设为首页
加入收藏
期刊导航
网站地图
首页
期刊
数学与物理
地球与环境
信息通讯
经济与管理
生命科学
工程技术
医药卫生
人文社科
化学与材料
会议
合作
新闻
我们
招聘
千人智库
我要投稿
办刊
期刊菜单
●领域
●编委
●投稿须知
●最新文章
●检索
●投稿
文章导航
●Abstract
●Full-Text PDF
●Full-Text HTML
●Full-Text ePUB
●Linked References
●How to Cite this Article
PureMathematics
n
Ø
ê
Æ
,2021,11(11),1763-1769
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1111199
f
(
e
)
˜
a
Marshall
.
4
Œ
Š
Ø
ª
°°°
äää
sss
§§§
¾¾¾
¤¤¤
§§§
___
Ü
“
‰
Œ
Æ
ê
Æ
†
Ú
O
Æ
§
[
‹
=
²
Â
v
F
Ï
µ
2021
c
9
24
F
¶
¹
^
F
Ï
µ
2021
c
10
27
F
¶
u
Ù
F
Ï
µ
2021
c
11
3
F
Á
‡
©
|
^
˜
‡
Ð
Ø
ª
§
f
(
e
)
{
S
n
,n
≥
1
}
˜
a
Marshall
.
4
Œ
Š
Ø
ª
§
Ó
ž
/
X
{
g
(
S
n
)
,n
≥
1
}
f
e
˜
‡
Marshall
.
4
Œ
Š
Ø
ª
"
'
…
c
f
(
e
)
§
Marshall
.
Ø
ª
§
4
Œ
Š
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
©
Ù
Ú
^
:
°
ä
s
§
¾
¤
§
_
.
f
(
e
)
˜
a
Marshall
.
4
Œ
Š
Ø
ª
[J].
n
Ø
ê
Æ
,2021,11(11):1763-
1769.DOI:10.12677/pm.2021.1111199
°
ä
s
{
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
«
½
Â
3
V
Ç
˜
m
(Ω,
F
,P)
þ
‘
Å
C
þ
S
.
P
S
0
= 0,
I
(
A
)
´
8
Ü
A
«
5
¼
ê
,
p>
0 ,
p
6
= 1
¿
…
1
p
+
1
q
=1.
½½½
ÂÂÂ
1
{
S
n
,n
≥
1
}
´
L
1
(Ω,
F
,P)
þ
‘
Å
C
þ
S
,
X
J
é
?
¿
1
≤
i
≤
j<
∞
,
k
E
[(
S
j
−
S
i
)
f
(
S
1
,
···
,S
n
)]
≥
0
,
K
¡
‘
Å
C
þ
S
{
S
n
,n
≥
1
}
´
˜
‡
f
(demimartingale),
Ù
¥
f
´¦
þ
ã
Ï
"
•
3
…
©
þ
Ø
~
¼
ê
.
?
˜
Ú
,
e
b
f
´
˜
‡
š
K
¼
ê
,
@
o
¡
{
S
n
,n
≥
1
}
´
˜
‡
f
e
(demisubmartingale).
f
V
g
´
d
Newman
Ú
Wright
3
©
z
[1]
¥
J
Ñ
,
ƒ
é
õ
Æ
ö
é
f
?
1
ï
Ä
,
¿
‰
Ñ
˜
k
¿Â
(
J
[2–12].
X
´
"
þ
Š
²
•
Œ
È
‘
Å
C
þ
,
@
o
é
u
?
¿
ε>
0 ,
k
P
{
X
≥
ε
}≤
EX
2
ε
2
+
EX
2
,
Marshall[13]
ò
þ
ã
Ø
ª
í
2
X
e
/
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.
3
þ
ã
^
‡
e
,
X
J
-
S
n
=
n
X
k
=1
X
k
DOI:10.12677/pm.2021.11111991764
n
Ø
ê
Æ
°
ä
s
@
o
{
S
n
,n
≥
1
}
Ò
´
˜
‡
. Mu
[14]
3
E
|
X
i
|
p
<
∞
,
i
≥
1,
…
p
≥
2
^
‡
e
,
ò
(1)
ª
í
2
,
X
e
/
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
¼
ê
•
Œ
Š
h
(
x
) = 1
−
x
+(1
−
x
)
2
−
q
x
q
−
1
,x
∈
[0
,
1]
.
