Limit laws for record values

Sidney I. Resnick

Stochastic Processes and their Applications, 1973, vol. 1, issue 1, 67-82

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.

Date: 1973
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