Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 838301, 11 pages
doi:10.1155/2010/838301
Research Article
Doob’s Type Inequality and Strong Law of
Large Numbers for Demimartingales
Wang Xuejun, Hu Shuhe, Zhao Ting, and Yang Wenzhi
School of Mathematical Science, Anhui University, Hefei 230039, China
Correspondence should be addressed to Hu Shuhe, hushuhe@263.net.com
Received 9 October 2009; Revised 14 December 2009; Accepted 28 January 2010
Academic Editor: Andrei Volodin
Copyright q 2010 Wang Xuejun et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We establish some maximal inequalities for demimartingales which generalize the result of Wang
2004. The maximal inequality for demimartingales is used as a key inequality to establish other
results including Doob’s type maximal inequality, strong law of large numbers, strong growth rate,
and integrability of supremum for demimartingales, which generalize and improve partial results
of Christofides 2000 and Prakasa Rao 2007.
1. Introduction
Definition 1.1. Let S1 , S2 , . . . be an L1 sequence of random variables. Assume that for j 1, 2, . . .
E
Sj1 − Sj f S1 , . . . , Sj ≥ 0
1.1
for all coordinatewise nondecreasing functions f such that the expectation is defined. Then
{Sj , j ≥ 1} is called a demimartingale. If in addition the function f is assumed to be
nonnegative, then the sequence {Sj , j ≥ 1} is called a demisubmartingale.
Definition 1.2. A finite collection of random variables X1 , X2 , . . . , Xm is said to be associated if
Cov fX1 , . . . , Xm , gX1 , . . . , Xm ≥ 0
1.2
2
Journal of Inequalities and Applications
for any two coordinatewise nondecreasing functions f, g on Rm such that the covariance
is defined. An infinite sequence {Xn , n ≥ 1} is associated if every finite subcollection is
associated.
Definition 1.3. A finite collection of random variables X1 , X2 , . . . , Xn is said to be strongly
positive dependent if
P X 1 ∈ Λ1 ; X 2 ∈ Λ2 ≥ P X 1 ∈ Λ1 P X 2 ∈ Λ2
1.3
for all Borel measurable and increasing or decreasing set pairs Λ1 , Λ2 ⊂ R1 × R2 A set Λ
is said increasing or decreasing if x ≤ or ≥y implies y ∈ Λ for any x ∈ Λ, where
X 1 Xi , i ∈ I,
R1 R|I| ,
X 2 Xi , i ∈ I c ,
R2 R|I
c
|
I ⊂ 1, 2, . . . , n, I c 1, 2, . . . , n \ I,
|I| stands for the base of I.
1.4
An infinite sequence {Xn , n ≥ 1} is strongly positive dependent if every finite subcollection is
strongly positive dependent.
Remark 1.4. Chow 1 proved a maximal inequality for submartingales. Newman and
Wright 2 extended Doob’s maximal inequality and upcrossing inequality to the case of
demimartingales, and pointed out that the partial sum of a sequence of mean zero associated
random variables is a demimartingale. Christofides 3 showed that the Chow’s maximal
inequality for submartingales can be extended to the case of demisubmartingales.
Wang 4 obtained Doob’s type inequality for more general demimartingales. Hu et al.
5 gave a strong law of large numbers and growth rate for demimartingales. Prakasa
Rao 6 established some maximal inequalities for demisubmartingales and N-demisupermartingales.
It is easily seen that the partial sum of a sequence of mean zero strongly positive
dependent random variables is also a demimartingale by the inequality 3 in Zheng 7,
that is, for all n ≥ 1,
E fS1 , . . . , Sn Sn1 − Sn E fX1 , X1 X2 , . . . , X1 X2 · · · Xn Xn1 ≥ 0
1.5
for all coordinatewise nondecreasing functions f such that the expectation is defined.
Therefore, the main results of this paper hold for the partial sums of sequences of mean zero
associated random variables and strongly positive dependent random variables.
Let {Xn , n ≥ 1} and {Sn , n ≥ 1} be sequences of random variables defined on a fixed
probability space Ω, F, P and IA the indicator function of the event A. Denote S0 0,
X max0, X, X − max0, −X, log x loge x ln x, log x lnmaxx, 1. The main
results of this paper depend on the following lemmas.
