Complete stability of large order statistics

Fix an integerr1. For eachnr, letM_nr be the rth largest ofX₁,...,X_n, where {X_n,n1} is a sequence of i.i.d. random variables. Necessary and sufficient conditions are given for the convergence of _n=rnP[|M_nr/a_n–1|<] for every >0, where {a_n} is a real sequence and –1. Moreover, it is shown that if this series converges for somer1 and some >–1, then it converges for everyr1 and every >–1.