Skip to main content
SpringerLink
Log in

Strong laws for exponential order statistics and spacings

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

We obtain upper and lower almost sure asymptotic bounds for spacings η K, N − η K + 1, N , where η 1, N > η 2, N > ⋯ > η N, N are the order statistics of independent exponentially distributed random variables η 1, η 2, ..., η N with mean 1; here N → ∞, and K = 1, 2, ... is fixed.

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. A. Astrauskas, Limit theorems for the maximal eigenvalues of the mean field Hamiltonian with random potential, Lith. Math. J., 39(2), 117–133 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Astrauskas, Extremal theory for spectrum of random discrete Schrödinger operator, Preprint (submitted for publication).

  3. A. Astrauskas and S. A. Molchanov, The ground state of a random stationary medium in the mean field approximation, in: New Trends in Probability and Statistics, Vol. 1, V. V. Sazonov and T. L. Shervashidze (Eds.), VSP, Utrecht/Mokslas, Vilnius (1991), pp. 668–682.

    Google Scholar 

  4. A. Astrauskas and S. A. Molchanov, Limit theorems for the ground states of the Anderson model, Functional Anal. Appl., 26, 305–307 (1992).

    Article  MathSciNet  Google Scholar 

  5. O. Barndorff-Nielsen, On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables, Math. Scand., 9, 383–394 (1961).

    MATH  MathSciNet  Google Scholar 

  6. K. L. Chung and P. Erdös, On the application of the Borel-Cantelli lemma, Trans. Amer. Math. Soc., 72, 179–186 (1952).

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Deheuvels, Upper bounds for the kth maximal spacings, Z. Wahrsch. verw. Geb., 62, 465–474 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Deheuvels, Strong limit theorems for maximal spacings from a general univariate distribution, Ann. Probab., 12, 1181–1193 (1984).

    MATH  MathSciNet  Google Scholar 

  9. P. Deheuvels, The limiting behaviour of the maximal spacing generated by an i.i.d. sequence of Gaussian random variables, J. Appl. Probab., 22, 816–827 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Deheuvels, Strong laws for the kth order statistics when k ⩽ c log2 n, Probab. Theory Related Fields, 72, 133–154 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Deheuvels, Strong laws for the kth order statistics when k ⩽ c log2 n. II, in: Extreme Value Theory, J. Hüsler and R. Reiss (Eds.), Lect. Notes in Statist., Vol. 51, Springer, Berlin (1989), pp. 21–35.

    Google Scholar 

  12. P. Deheuvels and D. M. Mason, The asymptotic behaviour of sums of exponential extreme values, Bull. Sci. Math. (2), 112, 211–233 (1988).

    MATH  MathSciNet  Google Scholar 

  13. P. Deheuvels, E. Haeusler, and D. M. Mason, On the almost sure behaviour of sums of extreme values from a distributionin the domain of attraction of a Gumbel law, Bull. Sci. Math. (2), 114, 61–95 (1990).

    MATH  MathSciNet  Google Scholar 

  14. L. Devroye, Laws of the iterated logarithm for order statistics of uniform spacings, Ann. Probab., 9, 860–867 (1981).

    MATH  MathSciNet  Google Scholar 

  15. L. Devroye, A loglog law for maximal uniform spacings, Ann. Probab., 10, 863–868 (1982).

    MATH  MathSciNet  Google Scholar 

  16. L. Devroye, Upper and lower class sequences for minimal uniform spacings, Z. Wahrsch. verw. Geb., 61, 237–254 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  17. L. Devroye, The largest exponential spacing, Utilitas Math., 25, 303–313 (1984).

    MATH  MathSciNet  Google Scholar 

  18. J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley, New York (1978).

    MATH  Google Scholar 

  19. J. Gärtner and W. König, The parabolic Anderson model, in: Interacting Stohastic Systems, J.-D. Deuschel and A. Greven (Eds.), Springer, Berlin (2005), pp. 153–179.

    Chapter  Google Scholar 

  20. J. Gärtner and S. A. Molchanov, Parabolic problems for the Anderson Hamiltonian. II. Second-order asymptotics and structure of high peaks, Probab. Theory Related Fields, 111, 17–55 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Kiefer, Iterated logarithm analogues for sample quantiles when p n ↓ 0, in: Proc. Sixth Berkeley Sympos. Math. Statist. Probab., Vol. 1, Univ. Calif. Press, Los Angeles (1972), pp. 227–244.

    Google Scholar 

  22. M. J. Klass, The minimal growth rate of partial minima, Ann. Probab., 12, 380–389 (1984).

    MATH  MathSciNet  Google Scholar 

  23. H. Robbins and D. Siegmund, On the law of the iterated logarithm for maxima and minima, in: Proc. Sixth Berkeley Sympos. Math. Statist. Probab., Vol. 3, Univ. Calif. Press, Los Angeles (1972), pp. 51–70.

    Google Scholar 

  24. A. N. Shiryaev, Probability, 2nd ed., Springer, New York (1996).

    Google Scholar 

  25. G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics, Wiley, New York (1986).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Akademijos 4, LT-08663, Vilnius, Lithuania

    A. Astrauskas

  2. Vilnius Gediminas Technical University, Saulėtekio 11, LT-10223, Vilnius, Lithuania

    A. Astrauskas

Authors
  1. A. Astrauskas
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 477–491, October–December, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Astrauskas, A. Strong laws for exponential order statistics and spacings. Lith Math J 46, 385–397 (2006). https://doi.org/10.1007/s10986-006-0036-9

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-006-0036-9

Keywords

Search

Navigation