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2005,20(4):416422
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up |Sk|≥ α)≤ A 2 .
k≥ m k
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*
*
*
Pr
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or
e
m 2.1.De
not
eΔτ = mi
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f
i
ne
j -τ
j
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ot
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r
mi
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i
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ej
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k
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k
k
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'
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nj
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i
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hef
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l
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nuquant
i
t
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s
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Jn(m
)= Qn(m
)- Qn(m
),
r
Kn(m
)=
nk,knk,k+ 1 2
1
p
k,
nΣ
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k= 1
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nk,k Sk,k Sk,k+ 1
1
nk,k+ 1
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r
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whe
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e,f
or1≤ i≤ j≤ r+ 1,Si,j=
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t
j
-
nk+ 1,k+ 1 Sk+ 1,k+ 1 Sk,k+ 1
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2
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nut
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s
enot
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t
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i
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c
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'
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.(3.9)
Σ
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m Cn
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WCLP s
u
p
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t
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g
Ln
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]
^ ]
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xt
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ee
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s
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t
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ons
t
ant
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hatdonotde
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tT)
s
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hatf
oral
lnWS,
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V
j
l C3\
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P max max
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p
W
C
Q∈ T t
∈ HI
Q,QN S
T) n
J,K,n(
kgW J\X V
g
m JC
]
^
g
P]max
max ‘UQ,QN S‘W C^L P]max max
XV
Q∈ T t
∈ HI
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J,K,n(
Q_ T 0L g
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h
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(3.SV)
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3.S3)
V. (
CV\
The
nby(3.S0)-(3.S3)behaqef
oranyJ>0,
P(t
n∈
HI
V
S
rN
VN
J
n\
]
)L KK
J,K,n
\
\
V
] ] ^^^,
V
.Thust
bhe
r
eKKst i
sac
ons
t
ant
hec
onc
l
us
i
oni
sobt
ai
ne
dbhe
nn\ut andJut.
vw xy
z{
|}{
~{
n}{
nc
{as
s
um~y
i
ons
;M∈ ZN }i
,
Suppos
e {ε
sas
t
at
i
onar
ys
e
que
nc
eofr
andom qar
i
abl
e
sbi
t
h Eε
M
MR 0
V
0sEε
,t
er
om t
hepr
oofbef
i
ndt
hat(3.3)hol
dsbhe
ne
qe
r(3.V)i
st
r
ue.Thus
he
Mst.
RATEOFCONVERGENCEFOR MULTI
PLECHANGEPOI
NTSESTI
MATI
ON...4
21
.
LiYunxi
a,e
tal
c
onc
l
us
i
onofThe
or
e
m 2.1i
sal
s
ot
r
ueunde
rt
hec
ondi
t
i
on(3.2).So,t
hec
onc
l
us
i
onof
The
or
e
m 2.1i
st
r
ue,i
foneoft
hef
ol
l
owi
ngc
ondi
t
i
onsi
ss
at
i
s
f
i
e
d:
(i
){ε
;t
∈Z+ }i
samar
t
i
ngal
es
e
que
nc
e.
t
(i
){ε
;t∈ Z+ }i
i
sas
e
que
nc
eofρmi
xi
ng (orφ
mi
xi
ng)r
andom var
i
abl
e
swi
t
h
t
Σ
∞
∞
1/2
ρ(2i)< ∞(orΣ i= 1φ
(2i)< ∞).
i
=1
(i
){ε
;t
∈Z+ }i
|2+ δ<∞ andα(n)
i
i
sas
e
que
nc
eofαmi
xi
ngr
andom var
i
abl
e
swi
t
hE|ε
t
t
2+δ
.
δ
(i
;t
∈Z+ }i
.
v){ε
sas
e
que
nc
eofne
gat
i
ve
l
yas
s
oc
i
at
e
dr
andom var
i
abl
e
s
t
=O(n- θ)f
>
ors
omeδ>0andθ
;t
∈Z+ }i
.
