Probability Theory and Related Fields

, Volume 77, Issue 4, pp 567–581 | Cite as

The structure of the class of subexponential distributions

  • Eric Willekens
Article

Summary

LetX1,X2, ...,X n be a sequence of positive, independent, identically distributed random variables with the same distribution function (d.f.)F and denote byX1:nX2:n≦...≦Xn:n the order statistics of the sample. We characterize the class of d.f.F for which
$$P(X_{1:n} + X_{2:n} + \ldots + X_{n - i:n} > x) \sim P(X_{n - i:n} > x) as x \to \infty $$
for fixedn andi (i≦n-1), and we show that it is independent ofn. This leads to the genesis of a new class of d.f.L i ; we show that the sequence (L i ) i =0 is strictly decreasing and we illustrate how the classesL i determine the probabilistic structure of the classL of subexponential distributions.

Keywords

Distribution Function Stochastic Process Probability Theory Order Statistic Mathematical Biology 

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References

  1. 1.
    Arov, D.Z., Bobrov, A.A.: The extreme terms of a sample and their role in the sum of independent variables. Theory Probab. Appl.5, 377–395 (1960)Google Scholar
  2. 2.
    Athreya, K.B., Ney P.: Branching processes, Berlin Heidelberg New York. Springer 1972Google Scholar
  3. 3.
    Chistyakov, V.P.: A theorem on sums of independent positive random variables and its applications to branching processes. Theory Probab. Appl.9, 640–648 (1964)Google Scholar
  4. 4.
    Chover, J., Ney, P., Wainger, S.: Degeneracy properties of subcritical branching processes. Ann. Probab.1, 663–673 (1973)Google Scholar
  5. 5.
    Cline, D.B.H.: Convolution tails, product tails and domains of attraction. Probab. Th. Rel. Fields72, 529–557 (1986)Google Scholar
  6. 6.
    de Haan, L.: On regular variation and its applications to the weak convergence of sample extremes. Amsterdam: Mathematical Centre Tracts 1970Google Scholar
  7. 7.
    Embrechts, P.: Subexponential distribution functions and their applications. A review. Proc. 7th Brasov Conf. Probability Theory, pp. 125–136 (1985)Google Scholar
  8. 8.
    Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 335–347 (1979)Google Scholar
  9. 9.
    Embrechts, P., Goldie, C.M.: On convolution tails. Stochastic Processes Appl.13, 263–279 (1981)Google Scholar
  10. 10.
    Embrechts, P., Omey, E.: A property of long tailed distributions. J. Appl. Probab.21, 80–87 (1984)Google Scholar
  11. 11.
    Goldie, C.M.: Subexponential distributions and dominated variation tails. J. Appl. Probab.15, 440–442 (1978)Google Scholar
  12. 12.
    Maller, R.A., Resnick, S.I.: Limiting behaviour of sums and the term of maximum modulus. Proc. Lond. Math. Soc. (3)49, 385–422 (1984)Google Scholar
  13. 13.
    Omey, E., Willekens, E.: Second order behaviour of the tail of a subordinated probability distribution. Stochastic Processes Appl.21, 339–353 (1986)Google Scholar
  14. 14.
    Pitman, E.J.G.: Subexponential distribution functions. J. Austral. Math. Soc. (Series A)29, 337–347 (1980)Google Scholar
  15. 15.
    Senata, E.: Regularly varying functions. Lect. Notes Math. vol. 508. Berlin Heidelberg New York: Springer 1976Google Scholar
  16. 16.
    Teugels, J.L.: The class of subexponential distributions. Ann. Probab.3, 1000–1011 (1975)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Eric Willekens
    • 1
  1. 1.Departement WiskundeKatholieke Universiteit LeuvenHeverleeBelgium

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