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

The Asymptotic Distributions of Sums of Records

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


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 


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    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

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