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Almost sure limit points of record values

Published online by Cambridge University Press:  14 July 2016

Laurens De Haan and
Sidney I. Resnick
Laurens De Haan*
Affiliation:
Stanford University
Sidney I. Resnick
Affiliation:
Stanford University
*
*Now at Erasmus University, Rotterdam.
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Abstract

{Xn, n ≧ 1} are i.i.d. unbounded random variables with continuous d.f. F(x) =1 —e –R(x). Xj is a record value of this sequence if Xj >max {X1, …, Xj-1} The almost sure behavior of the sequence of record values {XLn} is studied. Sufficient conditions are given for lim supn→∞XLn/R–l(n)=ec, lim inf n → ∞XLn/R−1 (n) = e−c, a.s., 0 ≦ c ≦ ∞, and also for lim supn→∞ (XLn—R–1(n))/an =1, lim infn→∞ (XLn—R–1(n))/an = − 1, a.s., for suitably chosen constants an. The a.s. behavior of {XLn} is compared to that of the sequence {Mn}, where Mn = max {X1, …, Xn}. The method is to translate results for the case where the Xn's are exponential to the general case by means of an extended theory of regular variation.

Keywords

RECORD VALUESMAXIMAEXTREME VALUESREGULAR VARIATIONITERATED LOGARITHMA.S. LIMIT POINTS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973, pp. 528 - 542
Copyright
Copyright © Applied Probability Trust 1973 

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Footnotes

Research supported by N.S.F. Grant GP-30711X.

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