciNet  MATH  Google Scholar 

  • Shimizu, R. and Davies, L. (1981). General characterization theorems for the Weibull and the stable distributions,Sankhyā, Series A,43 (3), 282–210.

    MathSciNet  MATH  Google Scholar 

  • Uchaikin, V. V. and Zolotarev, V. M. (1999).Chance and Stability Stable Distributions and Their Applications, Modern Probability and Statistics, VSP BV, Utrecht and Tokyo.

    MATH  Google Scholar 

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Authors and Affiliations

  1. LAGA, CNRS (UMR 7539), Institut Galilée, Université de Paris 13, 93430, Villetaneuse, France

    Mohamed Ben Alaya

  2. LPTM, Université de Cergy-Pontoise et CNRS (UMR 8089), 5, mail Gay-Lussac, Neuville sur Oise, 95031, Cergy-Pontoise Cedex, France

    Thierry Huillet

Authors
  1. Mohamed Ben Alaya
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  2. Thierry Huillet
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Alaya, M.B., Huillet, T. On a functional equation generalizing the class of semistable distributions. Ann Inst Stat Math 57, 817–831 (2005). https://doi.org/10.1007/BF02915441

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  • DOI: https://doi.org/10.1007/BF02915441

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