Journal of Inequalities and Applications
3
Lemma 1.5 see Wang 4, Theorem 2.1. Let {Sn , n ≥ 1} be a demimartingale and g a nonnegative
convex function on R with g0 0 and gSi ∈ L1 , i ≥ 1. Let {ck , k ≥ 1} be a nonincreasing sequence
of positive numbers. Then for any ε > 0,
n
εP max ck gSk ≥ ε ≤
cj E g Sj − g Sj−1 I max ck gSk ≥ ε .
1≤k≤n
1≤k≤n
j1
1.6
Lemma 1.6 see Fazekas and Klesov 8, Theorem 2.1 and Hu et al. 5, Lemma 1.5. Let
n
{Xn , n ≥ 1} be a random variable sequence and Sn i1 Xi for n ≥ 1. Let b1 , b2 , . . . be a
nondecreasing unbounded sequence of positive numbers and α1 , α2 , . . . nonnegative numbers. Let p
and C be fixed positive numbers. Assume that for each n ≥ 1,
p
n
E max |Sl | ≤ C αl ,
1≤l≤n
∞
αl
< ∞,
p
l1
l1
bl
1.7
then
lim
Sn
n → ∞ bn
1.8
0 a.s.,
and with the growth rate
βn
Sn
O
a.s.,
bn
bn
1.9
where
δ/p
βn max bk vk ,
1≤k≤n
∀0 < δ < 1,
vn ∞
αk
p
kn bk
,
lim
βn
n → ∞ bn
0.
1.10
In addition,
p
n
Sl αl
E max ≤ 4C
p < ∞,
1≤l≤n bl
b
l1
l
p
∞
Sl αl
E sup ≤ 4C
p < ∞.
b
l
b
l≥1
l1
1.11
l
If further assumes one that αn > 0 for infinitely many n, then
p
∞
Sl αl
E sup ≤ 4C
p < ∞.
β
l
β
l≥1
l1
1.12
l
Lemma 1.7 see Christofides 3, Lemma 2.1, Corollary 2.1. i If {Sn , n ≥ 1} is a
demisubmartingale (or a demimartingale) and g is a nondecreasing convex function such that gSi ∈
L1 , i ≥ 1, then {gSn , n ≥ 1} is a demisubmartingale.
ii If {Sn , n ≥ 1} is a demimartingale, then {Sn , n ≥ 1} is a demisubmartingale and {S−n , n ≥
1} is a demisubmartingale.
4
Journal of Inequalities and Applications
Lemma 1.8 see Hu et al. 9, Theorem 2.1. Let {Sn , n ≥ 1} be a demimartingale and {ck , k ≥ 1}
be a nonincreasing sequence of positive numbers. Let ν ≥ 1 and E|Sk |ν < ∞ for each k, then for any
ε > 0 and 1 ≤ n ≤ N,
max ck |Sk | ≥ ε
P
n≤k≤N
N
2
ν
ν
ν
ν
ν
.
ck E |Sk | − |Sk−1 |
≤ ν cn E|Sn | ε
kn1
1.13
Lemma 1.9 see Christofides 3, Corollary 2.4, Theorem 2.1. i Let {Sn , n ≥ 1} be a
demisubmartingale. Then for any ε > 0,
εP max Sk ≥ ε ≤
1≤k≤n
max Sk ≥ε
Sn dP.
1.14
1≤k≤n
ii Let {Sn , n ≥ 1} be a demisubmartingale and {ck , k ≥ 1} a nonincreasing sequence of
positive numbers. Then for any ε > 0,
n
cj E Sj − Sj−1 .
εP max ck Sk ≥ ε ≤
1≤k≤n
1.15
j1
Using Lemma 1.5, Wang 4 obtained the following inequalities for demimartingales.
Theorem 1.10 see Wang 4, Corollary 2.1. Let {Sn , n ≥ 1} be a demimartingale and {ck , k ≥ 1}
a nonincreasing sequence of positive numbers. Then
E max ck |Sk |
1≤k≤n
p
≤
p
p−1
p
⎤p
⎡
n
E⎣ cj Sj − Sj−1 ⎦ ,
p > 1,
1.16
j1
⎧
⎞
⎞⎤⎫
⎛
⎡⎛
n
n
⎬
e ⎨
1 E⎣⎝ cj Sj − Sj−1 ⎠log ⎝ cj Sj − Sj−1 ⎠⎦ .
E max ck |Sk | ≤
⎭
e − 1⎩
1≤k≤n
j1
j1
1.17
We point out that there is a mistake in the proof of 1.17, that is,
⎤
⎡⎛
⎞
n
E max cj Sj − 1 ≤ E⎣⎝ cj Sj − Sj−1 ⎠ log max cj Sj ⎦
1≤j≤n
j1
1≤j≤n
1.18
should be replaced by
⎞
⎡⎛
⎤
n
E max cj Sj − 1 ≤ E⎣⎝ cj Sj − Sj−1 ⎠log max cj Sj ⎦.
