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Proof. Note rst that Cum(Zq1Zq2) = E[Zq1Zq2] and Cum(Zq1Zq2Zq3) = E[Zq1Zq2Zq3]: Let
a (p)
j
= m;1=2 qp
j , p =1:::k: Then Zqp = S (p)
m ; ES (p)
m . From (7.14) and (7.11)
max
l j m ja(p)
j j Cm ;1=2 log 2 m jja (p) jj2 C p =1:::k: (7.35)
By diagram formalism (see e.g. Brillinger (1975), p.21), we can write
ck := Cum(Zq1:::Zqk)= X
Q
where ; c ; denotes a subset of connected partitions =(V1:::Vk) ofthetable (7.20), T ,and
Q is given in (7.22). We show that
where ; c 0
ck = X
2; c
0
;0 denotes the subset of connected partitions and
Q 0 =
mX
2; c
Q 0 + O(e 2
m) (7.36)
kY
( a
j1:::jk=l p=1
(p)
jp )q0 V1 :::q0 Vk with q 0 Vs (jp1jp2) =1fjp 1 =jp 2 gqVs(jp1jp2) forVs =((p1v2) (p2v2)).
The derivations (7.26)-(7.30) imply that for any connected partition 2 ; c , Q = Q 0 +r where
jr j
kX
pv=1:p6=v
jjq Vp2 jj2jjq Vv2 jj2
kY
j=1:j6=pv
By (7.25), jjq Vp jj2 C: With tapering, (7.28) and (7.35) imply
Without tapering, from (7.30) it follows that
jjq Vp jj2:
jjq Vp2jj 2
2 C((m=n) 2 + m ;1 log 4 m=l ;1 ) C e 2
m:
jjq Vp2jj 2
2 C((m=n) 2 + m ;1 log 7 m) C e 2
m:
Thus r = O(e2
m): Then (7.36) follows if we showthatQ0 = O(e2
m) for 2 ; cn; c 0.
In that case has at least two dierentVpVs 2V1. By the Cauchy inequality,
jQ 0 j jjq Vp1jj2jjq Vs1jj2
kY
j=1:j6=pl
jjq Vp jj2 = O(e 2
m)
since jjq Vp jj2 C j =1:::k and, for Vp 2V1, jjq Vp1 jj 2 2 = O(e2 m): Indeed, if Vs 2V1 then
jjqVs1jj 2
2 = max
1 ij m ja(p) i a(p0 )
j j
mX
j=l
jqVs(j j)j Cm ;1 (log m) 4
mX
j=l
jqVs(j j)j:
With tapering, from (b) of Lemma 2.2 it follows that jqVs(j j)j Cj ;2 ,sojjq Vs1 jj 2
2
C e 2 m:
37
Cm;1 log 4 m=l ;1