Skip to main content
SpringerLink
Log in

Inversion of Analytic Characteristic Functions and Infinite Convolutions of Exponential and Laplace Densities

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized Dirichlet series, which in turn is an infinite linear combination of exponential or Laplace densities. These results are applied to several examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Biane, P., Pitman, J., Yor, M.: Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Am. Math. Soc. (N.S.) 38(4), 435–465 (2001). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bondesson, L.: Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics, vol. 76. Springer, New York (1992)

    Book  MATH  Google Scholar 

  3. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Probability and Its Applications. Birkhäuser, Basel (2002)

    Book  MATH  Google Scholar 

  4. Ciesielski, Z., Taylor, S.J.: First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Am. Math. Soc. 103, 434–450 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  5. del Baño Rollin, S., Ferreiro-Castilla, A., Utzet, F.: On the density of log–spot in Heston volatility model. Stoch. Process. Appl. 120, 2037–2063 (2010)

    Article  MATH  Google Scholar 

  6. Dugué, D.: Sur les lois de Kolmogoroff et de van Mises. C. R. Acad. Sci. Paris Sér. A–B 262, A999–A1000 (1966)

    Google Scholar 

  7. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1966)

    MATH  Google Scholar 

  8. Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  9. Jessen, B., Wintner, A.: Distribution functions and the Riemann zeta function. Trans. Am. Math. Soc. 38(1), 48–88 (1935)

    Article  MathSciNet  Google Scholar 

  10. Kac, M.: On some connections between probability theory and differential and integral equations. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles, 1950, pp. 189–215. University of California Press, Berkeley (1951)

    Google Scholar 

  11. Katori, M., Tanemura, H.: Zeros of airy function and relaxation process. J. Stat. Phys. 136(6), 1177–1204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kent, J.: Some probabilistic properties of Bessel functions. Ann. Probab. 6(5), 760–770 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Levin, B.Y.: Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150. American Mathematical Society, Providence (1996). In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko

    MATH  Google Scholar 

  14. Lévy, P.: Wiener’s random function, and other Laplacian random functions. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles, 1950, pp. 171–187. University of California Press, Berkeley (1951)

    Google Scholar 

  15. Lukacs, E.: Characteristic Functions, 2nd end. Hafner, New York (1970)

    MATH  Google Scholar 

  16. Mandelbrojt, S.: Séries de Dirichlet. Principes et Méthodes. Monographies Internationales de Mathématiques Modernes, vol. 11. Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  17. Pitman, J., Yor, M.: Infinitely divisible laws associated with hyperbolic functions. Can. J. Math. 55(2), 292–330 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293. Springer, Berlin (1999)

    MATH  Google Scholar 

  19. Schilling, R., Song, L., Vondraček, Z.: Bernstein Functions, Theory and Applications. de Gruyter Studies in Mathematics, vol. 37. de Gruyter, Berlin (2010)

    MATH  Google Scholar 

  20. Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, London (2002)

    Google Scholar 

  21. Wintner, A.: On the differentiation of infinite convolutions. Am. J. Math. 57(2), 363–366 (1935)

    Article  MathSciNet  Google Scholar 

  22. Wintner, A.: On the densities of infinite convolutions. Am. J. Math. 59(2), 376–378 (1937)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre de Recerca Matemàtica, Apartat 50, 08193, Bellaterra (Barcelona), Spain

    Albert Ferreiro-Castilla

  2. Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

    Frederic Utzet

Authors
  1. Albert Ferreiro-Castilla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frederic Utzet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frederic Utzet.

Additional information

The authors were partially supported by grant MTM2009-08869 from the Ministerio de Ciencia e Innovacion and FEDER. A. Ferreiro-Castilla was also supported by a Ph.D. grant of the Centre de Recerca Matemàtica.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ferreiro-Castilla, A., Utzet, F. Inversion of Analytic Characteristic Functions and Infinite Convolutions of Exponential and Laplace Densities. J Theor Probab 25, 205–230 (2012). https://doi.org/10.1007/s10959-010-0313-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-010-0313-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation