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254 P.K. SEN whenever N and n are large. The regularity conditions underlying the Wald- Wolfowitz PCLT were rather stringent, and later on, these were relaxed by Noether [23], Hoeffding [16], Motoo [21], and others. The final say in this context is due to Hajek [6]. We present this evolutionary picture in a proper perspective. Let H N = {«jVi, • • •, a NN } and TN = {bNi, • • •, b NN } be two sequences of real numbers, and define a N , b N , A N and B N as in (2.9) and earlier. Then a linear permutation statistic is defined as LN = 2_j bmaNRi, (2.12) i
The Hajek Asymptotics for Finite Population Sampling and Their Ramifications 255 way (as is usually done in nonparametrics). Let U N: \ < • • • < U N:N be the order statistics corresponding to Ui,..., U N , and define a* Ni =E(U N:i ), \ 1). Also, let ,i , N A 2 = 2 (u)du ( 1. Then, so that E(L° N )\B N ) = L* N , VN>1, (2.20) E(L* N - L^) 2 = E(L% 2 ) - E(L* N 2 ) = NB 2 N(A 2 - A* N 2 ), (2.21) and it is easy to verify that A* N is | in N with lini/v—oo A* N = A 2 . Therefore, (2.17) follows readily from (2.21). Moreover, note that for an arbitrary U N , we have E VN (L N - L* N ) 2 /E VN L* 2 < ( £>„, - a* Ni ] 2 ) / ( ]T a& ) , (2.22) \i
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