ƒ
,Hu
[15]
ò
©
z
[14]
¥
e
Z
(
Ø
í
2
f
œ
/
e
,
f
Marshall
.
V
Ç
Ø
ª
.
©
z
[16]
ò
©
z
[15]
¥
'
u
f
{
S
n
,n
≥
1
}
Marshall
.
4
Š
Ø
ª
í
2
/
X
{
g
(
S
n
)
,n
≥
1
}
f
e
œ
/
.
É
©
z
[15]
é
u
,
©
|
^
˜
‡
Ð
Ø
ª
'
u
f
{
S
n
,n
≥
1
}
,
˜
a
Marshall
.
4
Œ
Š
Ø
ª
,
Ó
ž
/
X
{
g
(
S
n
)
,n
≥
1
}
f
e
˜
‡
Marshall
.
4
Œ
Š
Ø
ª
.
2.
f
(
e
)
˜
a
Marshall
.
4
Œ
Š
Ø
ª
ÚÚÚ
nnn
1
[17]
e
E
|
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)
ÚÚÚ
nnn
2
[15]
{
S
n
,n
≥
1
}
´
˜
‡
f
e
,
@
o
é
u
?
¿
ε>
0,
k
εP
( max
1
≤
k
≤
n
S
k
≥
ε
)
≤
E
(
S
n
I
( max
1
≤
k
≤
n
S
k
≥
ε
))
.
(4)
ÚÚÚ
nnn
3
[2]
{
S
n
,n
≥
1
}
´
˜
‡
f
(
½
f
e
),
g
(
·
)
´
R
þ
˜
‡
Ø
~
à
¼
ê
,
…
g
(0)=0,
K
{
g
(
S
n
)
,n
≥
1
}
´
˜
‡
f
e
.
ÚÚÚ
nnn
4
[5]
x ,
y
≥
0,
¿
…
q
≥
2,
K
k
y
q
≥
x
q
+
qx
q
−
1
(
y
−
x
)+(
y
−
x
)
q
.
ÚÚÚ
nnn
5
{
S
n
,n
≥
1
}
´
˜
‡
f
e
,
…
é
u
?
¿
n
≥
1 ,
k
ES
n
≤
0 ,
b
½
•
3
1
<p
≤
2,
¦
é
u
¤
k
n
≥
1 ,
Ñ
k
E
|
S
n
|
p
<
∞
.
@
o
é
u
?
¿
ε>
0,
k
[1
−
qP
(Λ)
q
−
1
(1
−
P
(Λ))]
1
q
(
E
|
S
n
|
p
)
1
p
≥
εP
(Λ)
.
ù
p
Λ =
{
max
1
≤
k
≤
n
S
k
≥
ε
}
.
yyy
²²²
P
Y
=
I
(Λ) ,
$
^
Ú
n
1
(2)
ª
Ú
Ú
n
2 ,
Œ
±
DOI:10.12677/pm.2021.11111991765
n
Ø
ê
Æ
°
ä
s
(
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 .
@
o
d
Ú
n
4,
K
k
(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))
q
Ï
•
E
|
Y
−
EY
|
q
=
P
(Λ)(1
−
P
(Λ))
q
+
P
(Λ)
q
(1
−
P
(Λ))
l
·
K
y
.
ííí
ØØØ
1
{
S
n
,n
≥
1
}
´
˜
‡
f
,
g
(
·
)
´
R
þ
˜
‡
Ø
~
à
¼
ê
,
g
(0) = 0,
…
é
u
?
¿
n
≥
1
,
k
Eg
(
S
n
)
≤
0 .
X
J
•
3
1
<p
≤
2 ,
¦
é
u
¤
k
n
≥
1 ,
Ñ
k
E
|
g
(
S
n
)
|
p
<
∞
.
@
o
é
u
?
¿
ε>
0,
k
[1
−
qP
(
A
)
q
−
1
(1
−
P
(
A
))]
1
q
[
E
|
g
(
S
n
)
|
p
]
1
p
≥
εP
(
A
)
,
ù
p
A
=
{
max
1
≤
k
≤
n
g
(
S
k
)
≥
ε
}
.
yyy
²²²
d
Ú
n
3
Œ
•
,
{
g
(
S
n
)
,n
≥
1
}
´
˜
‡
f
e
,
2
d
Ú
n
5,
(
Ø
y
.