1≤j≤n
j1
1≤j≤n
1.19
Journal of Inequalities and Applications
5
In fact, by Lemma 1.5 and Fubini Theorem, we can see that
∞ P max cj Sj ≥ x dx
E max cj Sj − 1 ≤
1≤j≤n
1
≤
∞
1
1≤j≤n
⎤
⎡
n
1 ⎣
E
cj Sj − Sj−1 I max cj Sj ≥ x ⎦dx
1≤j≤n
x
j1
1.20
⎡⎛
⎞
⎤
n
≤ E⎣⎝ cj Sj − Sj−1 ⎠log max cj Sj ⎦.
1≤j≤n
j1
The rest of the proof is similar to Corollary 2.1 in Wang 4.
The same problem exists in Shiryaev 10, page 495, in the proof of Theorem 2 and
Krishna and Soumendra 11, page 414, in the proof of Theorem 13.2.13. For example, the
following inequality
max |X |−1
j
dt
0≤j≤n
E max Xj − 1 ≤ E Xn
E Xn log max Xj 0≤j≤n
0≤j≤n
1t
0
1.21
in Shiryaev 10, page 495 should be revised as
$
%
max |Xj |−1
dt
0≤j≤n
E max Xj − 1 ≤ E Xn I max Xj ≥ 1
0≤j≤n
0≤j≤n
1t
0
E Xn log max Xj .
1.22
0≤j≤n
2. Main Results and Their Proofs
Theorem 2.1. Let {Sn , n ≥ 1} be a demimartingale and g a nonnegative convex function on R with
g0 0. Let {ck , k ≥ 1} be a nonincreasing sequence of positive numbers. p > 1. Suppose that
EgSk p < ∞ for each k ≥ 1, then for every n ≥ 1,
E max ck gSk 1≤k≤n
p
≤
p
p−1
p
⎤p
⎡
n
E⎣ cj g Sj − g Sj−1 ⎦ ,
2.1
j1
⎧
⎡⎛
⎛
⎞
⎞⎤⎫
n
n
⎬
e ⎨
1E⎣⎝ cj g Sj −g Sj−1 ⎠log ⎝ cj g Sj −g Sj−1 ⎠⎦ .
E max ck gSk ≤
⎭
e−1 ⎩
1≤k≤n
j1
j1
2.2
6
Journal of Inequalities and Applications
Proof. By Lemma 1.5 and Hölder’s inequality, we have
∞
p
E max ck gSk p xp−1 P max ck gSk ≥ x dx
1≤k≤n
0
1≤k≤n
0
j1
⎤
⎡
∞
n
≤ p xp−2 E⎣ cj g Sj − g Sj−1 I max ck gSk ≥ x ⎦dx
1≤k≤n
⎧⎡
⎫
⎤
p−1 ⎬
n
⎨ p
E ⎣ cj g Sj − g Sj−1 ⎦ max ck gSk ⎭
p − 1 ⎩ j1
1≤k≤n
2.3
⎧ ⎡
⎤p ⎫1/p
p 1/q
n
⎬
p ⎨ E⎣ cj g Sj − g Sj−1 ⎦
≤
,
E max ck gSk ⎭
p − 1 ⎩ j1
1≤k≤n
where q is a real number and satisfies 1/p 1/q 1. Since EgSk p < ∞ for each k ≥ 1, we
can obtain
p 1/p
E max ck gSk ≤
1≤k≤n
⎧ ⎡
⎤p ⎫1/p
n
⎨
⎬
p
E⎣ cj g Sj − g Sj−1 ⎦
,
⎭
p − 1 ⎩ j1
2.4
therefore,
E max ck gSk p
1≤k≤n
≤
p
p−1
p
⎤p
⎡
n
E⎣ cj g Sj − g Sj−1 ⎦ .
2.5
j1
Similar to the proof of 2.3 and using Lemma 1.5 again, we can see that
⎡⎛
⎞
⎤
n
E max ck gSk ≤ 1 E⎣⎝ cj g Sj − g Sj−1 ⎠log max ck gSk ⎦.
