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PureMathematicsnØêÆ,2021,11(11),1763-1769
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1111199
f(e)˜aMarshall.4ŒŠ
Øª
°°°ääässs§§§¾¾¾¤¤¤§§§___
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2021c924F¶¹^Fϵ2021c1027F¶uÙFϵ2021c113F
Á‡
©|^˜‡ÐØª§f(e){S
n
,n≥1}˜aMarshall.4ŒŠØª§Óž
/X{g(S
n
),n≥1}fe˜‡Marshall.4ŒŠØª"
'…c
f(e)§Marshall.Øª§4ŒŠ
AClassofMarshallTypeMaximal
InequalityforDemi(sub)martingales
YaliLu,DechengFeng,XiaLin
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Sep.24
th
,2021;accepted:Oct.27
th
,2021;published:Oct.3
rd
,2021
Abstract
Inthispaper,wegotaMarshalltypemaximalinequalityfordemi(sub)martingale
©ÙÚ^:°äs§¾¤§_.f(e)˜aMarshall.4ŒŠØª[J].nØêÆ,2021,11(11):1763-
1769.DOI:10.12677/pm.2021.1111199
°äs
{S
n
,n≥1}byusinganelementaryinequality.Atthesametime,wegotaMarshall
typemaximalinequalityfordemisubmartingale{g(S
n
),n≥1}.
Keywords
Demi(sub)martingale,MarshallTypeInequality,Maximal
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.ý•£
3©¥, {S
n
,n≥1}L«½Â3Vǘm(Ω,F,P) þ‘ÅCþS. PS
0
= 0, I(A) ´
8ÜA«5¼ê,p>0 ,p6= 1 ¿…
1
p
+
1
q
=1.
½½½ÂÂÂ1{S
n
,n≥1}´L
1
(Ω,F,P) þ‘ÅCþS,XJé?¿1 ≤i≤j<∞, k
E[(S
j
−S
i
)f(S
1
,···,S
n
)] ≥0,
K¡‘ÅCþS{S
n
,n≥1}´˜‡f(demimartingale),Ù¥f´¦þãÏ"•3…©þØ~
¼ê.?˜Ú,ebf´˜‡šK¼ê,@o¡{S
n
,n≥1}´˜‡fe(demisubmartingale).
fVg´dNewman ÚWright3©z[1]¥JÑ, ƒéõÆöéf?1ïÄ, ¿
‰Ñ˜k¿Â(J[2–12].
X ´"þŠ²•ŒÈ‘ÅCþ,@oéu?¿ε>0 ,k
P{X≥ε}≤
EX
2
ε
2
+EX
2
,
Marshall[13]òþãØªí2Xe/ª
P{max
1≤k≤n
(X
1
+X
2
+···+X
k
) ≥ε}≤
P
n
k=1
EX
2
k
ε
2
+
P
n
k=1
EX
2
k
,(1)
Ù¥,EX
k
= 0,E(X
k
|X
1
,X
2
,···,X
k−1
) = 0a.e., k≥2, …EX
2
k
<∞, k≥1.
3þã^‡e, XJ-
S
n
=
n
X
k=1
X
k
DOI:10.12677/pm.2021.11111991764nØêÆ
°äs
@o{S
n
,n≥1}Ò´˜‡. Mu[14]3E|X
i
|
p
<∞,i≥1, …p≥2^‡e, ò(1) ªí2, 
Xe/ªMarshall .Øª
P{max
1≤k≤n
S
k
≥ε}≤
E|S
n
|
p
α
1−p
ε
p
+E|S
n
|
p
,
Ù¥α´e¼ê•ŒŠ
h(x) = 1−x+(1−x)
2−q
x
q −1
,x∈[0,1].
ƒ,Hu[15]ò©z[14]¥eZ(Øí2fœ/e,fMarshall.VÇ
Øª.©z[16]ò©z[15]¥'uf{S
n
,n≥1}Marshall.4ŠØªí2/X
{g(S
n
),n≥1}feœ/.
