ABSTRACT
For the choices s 11 .J = B111 = B;(B,, U-lt'11 (V)) and s111 = Bn,J = B1( B,, U - 11'11 ( V) ), Eqns (SO)-(S3) are easily verified. Let us now verify Eq. (T). In view of Eqns (5.2), (8.4) and lliP11 IIo = 0(1), one finds
I II "" ' ~ - (. -l/2) N L....., Sn,•ln.j-Opo, ll
Butthisfollowsas
Ee,(T,7I7J11 ,•••,7)11 ,11 ,V,,...,V11 )=N 2LALrPn,j-iPn,j) ni=lj=d,vi
Theaboveshowthat I11
NowletS11 .j=z:=;::an.j-J'Y(Vi)- ) 11 (Vi))andS11 .i=0,wherethenumberset11.iaresuchthatI:;'=Ilall.il=0(I).ThenonecanagainverifyEq.(T) andEqns(SO)-(S3).Takingan.i=p,;,-landthenan.i=p11 (i-l)tL;,-2 - ip,;,-l, onegets
withA11 .J=Ai(B11 ,)11 )andA 11 ,J=Ai(B11 ,"'f). Finally,letusshowthat III
(8.9)
InviewofEq.(8.6)andthefactthatA 11••=Op""(n-112 ),oneneedsonly showthat
SinceeachcoordinateofA11 ,;canbewrittenas2::~:~1Jn.tJx11 .j-kwhere EZ=Iian,ki=0(1),itsufficestoprove
(8.10)
where II
UsingthedefinitionofL 11 ,iandLemmas10.1,10.2and10.5ofSchick(1993) onefindsthattherearerandomvariablesL11 .JandLn.J.ithatarenotbased on1Jn.Jand(1J11 .J,1Jn,i),respectively,suchthat
Il-l L Ee,(ILII,}- L,l./)I2IZ11 ) ~ 0, i=l
The desired result in Eq. (8.10) now follows from these and an argument sjmilar to tl!e one given in Lemma 10.3 of Schick (1993) with lin.J(Y) = yLn,J• h11,j(y) = yL11 ,; and r 11 = 0.