1≤k≤n
j1
1≤k≤n
2.6
For constants a ≥ 0 and b > 0, it follows that
alog b ≤ alog a be−1 .
2.7
Journal of Inequalities and Applications
7
Combining 2.6 and 2.7, we have
⎡⎛
⎛
⎞
⎞⎤
n
n
E max ck gSk ≤ 1 E⎣⎝ cj g Sj − g Sj−1 ⎠log ⎝ cj g Sj − g Sj−1 ⎠⎦
1≤k≤n
j1
j1
e−1 E max ck gSk .
1≤k≤n
2.8
Thus, 2.2 follows from 2.8 immediately. The proof is complete.
Remark 2.2. If we take gx |x| in Theorem 2.1, then Theorem 2.1 implies Corollary 2.1 in
Wang 4.
Corollary 2.3. Let the conditions of Theorem 2.1 be satisfied with ck ≡ 1 for each k ≥ 1. Then for
every n ≥ 1,
p p
p
p
E max gSk ≤
E gSn ,
p−1
1≤k≤n
'
&
e E max gSk ≤
1 E gSn log gSn .
e−1
1≤k≤n
2.9
2.10
Corollary 2.4 Doob’s type maximal inequality for demimartingales. Let p > 1 and {Sn , n ≥
1} be a demimartingale. Suppose that E|Sk |p < ∞ for each k ≥ 1, then for every n ≥ 1,
p p
p
E max |Sk | ≤
E|Sn |p ,
p−1
1≤k≤n
'
e &
E max |Sk | ≤
1 E |Sn |log |Sn | .
e−1
1≤k≤n
2.11
Theorem 2.5. Let {Sn , n ≥ 1} be a demimartingale and g a nonnegative convex function on R with
g0 0. Let {bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers. p > 1. Suppose
that EgSk−1 p ≤ EgSk p < ∞ for each k ≥ 1 and
p
∞ E gS p − E gS
n
n−1 p
bn
n1
< ∞,
2.12
then limn → ∞ gSn /bn 0 a.s., and 1.9-1.10 hold (Sn is replaced by gSn ), where
αn p
p−1
p
p
p '
& E gSn − E gSn−1 ,
n ≥ 1.
2.13
8
Journal of Inequalities and Applications
In addition,
∞
gSn p
αn
≤4
E sup
p < ∞.
b
n
n≥1
n1 bn
n
gSk p
≤ 4 αk < ∞,
E max p
bk
1≤k≤n
k1 b
k
2.14
If further one assumes that αk > 0 for infinitely many k, then
∞
gSn p
αn
≤
4
E sup
p < ∞.
βn
n≥1
n1 βn
2.15
Proof. By the condition of the theorem, we can see that αn ≥ 0 for all n ≥ 1. Thus,
p
p n
p p
E max gSk ≤
E gSn αk
p−1
1≤k≤n
k1
2.16
follows from 2.9 for each n ≥ 1. By 2.12, we have
∞
αn
p
n1 bn
p
p−1
p
p ∞ E gS p − E gS
n
n−1 p
bn
n1
< ∞.
2.17
Therefore, limn → ∞ gSn /bn 0 a.s. follows from Lemma 1.6, 2.16, and 2.17; 1.9,
1.10, 2.14, 2.15 hold. This completes the proof of the theorem.
In Theorem 2.5, if we assume that gx is a nonnegative and nondecreasing convex
function on R with g0 0, then the condition “EgSk−1 p ≤ EgSk p for each k ≥ 1” is
satisfied.
Remark 2.6. Theorem 2.5 generalizes and improves the results of Theorem 2.2 in Christofides
3 and Theorem 2.7 in Prakasa Rao 6.
