Expansions for Quantiles and Multivariate Moments of Extremes for Heavy Tailed Distributions

Abstract

(...)The study of the asymptotes of the moments of Xn,r has been of considerable interest. McCord [12] gave a first approximation to the moments of X_n,1 for three classes. This showed that a moment of X_n,1 can behave like any positive power of n or n₁ = log n. (Here, log is to the base e.) Pickands [15] explored the conditions under which various moments of (X_n,1 − b_n) /a_n converge to the corresponding moments of the extreme value distribution. It was proved that this is indeed true for all F in the domain of attraction of an extreme value distribution provided that the moments are finite for sufficiently large n. Nair [13] investigated the limiting behavior of the distribution and the moments of X_n,1 for large n when F is the standard normal distribution function. (...)