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To prove (4.18), note that j d
d R(1)
m 0 ( )j C log2 m 0 (see the proof of (5.12)). Then, by (4.19)
p2m C(n=m 0 ) m ;1=2 jUmj log 2 m 0 Cm ; m
since by Lemma 5.7, Um log 2 m 0 = m 1=2 m m log 2 m 0 = m log 2 m 0 2X, bearing in mind that under
Assumption m, m = O(m ;1=2 ). Thus (4.18) holds and the proof for b `1 is completed.
The proof for b `2 follows on showing that dim(b) = bci ; ci ! 0, a.s. i = 1 2. As before it
su ces to show that, for all >0
1X
m=1
P fjdim(b)j g < 1 i =0 1: (4.20)
By Lemma 5.8 with k = p =2,P fjb ; 0j (log n) ;2 g = o(m ;2 ) it su ces to prove (4.20) in case
jb ; 0j (log n) ;2 . Similarly to the proof of Lemma 3.1 it remains to show that
for some >0where m 2X. By the mean value theorem:
jdim(b)j = o(1) + m ; m i =1 2 (4.21)
dim(b) =dim( 0)+(b ; 0) d
d dim( )
where j ; 0j jb ; 0] (log m) ;2 . We showthatasn !1for i =1 2,
Edim( 0) =o(1) (4.22)
p 0 im := dim( 0) ; Edim( 0) =m ; m (4.23)
p 00
im := jb ; 0jj d
d dim( )j = m ; m (4.24)
which yield (4.21).
First, (4.22) follows approximating sums by integrals and Lemma 7.1. We have
jp 0 1m j C m0;1=2 jZ0m 0( 0)j +(m 0 =n) ; m 0;1=2 jZ0m 0( 0 + )j
p 0 2m C (m 0 =n) ; m 0;1=2 jZ0m 0( 0)j +(m 0 =n) ;2 m 0;1=2 jZ0m 0( 0 + )j :
By (3.11), (m 0 =n) ; m 0;1=2 m ; ,andby Lemma 5.3, Z0m 0( 0) 2X. Using Lemma 7.3 it is easy to
show that Ej(m 0 =n) ; Z0m 0( 0 + )j k < 1 as n !1for any k 1, so (m 0 =n) ; Z0m 0( 0 + ) 2X,
to imply (4.23). It remains to show (4.24). Since j ; 0j log ;2 m, it is easy to see that
Thus
j d
d S0m0( )j C(log m)S0m0( 0) (m 0 ; d
=n)
d S0m0( + ) C(log m)S0m0( 0):
p 00
im Cjbm ; 0j(m 0 =n) ; log mS0m 0( 0) C(m 0 =n) ; m ;1=2 jUmj log mS0m 0( 0):
Since under Assumption m, m = O(m ;1=2 ), from (4.19) and Lemma 5.7 it follows that
(m 0 =n) ; m ;1=2 jUmj log m = m ; m log m 2X:
This and S0m 0( 0) 2X imply that p 00
im = m ; m.
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