Kiefer’s law of the iterated logarithm for the vector of upper order statistics

Let {Xn} be a sequence of independent identically distributed random variables with a common continuous distribution function and let Mj;n denote the jth upper order statistic among X1,X2, . . . ,Xn, n ≥ j. For a large class of distributions, we obtain the law of the iterated logarithm for {M1,n,M2,n}, properly normalized. As a consequence, we establish a law of the iterated logarithm for the spacings {M1,n −M2,n}.