Almost sure limit points of record values

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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 {X 1, …, Xj- 1} The almost sure behavior of the sequence of record values {XL n } is studied. Sufficient conditions are given for lim sup n→∞ X Ln /R –l(n)=e c, lim inf n → ∞ X Ln /R −1 (n) = e −c, a.s., 0 ≦ c ≦ ∞, and also for lim sup n→∞ (XLn—R– 1(n))/an =1, lim inf n→∞ (XLn—R– 1(n))/an = − 1, a.s., for suitably chosen constants an . The a.s. behavior of {XL n } is compared to that of the sequence {Mn }, where Mn = max {X 1, …, 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.

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