The Appropriate Reduction of the Weighted Empirical Process

Us a new transformation that the weighted This is shown to arise inequality of Marcus and to reduce dow~ to [0,1], it is shown processes [0,1] are always t a discussion of a new 1. ras PROCESS Let xn 1, ... , Xnn be independent with completely arbitrary df's Fn 1, ... ,Fnn' Let Cn 1> •.• ,Cnn denote known constants satisfying :Er cn~ = 1. Then is called the weighted empirical process. for 00 < x < 00 ( 1.1) Let Xn1, ... ,Xnn be the associated continuous rv's of van Zuijlen (1978), and let Hn t, ... ,Hnn denote their df's. Then (1.2) = n where c; == :E cn~ Gni = I t (1.3) for 0 ~ t ~ 1 a.s., (1.4) [Xni~X]=[Pni~H:(X)]=[~ni~Gni(Hn(x))] for -oo 7l on (D, J) ,I /q[i), will be taken to mean that Skorokhod constructions of these processes (also labeled tln and 7l) satisfy 1(71n 7l)/q II -l'p 0 as n -l' 00. The statement that tln is weakly compact on (D, /q II) is taken to mean that every subsequence ri' has a further subsequence n" for which 7ln" => (some 7l) on (D, ,1), /q). THEOREM 1, Suppose q E Q and the Cni'S are u.a.n. Then (i) 7Ln. is necessarily weakly compact on (D, , I /q ). (Ii) Any possible limit process 7l must satisfy P(71/q E:C) = 1. condition that ,t as n -l' 00 for 0 ~ s t ~ is necessary wid sufficient for to ",,,,t",,,,!v =