Advertisement

Lithuanian Mathematical Journal

, Volume 58, Issue 4, pp 480–499 | Cite as

Semi-Heavy Tails

  • Edward OmeyEmail author
  • Stefan Van GulckEmail author
  • Rein Vesilo
Article
  • 26 Downloads

Abstract

In this paper, we study properties of functions and sequences with a semi-heavy tail, that is, functions and sequences of the form w(x) = e−βxf(x), β > 0, resp., wn = cnfn, 0 < c < 1, where the function f(x), resp., the sequence (fn), is regularly varying. Among others, we give a representation theorem and study convolution properties. The paper includes several examples and applications in probability theory.

Keywords

semi-heavy tail regular variation convolutions asymptotic behaviour subordination 

MSC

26A12 33B99 60K05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.M.P. Albin and M. Sunden, On the asymptotic behaviour of Lévy processes. Part 1: Subexponential and exponential processes, Stochastic Processes Appl., 119:281–304, 2009.MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Asmussen, Applied Probability and Queues, 2nd ed., Springer, New York, 2003.zbMATHGoogle Scholar
  3. 3.
    O.E. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance Stoch., 2:41–68, 1998.MathSciNetCrossRefGoogle Scholar
  4. 4.
    N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation, Encycl. Math. Appl., Vol. 27, Cambridge Univ. Press, Cambridge, 1987.Google Scholar
  5. 5.
    S. Borak, A. Misiorek, and R. Weron, Models for heavy tailed asset returns, in Statistical Tools for Finance and Insurance, Springer, Berlin, Heidelberg, 2011, pp. 21–55.CrossRefGoogle Scholar
  6. 6.
    J. Chover, P. Ney, and S. Wainger, Functions of probability measures, J. Anal. Math., XXVI:255–302, 1973.MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. Eberlein and U. Keller, Hyperbolic distributions in finance, Bernoulli, 1:281–299, 1995.CrossRefGoogle Scholar
  8. 8.
    P. Embrechts and E. Omey, Functions of power series, Yokohama Math. J., 32:77–88, 1984.MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 2011.Google Scholar
  10. 10.
    J.L. Geluk and L. de Haan, Regular Variation, Extensions and Tauberian Theorems, CWI Tracts, Vol. 40, Centrum voorWiskunde en Informatica, Amsterdam, 1987.Google Scholar
  11. 11.
    B.G. Hansen and E. Willekens, The generalized logarithmic series distribution, Stat. Probab. Lett., 9:311–316, 1990.MathSciNetCrossRefGoogle Scholar
  12. 12.
    C. Klüppelberg, Subexponential distributions and characterizations of related classes, Probab. Theory Relat. Fields, 82:259–269, 1989.MathSciNetCrossRefGoogle Scholar
  13. 13.
    D.V. Lindley, Fiducial distributions and Bayes’ theorem, J. R. Stat. Soc., Ser. B, 20:102–107, 1958.Google Scholar
  14. 14.
    S. Nadarajah, H.S. Bakouch, and R. Tahmasbi, A generalized Lindley distribution, Sankhyā, Ser. B, 73:331–359, 2011.MathSciNetCrossRefGoogle Scholar
  15. 15.
    N.U. Prabhu, Stochastic Storage Models: Queues, Insurance Risk, Dams, and Data Communication, 2nd ed., Springer, 1998.Google Scholar
  16. 16.
    K. Prause, The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures, PhD thesis, Freiburg University, 1999, available from: http://www.freidok.uni-freiburg.de/volltexte/15.
  17. 17.
    H. Schmidli, Compound sums and subexponentiality, Bernoulli, 5(6):999–1012, 1999.MathSciNetCrossRefGoogle Scholar
  18. 18.
    W. Schoutens, Lévy Processes in Finance, Wiley, New York, 2003.CrossRefGoogle Scholar
  19. 19.
    V. Seshadri, Halphen’s laws, in S. Kotz, C.B. Read, and D. L. Banks (Eds.), Encyclopedia of Statistical Sciences, Update Vol. 1, Wiley, New York, 1997, pp. 302–306.Google Scholar
  20. 20.
    H. Xu, M. Scheutzow, and Y. Wang, On a transformation between distributions obeying the principle of a single big jump, J. Math. Anal. Appl., 430:672–684, 2015.MathSciNetCrossRefGoogle Scholar
  21. 21.
    H. Zakerzadeh and A. Dolati, Generalized Lindley distribution, J. Math. Ext., 3(2):13–25, 2009.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Economics and Business, KU LeuvenBrusselsBelgium
  2. 2.Department of EngineeringMacquarie UniversityNorth RydeAustralia

Personalised recommendations