On the limiting distribution of fractional parts of extreme order statistics

Let X_2, X_2, ... be a sequence of independent and identically distributed random variables with distribution function F, and suppose that F belongs to the domain of attraction of an extreme value distribution. We give (additional) sufficient conditions on F for the fractional part of max (X_1, X_2, ..., X_n) to converge in distribution to a random variable with a uniform distribution on [0,1), or to diverge in distribution as n -> infinity. Key Words & Phrases: distribution (modulo 1), extreme value theory, Fourier-Stieltjes coefficients, fractional part.