Advertisement

Extremes

, Volume 1, Issue 3, pp 351–363 | Cite as

The Asymptotic Distributions of Sums of Records

  • Barry C. Arnold
  • Jose´ A. Villasen~or
Article

Abstract

LetX1,X2... be a sequence of i.i.d. random variables and let \(X^{\left( 0 \right)} \left( { = X_1 } \right),X^{\left( 1 \right)} ,X^{\left( 2 \right)} ...\) be the associated (upper) record sequence. Resnick (1973) identified the class of all possible limit distributions for \(X^{\left( n \right)} - b_n /a_n\). Here we focus on sums of records, \(T_n = \sum\nolimits_{j = 0}^n {X^{\left( j \right)} }\) . We describe three cases in which Tn can be normalized to have a non-trivial limiting distribution. The problem of identifying all possible limit laws for normalized sums of records remains open.

Beta records Gumbel records exponential records infinitely divisible distributions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Breiman, L., Probability, Addison-Wesley, Reading, Massachusetts, 1968.Google Scholar
  2. [2]
    Chung, K.L., A Course in Probability Theory, 2nd Ed., Academic Press, New York, 1974.Google Scholar
  3. [3]
    Darling, D.A., “Problem #10436,” American Mathematical Monthly 102, 170, (1995).Google Scholar
  4. [4]
    Feller, W., An Introduction to Probability Theory and Its Applications, Volume II, 2nd Ed., Wiley, New York, 1971.Google Scholar
  5. [5]
    Resnick, S.I., “Limit laws for record values,” Stochastic Processes and their Applications 1, 67–82, (1973).Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Barry C. Arnold
    • 1
  • Jose´ A. Villasen~or
    • 2
  1. 1.University of CaliforniaRiversideUSA
  2. 2.Colegio de PostgraduadosMontecilloMexico

Personalised recommendations