Asymptotic Behavior of Intermediate Order Statistics: The Infinite Endpoint Case.

摘要

Suppose X1, X2, ... is a sequence of independent and identically distributed random variables with marginal distribution function F(x) = P(X1 0 for all real x. Let X(k sub n) superscript n denote the (k sub n)th smallest order statistic of the sample X1, ..., Xn, where k sub n/n approaches 0 as n approaches infinity. An almost sure representation of X(k sub n) superscript n in terms of the empirical distribution function is established. The conditions imposed upon F include those under which it is known that X(k sub n) superscript n is asymptotically normal. From the representation the law of the iterated logarithm for X(k sub n) superscript n is obtained. Examples illustrating the general result are presented. (Author)