É©z[15]éu,©|^˜‡ÐØª'uf{S
n
,n≥1},˜aMarshall
.4ŒŠØª, Óž/X{g(S
n
),n≥1}fe˜‡Marshall.4ŒŠØª.
2.f(e)˜aMarshall.4ŒŠØª
ÚÚÚnnn1[17]eE|X|
p
<∞,E|Y|
q
<∞,K
E|XY|≤(E|X|
p
)
1
p
(E|Y|
q
)
1
q
,p>1,(2)
E|XY|≥(E|X|
p
)
1
p
(E|Y|
q
)
1
q
,0 <p<1.(3)
ÚÚÚnnn2[15]{S
n
,n≥1}´˜‡fe, @oéu?¿ε>0, k
εP( max
1≤k≤n
S
k
≥ε) ≤E(S
n
I( max
1≤k≤n
S
k
≥ε)) .(4)
ÚÚÚnnn3[2]{S
n
,n≥1}´˜‡f(½fe), g(·)´Rþ˜‡Ø~à¼ê,…g(0)=0,
K{g(S
n
),n≥1}´˜‡fe.
ÚÚÚnnn4[5]x ,y≥0, ¿…q≥2,Kk
y
q
≥x
q
+qx
q −1
(y−x)+(y−x)
q
.
ÚÚÚnnn5{S
n
,n≥1}´˜‡fe, …éu?¿n≥1 ,kES
n
≤0 ,b½•31 <p≤2, ¦
éu¤kn≥1 ,ÑkE|S
n
|
p
<∞.@oéu?¿ε>0,k
[1−qP(Λ)
q −1
(1−P(Λ))]
1
q
(E|S
n
|
p
)
1
p
≥εP(Λ) .
ùpΛ = {max
1≤k≤n
S
k
≥ε}.
yyy²²²PY= I(Λ) ,$^Ún1 (2)ªÚÚn2 ,Œ±
DOI:10.12677/pm.2021.11111991765nØêÆ
°äs
(E|Y−EY|
q
)
1
q
(E|S
n
|
p
)
1
p
≥E[(Y−EY)S
n
]
=E(YS
n
)−EYES
n
=E[I(Λ)S
n
]−EI(Λ)ES
n
≥E[I(Λ)S
n
]
≥E[εI(Λ)]
=εP(Λ).
Ï•1 <p≤2,…
1
p
+
1
q
= 1,Œq≥2 .@odÚn4, Kk
(1−P(Λ))
q
≤1−P(Λ)
q
−qP(Λ)
q −1
(1−P(Λ))
=
(1−P(Λ))
q
+P(Λ)
q
≤1−qP(Λ)
q −1
(1−P(Λ))
¤±
P(Λ)(1−P(Λ))
q
+P(Λ)
q
(1−P(Λ)) ≤1−qP(Λ)
q −1
(1−P(Λ))
qϕ
E|Y−EY|
q
= P(Λ)(1−P(Λ))
q
+P(Λ)
q
(1−P(Λ))
l·Ky.
íííØØØ1{S
n
,n≥1}´˜‡f,g(·)´Rþ˜‡Ø~à¼ê,g(0) = 0,…éu?¿n≥1
, kEg(S
n
)≤0 . XJ•31<p≤2 , ¦éu¤kn≥1 , ÑkE|g(S
n
)|
p
<∞. @oéu?¿
ε>0,k
[1−qP(A)
q−1
(1−P(A))]
1
q
[E|g(S
n
)|
p
]
1
p
≥εP(A) ,
ùpA= {max
1≤k≤n
g(S
k
) ≥ε}.
yyy²²²dÚn3 Œ•,{g(S
n
),n≥1}´˜‡fe,2dÚn5, (Øy.
ÚÚÚnnn6{S
n
,n≥1}´˜‡f, …ES
1
≤0 .XJ•31 <p≤2, ¦éu¤kn≥1 ,Ñ
kE|S
n
|
p
<∞,@oéu?¿ε>0,k
[1−qP(Λ)
q −1
(1−P(Λ))]
1
q
(E|S
n
|
p
)
1
p
≥εP(Λ) .