(v){ε
sas
e
que
nc
eofas
s
oc
i
at
e
dr
andom var
i
abl
e
s
t
.I
)-(v)r
)
Pr
oof
tne
e
dst
ove
r
i
f
yt
hec
ondi
t
i
on(3.2)f
ort
hes
e
que
nc
ei
n(i
e
s
pe
c
t
i
ve
l
y:(i
1/2
)Fi
Eas
y,(i
i
r
s
tnot
i
c
eρ(n)≤φ (n).Andt
he
nwer
e
f
e
rt
o[8,9]f
or(3.2).(i
v)Wer
e
f
e
r
)ByThe
≤∞ andθ
≥pr
/(2(r
-p)),t
t
o[11].(i
i
i
or
e
m 4.1of[10]i
f2<p<r
he
n
n
E
Σε
i
+m
p
r p/r
≤ Knp/2 max(E|ε
∀n≥ 1,m ≥ 1
i
+ m|) ,
i
=1
1≤ i
≤n
andt
he
nbyat
he
or
e
m of[6]
p
r p/r
E max|ε
..+ ε
np/2 max(E|ε
∀n≥ 1,m ≥ 1.
m+ 1 +.
m+ k| ≤ K'
i
+ m|) .
1≤ k≤ n
1≤ i
≤n
2+δ
.The
n
δ
2
p 2/p
Emax|ε
..+ ε
Emax|ε
..+ ε
≤ Cn
m+ 1 +.
m+ k| ≤ (
m+ 1 +.
m+ k|)
=2+δ,andps
>pr
/2(r
-p)>
Now,c
hoos
er
uf
f
i
c
i
e
nt
l
yne
ar2s
uc
ht
hatθ
k≤ n
k≤ n
[7]
and(3.2)i
sve
r
i
f
i
e
d.(v)Ne
wmanandWr
i
ght e
s
t
abl
i
s
he
dt
hef
ol
l
owi
ngi
ne
qual
i
t
y:
2
2
..+ ε
E(ε
..+ ε
Emax(ε
1 +.
k) ≤ 2
1 +.
n),
k≤ n
whi
c
hl
e
adst
o
n- 1
n
2
2
E(ε
..+ ε
Eε
1 +.
n) = n
1+ 2
Σ Σ Cov(εi,εj)≤ 2nσ2.
i
= 1j
=i
+1
I
tf
ol
l
owst
hat
2
2
Emax(ε
..+ ε
x(ε
..+ ε
nσ2,
m+ 1 +.
m+ k) = Ema
1 +.
k) ≤ 4
k≤ n
k≤ n
t
hus(3.2)i
sve
r
i
f
i
e
d.
Re
f
e
r
e
nc
e
s
1 Ba
.Le
,JTi
,1994,15:453472.
iJ
a
s
ts
qua
r
e
se
s
t
i
ma
t
i
onofas
hi
f
ti
nl
i
ne
a
rpr
oc
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s
s
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s
meSe
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iJ
r
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ma
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i
nga
ndt
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s
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ngl
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ne
a
rmode
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swi
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hmul
t
i
pl
es
t
r
uc
t
ur
a
lc
ha
nge
s
onome
t
r
i
c
a,
1998,66:4778.
3 Ki
.A c
m T S,Ba
e
k JI
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nt
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tt
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sge
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dbyl
i
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a
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,St
,2001,51:299305.
pos
i
t
i
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l
yqua
dr
a
nt
de
pe
nde
ntpr
oc
e
s
s
a
t
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s
tPr
oba
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t
t
4 La
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l
l
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nas
e
que
nc
eofde
pe
nde
ntva
r
i
a
bl
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s
ocPr
oca
ndt
he
i
r
,1999,83:79102.
Appl
O22
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ij
k
jlih
m.nj
o
.20,Qo.O
qol
5 La
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sE,Le
a
s
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qua
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s
t
i
ma
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i
onofa
nunknownnumbe
rofs
hi
f
t
si
nat
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mes
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r
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s
,2000,21:3359.
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