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Limit laws for record values

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  • Resnick, Sidney I.

Abstract

{Xn,n[greater-or-equal, slanted]1} are i.i.d. random variables with continuous d.f. F(x). Xj is a record value of this sequence if Xj>max{X1,...,Xj-1}. Consider the sequence of such record values {XLn,n[greater-or-equal, slanted]1}. Set R(x)=-log(1-F(x)). There exist Bn > 0 such that XLn/Bn-->1. in probability (i.p.) iff XLn/R-1(n)-->1 i.p. iff {R(kx)-R(x)}/R1/2(kx) --> [infinity] as x-->[infinity] for all k>1. Similar criteria hold for the existence of constants An such that XLn-An --> 0 i.p. Limiting record value distributions are of the form N(-log(-logG(x))) where G(·) is an extreme value distribution and N(·) is the standard normal distribution. Domain of attraction criteria for each of the three types of limit laws can be derived by appealing to a duality theorem relating the limiting record value distributions to the extreme value distributions. Repeated use is made of the following lemma: If P{Xn[less-than-or-equals, slant]x}=1-e-x,x[greater-or-equal, slanted]0, then XLn=Y0+...+Yn where the Yj's are i.i.d. and P{Yj[less-than-or-equals, slant]x}=1-e-x.

Suggested Citation

  • Resnick, Sidney I., 1973. "Limit laws for record values," Stochastic Processes and their Applications, Elsevier, vol. 1(1), pages 67-82, January.
  • Handle: RePEc:eee:spapps:v:1:y:1973:i:1:p:67-82
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    Cited by:

    1. L. Gajek & U. Gather, 1991. "Moment inequalities for order statistics with applications to characterizations of distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 38(1), pages 357-367, December.
    2. Barakat, H.M. & Abd Elgawad, M.A., 2017. "Asymptotic behavior of the joint record values, with applications," Statistics & Probability Letters, Elsevier, vol. 124(C), pages 13-21.
    3. Nayak, S. S. & Zalki, Madhusudhan, 2001. "Almost sure limit points of record values from two independent populations," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 181-187, January.
    4. Mohammad Raqab, 2009. "Distribution-free prediction intervals for the future current record statistics," Statistical Papers, Springer, vol. 50(2), pages 429-439, March.
    5. M. Emadi & J. Ahmadi & N. Arghami, 2007. "Comparison of record data and random observations based on statistical evidence," Statistical Papers, Springer, vol. 48(1), pages 1-21, January.
    6. M. A. Abd Elgawad & H. M. Barakat & Ting Yan, 2020. "Bivariate Limit Theorems for Record Values Based on Random Sample Sizes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(1), pages 50-67, February.

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