ÚÚÚ
nnn
6
{
S
n
,n
≥
1
}
´
˜
‡
f
,
…
ES
1
≤
0 .
X
J
•
3
1
<p
≤
2,
¦
é
u
¤
k
n
≥
1 ,
Ñ
k
E
|
S
n
|
p
<
∞
,
@
o
é
u
?
¿
ε>
0,
k
[1
−
qP
(Λ)
q
−
1
(1
−
P
(Λ))]
1
q
(
E
|
S
n
|
p
)
1
p
≥
εP
(Λ)
.
ù
p
Λ =
{
max
1
≤
k
≤
n
S
k
≥
ε
}
.
yyy
²²²
{
S
n
,n
≥
1
}
´
˜
‡
f
ž
,
k
ES
n
=
ES
1
,
n
≥
2,
¤
±
d
Ú
n
5
´
.
½½½
nnn
1
{
S
n
,n
≥
1
}
´
˜
‡
f
e
,
ES
n
≤
0,
n
≥
1.
e
•
3
1
<p
≤
2,
¦
é
u
?
¿
n
≥
1,
þ
k
E
|
S
n
|
p
<
∞
,
@
o
é
u
?
¿
ε>
0,
k
P
(Λ)
≤
1
1+
M
.
DOI:10.12677/pm.2021.11111991766
n
Ø
ê
Æ
°
ä
s
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
}
.
yyy
²²²
´
•
§
(5)
k
•
˜
)
.
d
Ú
n
5
Œ
•
[1
−
qP
(Λ)
q
−
1
(1
−
P
(Λ))](
E
|
S
n
|
p
)
q
p
≥
ε
q
P
(Λ)
q
.
P
(Λ) = 0
ž
,
(
Ø
w
,
¤
á
.
Ø
”
P
(Λ)
6
= 0 ,
K
ü
>
Ó
ž
Ø
±
P
(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
,
K
k
P
(Λ) =
1
1+
x
0
.
Ï
d
(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
´
˜
‡
à
¼
ê
,
ù
¿
›
X
é
u
?
¿
x
∈
(0
,M
),
k
h
(
x
)
−
h
(0)
x
−
0
≤
h
(
M
)
−
h
(
x
)
M
−
x
,
Ï
•
h
(0) =
−
β<
0
…
h
(
M
) = 0,
¤
±
é
u
?
¿
x
∈
(0
,M
)
k
h
(
x
)
<
0,
=
M
´¦
•
§
(7)
¤
á
•
Š
,
·
K
y
.
½½½
nnn
2
{
S
n
,n
≥
1
}
´
˜
‡
f
,
ES
1
≤
0.
e
•
3
1
<p
≤
2,
¦
é
u
?
¿
n
≥
1,
þ
k
E
|
S
n
|
p
<
∞
,
@
o
é
u
?
¿
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.2021.11111991767
n
Ø
ê
Æ
°
ä
s
yyy
²²²
†
½
n
1
y
²
L
§
a
q
,
(
Ü
Ú
n
6 ,
=
Œ
y
(
Ø
.
ííí
ØØØ
2
{
S
n
,n
≥
1
}
´
˜
‡
f
,
g
(
·
)
´
R
þ
˜
‡
Ø
~
à
¼
ê
,
…
é
u
?
¿
n
≥
1,
Eg
(
S
n
)
≤
0 .
e
•
3
1
<p
≤
2,
¦
é
u
?
¿
n
≥
1,
þ
k
E
|
g
(
S
n
)
|
p
<
∞
,
@
o
é
u
?
¿
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
}
.
yyy
²²²
(
Ü
í
Ø
1
Ú
½
n
1
y
²
L
§
,
´
.
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
]
Ï
‘
8
(11861057,11761064),
[
‹
Ž
p
Æ
M
#
U
å
J
,
‘
8
(2019A-
003),
Ü
“
‰
Œ
Æï
Ä
)
‰
ï
]
Ï
‘
8
(2020KYZZ001113),
[
‹
Ž
`
Dï
Ä
)
/
M
#
ƒ
(
0
‘
8
(2021CXZX-262).
ë
•
©
z
[1]Newman,C.M.andWright,A.L.(1982)AssociatedRandomVariablesandMartingaleIn-
equalities.
