SOME CONTRIBUTIONS TO THE THEORY OF ORDER STATISTICS PETER J. BICKEL UNIVERSITY OF CALIFORNIA, BERKELEY 1. Introduction and summary This paper arose from the problem of proving the asymptotic normality of linear combinations of order statistics which was first posed by Jung [9]. In the course of this investigation, several facts of general interest in the study of mo- ments of order statistics, which either had not been stated or had not been proved in their most satisfactory form, were established. These are collected in theorems 2.1 and 2.2 of section 2. Briefly we show in theorem 2.1 that any two order sta- tistics are positively correlated, and in theorem 2.2 we give necessary and suffi- cient conditions for the existence of moments of quantiles and the convergence of the suitably normalized moments to those of the appropriate normal distribu- tion. Section 3 contains an "invariance principle" for order statistics more elemen- tary than the one given by Hajek [7] but requiring fewer regularity conditions and adequate for our purposes in section 4. In an as yet unpublished paper, J. L. Hodges and the author give another application of this principle in deriving the asymptotic distribution of an estimate of location in the one sample problem. Section 4 contains the principal results of the paper. We consider linear com- binations of order statistics which do not involve the extreme statistics to a more significant extent than the sample mean does. For this class of statistics we estab- lish asymptotic normality and convergence of normalized moments to those of the appropriate Gaussian distribution. 2. Some properties of moments of order statistics Let Xi, * * *, Xn be a sample from a population with distribution F and density f which is continuous and strictly positive on {xlO < F(x) < 1}. Then F-1(t) is well-defined and continuous for 0 < t < 1, and for those values of t we may define 4'(t) = f[F-'(t)]. We denote by Z1j, < ... < Z,nn the order statistics of the sample. The following two theorems will be proved in this section. THEOREM 2.1. Suppose that E(Z'.) + E(Zk,n) <O. Then, co (Zi Zj,n) O. THEOREM 2.2. Suppose that limx- IxI'|[1 - F(x) + F(-x)] = 0 for some e > 0. Then, Prepared with the-partial support of the National Science Foundation Grant GP 2593. 575