ùpΛ = {max
1≤k≤n
S
k
≥ε}.
yyy²²²{S
n
,n≥1}´˜‡fž, kES
n
= ES
1
,n≥2, ¤±dÚn5´.
½½½nnn1{S
n
,n≥1}´˜‡fe,ES
n
≤0,n≥1.e•31 <p≤2,¦éu?¿n≥1,
þkE|S
n
|
p
<∞,@oéu?¿ε>0, k
P(Λ) ≤
1
1+M
.
DOI:10.12677/pm.2021.11111991766nØêÆ
°äs
Ù¥M ´e¡•§):
(1+x)
q
= qx+β,x∈(0,+∞).(6)
Ù¥β=
ε
q
(E|S
n
|
p
)
q
p
,
1
p
+
1
q
= 1, Λ = {max
1≤k≤n
S
k
≥ε}.
yyy²²²´•§(5) k•˜).dÚn5 Œ•
[1−qP(Λ)
q −1
(1−P(Λ))](E|S
n
|
p
)
q
p
≥ε
q
P(Λ)
q
.
P(Λ) = 0 ž,(Øw,¤á.
Ø”P(Λ) 6= 0 ,Kü>ӞرP(Λ)
q
Œ
[
1
P(Λ)
q
−q
1−P(Λ)
P(Λ)
](E|S
n
|
p
)
q
p
≥ε
q
-x
0
=
1−P(Λ)
P(Λ)
,β=
ε
q
(E|S
n
|
p
)
q
p
,KkP(Λ) =
1
1+x
0
.
Ïd
(1+x
0
)
q
−qx
0
≥β
=
(1+x
0
)
q
≥qx
0
+β(7)
-h(x) = (1+x)
q
−qx−β,M´•§(6)).Ï•h
00
(x) = q(q−1)(1+x)
q −2
>0,x∈(0,+∞)
,´•h(x)3«m(0,+∞)þ´˜‡à¼ê,ù¿›Xéu?¿x∈(0,M),k
h(x)−h(0)
x−0
≤
h(M)−h(x)
M−x
,
Ï•h(0) = −β<0…h(M) = 0,¤±éu?¿x∈(0,M) kh(x) <0,=M´¦•§(7)¤á
•Š, ·Ky.
½½½nnn2{S
n
,n≥1}´˜‡f,ES
1
≤0.e•31<p≤2,¦éu?¿n≥1, þk
E|S
n
|
p
<∞,@oéu?¿ε>0, k
P(Λ) ≤
1
1+M
.
Ù¥M´e¡•§)
(1+x)
q
= qx+β,x∈(0,+∞).
Ù¥β=
ε
q
(E|S
n
|
p
)
q
p
,
1
p
+
1
q
= 1, Λ = {max
1≤k≤n
S
k
≥ε}
DOI:10.12677/pm.2021.11111991767nØêÆ
°äs
yyy²²²†½n1 y²L§aq,(ÜÚn6 ,=Œy(Ø.
íííØØØ2{S
n
,n≥1}´˜‡f,g(·)´Rþ˜‡Ø~à¼ê,…éu?¿n≥1,
Eg(S
n
) ≤0 .e•31 <p≤2,¦éu?¿n≥1, þkE|g(S
n
)|
p
<∞,@oéu?¿ε>0, k
P(A) ≤
1
1+M
.
Ù¥M ´e¡•§)
(1+x)
q
= qx+β,x∈(0,+∞).
Ù¥β=
ε
q
(E|g(S
n
)|
p
)
q
p
,
1
p
+
1
q
= 1, A= {max
1≤k≤n
g(S
k
) ≥ε}.
yyy²²²(ÜíØ1 Ú½n1y²L§, ´.
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11861057,11761064),[‹ŽpÆM#UåJ,‘8(2019A-
003),Ü“‰ŒÆïÄ)‰ï]Ï‘8(2020KYZZ001113),[‹Ž`DïÄ)/M#ƒ(0‘
8(2021CXZX-262).
ë•©z
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DOI:10.12677/pm.2021.11111991769nØêÆ

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