On the relative stability of large order statistics

Abstract

AbstractLet X1, X2, … be independent and identically distributed (i.i.d.) random variables and let Xn, r denote the rth largest of X1, X2, …, Xn (so that Xn, r is the (n − r + l)th-order statistic of X1, X2, …, Xn). It is well known that if Xn, l/cn→1 in probability or with probability 1, for some sequence of constants cn, then Xn, r/cn→1 for each r ≥ 1. Therefore if r(n) → ∞ sufficiently slowly, Xn, r(n)/cn→1 for the same sequence of constants cn. In this paper we study behaviour of this type in considerable detail. We find necessary and sufficient conditions on the rate of increase of r(n), n ≤ 1, for the limit theorem Xn, r(n)/cn→1 to hold, and we investigate the rate of convergence in terms of a central limit theorem and a law of the iterated logarithm (LIL). The LIL takes a particularly interesting form, and there are five distinctly different modes of behaviour.