Zeitschriftf¨urWahrscheinlichkeitstheorieundVerwandteGebiete
,
59
,361-371.
https://doi.org/10.1007/BF00532227
[2]Christofides,T.C.(2000)Maximal Inequalities forDemimartingalesand a StrongLargeNum-
bers.
StatisticsandProbabilityLetters
,
50
,357-363.
https://doi.org/10.1016/S0167-7152(00)00116-4
[3]Christofides,T.C.(2004)U-StatisticsonAssociatedRandomVariables.
JournalofStatistical
PlanningandInference
,
119
,1-15.https://doi.org/10.1016/S0378-3758(02)00418-4
[4]Christofides, T.C. and Hadjikyriakou, M. (2009) Exponential Inequalities for Demimartingales
and
N
-DemimartingalesandNegativelyAssociatedRandomVariables.
StatisticsandProba-
bilityLetters
,
79
,2060-2065.https://doi.org/10.1016/j.spl.2009.06.013
[5]Christofides,T.C.andHadjikyriakou,M. (2012)MaximalandMomentInequalitiesforDemi-
martingalesand
N
-Demimartingales.
StatisticsandProbabilityLetters
,
82
,683-691.
https://doi.org/10.1016/j.spl.2011.12.009
[6]Dai,P.-P.,Shen,Y.andHu,S.-H.(2014)SomeResultsforDemimartingalesand
N
-
Demimartingales.
JournalofInequalitiesandApplications
,
2014
,489-501.
https://doi.org/10.1186/1029-242X-2014-489
DOI:10.12677/pm.2021.11111991768
n
Ø
ê
Æ
°
ä
s
[7]PrakasaRao,B.L.S.(2007)OnSomeMaximalInequalitiesforDemisubmartingalesand
N
-
DemisuperMartingales.
JournalofInequalitiesinPureandAppliedMathematics
,
8
,Article
112.
[8]PrakasaRao, B.L.S.(2012)Remarkson MaximalInequalitiesforNonnegativeDemisubmartin-
gales.
StatisticsandProbabilityLetters
,
82
,1388-1390.
https://doi.org/10.1016/j.spl.2012.03.019
[9]
Ó
Ü
,
©
“
,
Æ
,
.
'
u
N-
f
Ú
f
Ø
ª
˜
‡
5
P
[J].
X
Ú
‰
Æ
†
ê
Æ
,2010,
30(8):1052-1058.
[10]Wang,X.H.andWang,X.J.(2013)SomeInequalitiesforConditionalDemimartingalesand
Conditional
N
-Demimartingales.
StatisticsandProbabilityLetters
,
83
,700-709.
https://doi.org/10.1016/j.spl.2012.11.017
[11]Wang,J.F.(2004)MaximalInequalitiesforAssociatedRandomVariablesandDemimartin-
gales.
StatisticsandProbabilityLetters
,
66
, 347-354. https://doi.org/10.1016/j.spl.2003.10.021
[12]Wang,X.J.andHu,S.H.(2009)MaximalInequalitiesforDemimartingalesandTheirAppli-
cations.
ScienceinChinaSeriesA:Mathematics
,
52
,2207-2217.
https://doi.org/10.1007/s11425-009-0067-x
[13]Marshall,A.W.(1960)A One-SidedAnalogofKolmogorov’sInequality.
TheAnnalsofMath-
ematicalStatistics
,
31
,483-487.https://doi.org/10.1214/aoms/1177705912
[14]Mu,J.Y.andMiao,Y.(2011)GeneralizingtheMarshall’sInequality.
Communicationsin
StatisticsTheoryandMethods
,
40
,2809-2817.https://doi.org/10.1080/03610926.2010.493276
[15]Hu,S.H.,Wang,X.J.andYang,W.Z.(2012)SomeInequalitiesforDemimartingalesand
N
-
Martingales.
StatisticsandProbabilityLetters
,
82
,232-239.
https://doi.org/10.1016/j.spl.2011.10.021
[16]
¾
¤
,
=
,
o
Œ
.
f
(
e
)
˜
a
Marshall
.
Ø
ª
[J].
o
A
“
‰
Œ
ÆÆ
(
g
,
‰
Æ
‡
),
2018,41(4):495-499.
[17]
ö
,
x
“
À
.
V
Ç
Ø
ª
[M].
®
:
‰
Æ
Ñ
‡
,2007.
DOI:10.12677/pm.2021.11111991769
n
Ø
ê
Æ