Theorem 2.7. Let p > 1 and {Sn , n ≥ 1} a demimartingale with E|Sk |p < ∞ for each k ≥ 1. Let
{bn , n ≥ 1} be a nondecreasing sequence of positive numbers. If
∞
E|Sk |p − E|Sk−1 |p
p
k1
bk
< ∞,
2.18
then for any 0 < r < p,
r ∞
Sk 2r E|Sk |p − E|Sk−1 |p
E sup ≤1
< ∞.
p
p − r k1
b
k≥1 bk
k
2.19
Journal of Inequalities and Applications
9
Proof. Taking n 1, ν p and ck 1/bk in Lemma 1.8, we have
N
Sk 2
E|Sk |p − E|Sk−1 |p
P max ≥ ε ≤ p
.
p
ε k1
1≤k≤N bk
b
2.20
k
Thus, by 2.20 and 2.18, we can get
∞ ∞
r r
Sk Sk Sk ≤1
E sup P sup > t dt 1 lim P max > t1/r dt
1≤k≤N bk
1
1 N →∞
k≥1 bk
k≥1 bk
∞
2r E|Sk |p − E|Sk−1 |p
≤1
< ∞.
p
p − r k1
b
2.21
k
Theorem 2.8. Let {Sn , n ≥ 1} be a demisubmartingale and g a nondecreasing and nonnegative
convex function on R with g0 0 and gSi ∈ L1 , i ≥ 1. Let {ck , k ≥ 1} be a nonincreasing
sequence of positive numbers. Then for all 0 < p < 1 and each n ≥ 1,
E max ck gSk 1≤k≤n
p
≤ cn EgSn p−1
⎤
n
p ⎣cn EgSn cj E g Sj − g Sj−1 ⎦.
1 − p j1
⎡
2.22
Proof. By Fubini theorem, it is easy to check that
∞
p
E max ck gSk p xp−1 P max ck gSk ≥ x dx
1≤k≤n
1≤k≤n
0
≤p
cn EgSn x
p−1
dx p
0
cn EgSn p
p
∞
∞
cn EgSn cn EgSn x
xp−1 P max ck gSk ≥ x dx
p−1
1≤k≤n
2.23
P max ck gSk ≥ x dx.
1≤k≤n
It follows from Lemma 1.7i and Lemma 1.9ii that
∞
p
cn EgSn ∞
xp−1 P max ck gSk ≥ x dx ≤ p
1≤k≤n
cn EgSn xp−2
n
cj E g Sj − g Sj−1 dx
j1
n
p−1 p cj E g Sj − g Sj−1 .
cn EgSn 1−p
j1
2.24
Therefore, 2.22 follows from the above statements immediately.
10
Journal of Inequalities and Applications
Corollary 2.9. Let the conditions of Theorem 2.8 be satisfied with ck ≡ 1 for each k ≥ 1. Then for all
0 < p < 1 and each n ≥ 1,
E max gSk 1≤k≤n
p
≤
p
1 EgSn .
1−p
2.25
By Corollary 2.9, we can get the following theorem.
Theorem 2.10. Let {Sn , n ≥ 1} be a demisubmartingale and g a nondecreasing and nonnegative
convex function on R with g0 0 and gSi ∈ L1 , i ≥ 1. Let {bn , n ≥ 1} be a nondecreasing
unbounded sequence of positive numbers. If there exists some 0 < p < 1 such that
p
∞ EgS p − EgS
n
n−1 p
bn
n1
< ∞,
2.26
then limn → ∞ gSn /bn 0 a.s., and 1.9-1.10 hold (Sn is replaced by gSn ), where
αn p p '
1 &
EgSn − EgSn−1 ,
1−p
n ≥ 1.
2.27
In addition,
n
gSk p
≤ 4 αk < ∞,
E max p
bk 1≤k≤n
k1 b
k
∞
gSn p
αn
≤
4
E sup
p < ∞.
bn
n≥1
n1 bn
2.28
If further one assumes that αk > 0 for infinitely many k, then
∞
gSn p
αn
≤4
E sup
p < ∞.
β
n
n≥1
n1 βn
2.29
Similar to the proof of Theorem 2.8 and using Lemma 1.9i, we can get the following.
Theorem 2.11. Let {Sn , n ≥ 1} be a nonnegative demisubmartingale. Then for all 0 < p < 1,
Emax1≤k≤n Sk p ≤ 1/1 − pESn p .
Acknowledgments
The authors are most grateful to the Editor Andrei Volodin and an anonymous referee for
the careful reading of the manuscript and valuable suggestions which helped in significantly
improving an earlier version of this paper. This work was supported by the National Natural
Science Foundation of China Grant no. 10871001, 60803059, Talents Youth Fund of Anhui
Province Universities Grant no. 2010SQRL016ZD, Youth Science Research Fund of Anhui
University Grant no. 2009QN011A, Provincial Natural Science Research Project of Anhui
Colleges and the Innovation Group Foundation of Anhui University.
Journal of Inequalities and Applications
11
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