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A direct approach to the stable distributions

  • E. J. G. Pitman and Jim Pitman (a1)


The explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.


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Statistics Department, University of California, Berkeley, 367 Evans Hall, Berkeley, CA94720‒3860, USA. Email address:


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[1] Aczél, J. and Dhombres, J. (1989).Functional Equations in Several Variables (Encyclopaedia Math. Appl. 31).Cambridge University Press.
[2] Bergström, H. (1963).Limit Theorems for Convolutions.John Wiley,New York.
[3] Bertoin, J. (1996).Lévy Processes(Camb. Tracts Math. 121).Cambridge University Press.
[4] Bingham, N. H.,Goldie, C. M. and Teugels, J. L. (1989).Regular Variation (Encyclopaedia Math. Appl. 27), revised edn.Cambridge University Press.
[5] Courant, R. (1936).Differential and Integral Calculus, Vol. II.Blackie,London.
[6] Durrett, R. (2010).Probability: Theory and Examples,4th edn.Cambridge University Press.
[7] Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II,2nd edn.John Wiley,New York.
[8] Geluk, J. L. and de Haan, L. (1987).Regular Variation, Extensions and Tauberian Theorems (CWI Tract 40).Mathematisch Centrum,Amsterdam.
[9] Geluk, J. L. and de Haan, L. (2000).Stable probability distributions and their domains of attraction: a direct approach.Prob. Math. Statist. 20,169188.
[10] Gil-Pelaez, J. (1951).Note on the inversion theorem.Biometrika 38,481482.
[11] Gnedenko, B. V. and Kolmogorov, A. N. (1954).Limit Distributions for Sums of Independent Random Variables.Addison‒Wesley,Cambridge, MA.
[12] Gupta, A. K.,Nguyen, T. T. and Zeng, W.-B. (1997).Characterization of multivariate distributions through a functional equation of their characteristic functions.J. Statist. Planning Infer. 63,187201.
[13] Gupta, A. K.,Jagannathan, K.,Nguyen, T. T. and Shanbhag, D. N. (2006).Characterizations of stable laws via functional equations.Math. Nachr. 279,571580.
[14] Hall, P. (1981).A comedy of errors: the canonical form for a stable characteristic function.Bull. London Math. Soc. 13,2327.
[15] Kagan, A. M.,Linnik, Yu. V. and Rao, C. R. (1973).Characterization Problems in Mathematical Statistics.John Wiley,New York.
[16] Kallenberg, O. (2002).Foundations of Modern Probability,2nd edn.Springer,New York.
[17] Khintchine, A. Ya. and Lévy, P. (1936).Sur les lois stables.C. R. Acad. Sci. Paris Sér. A 202,374376.
[18] Kuczma, M. (2009).An Introduction to the Theory of Functional Equations and Inequalities, 2nd edn., ed. A. Gilányi.Birkhäuser,Basel.
[19] Lévy, P. (1924).Théorie des erreurs. La loi de Gauss et les lois exceptionnelles.Bull. Soc. Math. France 52,4985.
[20] Lévy, P. (1954).Théorie de l'addition des variables aléatoires,2nd edn.Gauthier-Villars,Paris.
[21] Mijnheer, J. L. (1975).Sample Path Properties of Stable Processes(CWI Tract 59).Mathematisch Centrum,Amsterdam.
[22] Pitman, E. J. G. (1968).On the behavior of the characteristic function of a probability distribution in the neighborhood of the origin.J. Austral. Math. Soc. 8,423443.
[23] Ramachandran, B. and Lau, K.-S. (1991).Functional Equations in Probability Theory.Academic Press,Boston, MA.
[24] Samorodnitsky, G. and Taqqu, M. S. (1994).Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance.Chapman & Hall/CRC,New York.
[25] Sato, K.-I. (1999).Lévy Processes and Infinitely Divisible Distributions(Camb. Stud. Adv. Math. 68).Cambridge University Press.
[26] Seneta, E. (2002).Karamata's characterization theorem, Feller, and regular variation in probability theory.Publ. Inst. Math. (Beograd) (N.S.) 71(85),7989.
[27] Soni, K. and Soni, R. P. (1975).Slowly varying functions and asymptotic behavior of a class of integral transforms I, II, III.J. Math. Anal. Appl. 49,166179,477495,612628.
[28] Steutel, F. W. and van Harn, K. (2004).Infinite Divisibility of Probability Distributions on the Real Line(Monogr. Textbooks Pure Appl. Math. 259).Marcel Dekker,New York.
[29] Uchaikin, V. V. and Zolotarev, V. M. (1999).Chance and Stability: Stable Distributions and Their Applications.VSP,Utrecht.
[30] Von Bahr, B. (1965).On the convergence of moments in the central limit theorem.Ann. Math. Statist. 36,808818.
[31] Wendel, J. G. (1961).The non-absolute convergence of Gil-Pelaez' inversion integral.Ann. Math. Statist. 32,338339.
[32] Whittaker, E. T. and Watson, G. N. (1927).A Course of Modern Analysis,4th edn.Cambridge University Press.
[33] Zolotarev, V. M. (1986).One-Dimensional Stable Distributions(Transl. Math. Monogr. 65).American Mathematical Society,Providence, RI.
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Advances in Applied Probability
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  • EISSN: 1475-6064
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