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Lemma 5.4 applied to S0m 0( 0) ;1 implies that
whereas by Lemma 7.2 and Lemma 5.3
S0m 0( 0) ;1 =2; S0m 0( 0)+o(tm 0)+m0;1=2
S0m0( 0) =ES0m0( 0)+m 0;1=2
Z0m0( 0) =1+atm0 + o(tm0)+m0;1=2 Since (m=m 0 ) 2 tm 0 = tm we get
d1m = m 1=2 (m=m 0 ) 2
tm0 + vt2 0 m + o(t2m
0)+m0;1=2
= p mtm +(v ; a) p 1+
mtmtm0 + o(km)+vm for some >0. Thus (4.29) holds.
We now estimate d2m: By Lemma 5.4 and Lemma 7.2,
m
m
m:
m 1 ; atm0 + o(tm0)+m0;1=2 R (2)
m 0 ( 0) (S2m 0( 0)S0m 0( 0) ; S1m 0( 0) 2 )S0m 0( 0) ;2 =1+O((m 0 =n) + m 0;1=2 ) m
and by (4.25)-(4.26),
Um = p m(b ; 0) =; p mtm + Vm + o(rm)+O(m ;1=2 ml)+ 1+
m
From (4.8), Lemma 5.3 and (4.32) it follows that Vm = ;Z1 + e m m. Thus Um = ; p mtm ; Z1 +
o(rm)+O(m ;1=2 ml)+e m m and
d2m = vm(; p mtm ; Z1 + o(rm)+O(m ;1=2 ml)+e m m)(1 + e m m)
= ; p mtmvm ; vmZ1 + o(km + vm)+v 1+
m
since e m v m for some >0. This proves (4.30).
Finally, note that
jd3mj vmm ;1=2 U 2
mjR (3)
1=2 2
0 m ( )j = vmm
m 2
m
mjR (3)
0 m ( )j
by Lemma 5.7. Similarly to (5.12) we can show that jR (3)
m 0 ( )j C(log m0 ) 4 : Since under (3.18)
m 1=2 2 m v m for some >0we get jd3mj v 1+
m
5 Approximation lemmas
m: Thus (4.31) holds.
To characterize negligible terms of our the expansions we shall use the following version of Chibisov's
(1972) Theorem 2, which we present without proof.
Lemma 5.1 Let Ym = Vm + 2 m m, where m ! 0 as m ! 0,
P (j mj
and Vm has the asymptotic expansion
;
m )=o( m) some 0 <