Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-16T16:11:07.720Z Has data issue: false hasContentIssue false
  • English
  • Français

A direct approach to the stable distributions

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  25 July 2016

E. J. G. Pitman  and
Jim Pitman
Jim Pitman*
Affiliation:
University of California, Berkeley
*
Statistics Department, University of California, Berkeley, 367 Evans Hall, Berkeley, CA94720‒3860, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Stable distributioncharacteristic functionregular variationinfinitely divisible distributionLévy‒Khintchine formulaEulerian integral

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 261 - 282
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aczél, J. and Dhombres, J. (1989).Functional Equations in Several Variables (Encyclopaedia Math. Appl. 31).Cambridge University Press.Google Scholar
[2] Bergström, H. (1963).Limit Theorems for Convolutions.John Wiley,New York.Google Scholar
[3] Bertoin, J. (1996).Lévy Processes(Camb. Tracts Math. 121).Cambridge University Press.Google Scholar
[4] Bingham, N. H.,Goldie, C. M. and Teugels, J. L. (1989).Regular Variation (Encyclopaedia Math. Appl. 27), revised edn.Cambridge University Press.Google Scholar
[5] Courant, R. (1936).Differential and Integral Calculus, Vol. II.Blackie,London.Google Scholar
[6] Durrett, R. (2010).Probability: Theory and Examples,4th edn.Cambridge University Press.Google Scholar
[7] Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II,2nd edn.John Wiley,New York.Google Scholar
[8] Geluk, J. L. and de Haan, L. (1987).Regular Variation, Extensions and Tauberian Theorems (CWI Tract 40).Mathematisch Centrum,Amsterdam.Google Scholar
[9] Geluk, J. L. and de Haan, L. (2000).Stable probability distributions and their domains of attraction: a direct approach.Prob. Math. Statist. 20,169–188.Google Scholar
[10] Gil-Pelaez, J. (1951).Note on the inversion theorem.Biometrika 38,481–482.Google Scholar
[11] Gnedenko, B. V. and Kolmogorov, A. N. (1954).Limit Distributions for Sums of Independent Random Variables.Addison‒Wesley,Cambridge, MA.Google Scholar
[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. 63,187–201.CrossRefGoogle Scholar
[13] Gupta, A. K.,Jagannathan, K.,Nguyen, T. T. and Shanbhag, D. N. (2006).Characterizations of stable laws via functional equations.Math. Nachr. 279,571–580.Google Scholar
[14] Hall, P. (1981).A comedy of errors: the canonical form for a stable characteristic function.Bull. London Math. Soc. 13,23–27.CrossRefGoogle Scholar
[15] Kagan, A. M.,Linnik, Yu. V. and Rao, C. R. (1973).Characterization Problems in Mathematical Statistics.John Wiley,New York.Google Scholar
[16] Kallenberg, O. (2002).Foundations of Modern Probability,2nd edn.Springer,New York.CrossRefGoogle Scholar
[17] Khintchine, A. Ya. and Lévy, P. (1936).Sur les lois stables.C. R. Acad. Sci. Paris Sér. A 202,374–376.Google Scholar
[18] Kuczma, M. (2009).An Introduction to the Theory of Functional Equations and Inequalities, 2nd edn., ed. A. Gilányi.Birkhäuser,Basel.Google Scholar
[19] Lévy, P. (1924).Théorie des erreurs. La loi de Gauss et les lois exceptionnelles.Bull. Soc. Math. France 52,49–85.Google Scholar
[20] Lévy, P. (1954).Théorie de l'addition des variables aléatoires,2nd edn.Gauthier-Villars,Paris.Google Scholar
[21] Mijnheer, J. L. (1975).Sample Path Properties of Stable Processes(CWI Tract 59).Mathematisch Centrum,Amsterdam.Google Scholar
[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. 8,423–443.CrossRefGoogle Scholar
[23] Ramachandran, B. and Lau, K.-S. (1991).Functional Equations in Probability Theory.Academic Press,Boston, MA.Google Scholar
[24] Samorodnitsky, G. and Taqqu, M. S. (1994).Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance.Chapman & Hall/CRC,New York.Google Scholar
[25] Sato, K.-I. (1999).Lévy Processes and Infinitely Divisible Distributions(Camb. Stud. Adv. Math. 68).Cambridge University Press.Google Scholar
[26] Seneta, E. (2002).Karamata's characterization theorem, Feller, and regular variation in probability theory.Publ. Inst. Math. (Beograd) (N.S.) 71(85),79–89.Google Scholar
[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. 49,166–179,477–495,612–628.Google Scholar
[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.Google Scholar
[29] Uchaikin, V. V. and Zolotarev, V. M. (1999).Chance and Stability: Stable Distributions and Their Applications.VSP,Utrecht.Google Scholar
[30] Von Bahr, B. (1965).On the convergence of moments in the central limit theorem.Ann. Math. Statist. 36,808–818.Google Scholar
[31] Wendel, J. G. (1961).The non-absolute convergence of Gil-Pelaez' inversion integral.Ann. Math. Statist. 32,338–339.CrossRefGoogle Scholar
[32] Whittaker, E. T. and Watson, G. N. (1927).A Course of Modern Analysis,4th edn.Cambridge University Press.Google Scholar
[33] Zolotarev, V. M. (1986).One-Dimensional Stable Distributions(Transl. Math. Monogr. 65).American Mathematical Society,Providence, RI.Google Scholar