Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 569759, 14 pages
doi:10.1155/2010/569759
Research Article
AH´ajek-R´enyi-Type Maximal Inequality and
Strong Laws of Large Numbers for
Multidimensional Arrays
Nguyen Van Quang
1
and Nguyen Van Huan
2
1
Department of Mathematics, Vinh University, Nghe An 42000, Vietnam
2
Department of Mathematics, Dong Thap University, Dong Thap 871000, Vietnam
Correspondence should be addressed to Nguyen Van Huan,
Received 1 July 2010; Accepted 27 October 2010
Academic Editor: Alexander I. Domoshnitsky
Copyright q 2010 N. V. Quang and N. Van Huan. 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.
AH
´
ajek-R
´
enyi-type maximal inequality is established for multidimensional arrays of random
elements. Using this result, we establish some strong laws of large numbers for multidimensional
arrays. We also provide some characterizations of Banach spaces.
1. Introduction and Preliminaries
Throughout this paper, the symbol C will denote a generic positive constant which is not
i
for all i 1, 2, ,d,the
limit n →∞is interpreted as n
i
→∞for a ll i 1, 2, ,d this limit is equivalent to
min{n
1
,n
2
, ,n
d
}→∞,andwedefine|n|
d
i1
n
i
.
Let {b
n
, n ∈
d
} be a d-dimensional array of real numbers. We define Δb
n
to be the
dth-order finite difference of the b’s at the point n.Thus,b
n
1kn
b
l
for any
points k l.
H
´
ajek and R
´
enyi 1 proved the following important inequality: If X
j
,j 1 is a
sequence of real-valued independent random variables with zero means and finite second
2 Journal of Inequalities and Applications
moments, and b
j
,j 1 is a nondecreasing sequence of positive real numbers, then for any
ε>0 and for any positive integers n, n
0
n
0
<n,
⎛
⎝
max
n
0
i n
1
b
i
2
j
b
2
n
0
n
jn
0
1
X
2
j
b
2
j
⎞
⎠
. 1.1
This inequality is a generalization of the Kolmogorov inequality and is a useful tool to
prove the strong law of large numbers. Fazekas and Klesov 2 gave a general method for
obtaining the strong law of lar ge numbers for sequences of random variables by using a
H
´
ajek-R
´
enyi-type maximal inequality. Afterwards, Nosz
´
,where{b
i
n
i
,n
i
1} is a nondecreasing sequence of positive real numbers for each
i 1 , 2, ,d. Then, we have
b
n
1kn
Δb
k
b
1
n
1
b
2
n
2
···b
d
n
d
, n ∈
d
. 1.3
d
− b
d
n
d
−1
, n ∈
d
. 1.4
Therefore,
Δb
n
0, n ∈
d
,
1.5
Δb
n
Δb
n1
Δb
n
1
n
2
···n
d−1
,n
d
d
} is an array of positive real
numbers satisfying 1.5 and continue to study the problem of finding the sufficient condition
for the strong law of large numbers 1.2. We also establish a H
´
ajek-R
´
enyi-type maximal
inequality for multidimensional arrays of random elements and some maximal moment
inequalities for arrays of dependent random elements.
Journal of Inequalities and Applications 3
The paper is organized as follows. In the rest of this section, we recall some definitions
and present some lemmas. Section 2 is devoted to our main results and their proofs.
Let Ω, F,
be a probability space. A family {F
n
, n ∈
d
0
} of nondecreasing sub-σ-
algebras of F related to the partial order on
d
0
is said to be a stochastic basic.
Let {F
n
, n ∈
d
0
} be a stochastic basic such that F
F
1
n
k
i
1 2 i d
F
n
1
k
2
k
3
···k
d
:
∞
k
2
1
∞
k
3
1
···
∞
j1 i d
F
k
1
···k
j−1
n
j
k
j1
···k
d
if 1 <j<d,
F
d
n
k
i
1 1 i d−1
F
k
1
k
2
···k
d−1
n
τ
sup
x y
x − y
2
− 1, ∀x, y ∈ E,
x
1,
y
τ
O
p
C
n
j1
X
j
p
. 1.9
In Quang and Huan 5, this inequality was used to define p-uniformly smooth Banach
spaces.
Let {Y
j
,j 1} be a sequence of independent identically distributed random variables
with
Y
1
1 Y
1
−11/2. Let E
∞
E × E × E ×··· and define
E
∞
j1
Y
j
v
j
p
C
∞
j1
v
j
´
om
´
acs 10 and is a multidimensional version of
the Kronecker lemma.
Lemma 1.1. Let {x
n
, n ∈
d
} be an array of nonnegative real numbers, and let {b
n
, n ∈
d
} be a
nondecreasing array of positive real numbers such that b
n
→∞as n →∞.If
n1
x
n
< ∞, 1.12
then
1
b
n
1kn
b
k
k
x
k
−
1kn
0
b
k
x
k
1kn
x
k
−
1kn
0
x
k
ε. 1.15
It means that
lim
n →∞
1
b
n
k
x
k
0. 1.17
Combining the above arguments, this completes the proof of Lemma 1.1.
The proof of the next lemma is very simple and is therefore omitted.
Lemma 1.2. Let Ω, F,
be a probability space, and let {A
n
, n ∈
d
} be an array of sets in F such
that A
n
⊂ A
m
for any points m n.Then,
n1
A
n
lim
n →∞
A
n
. 1.18
k
1
i
lim
n →∞
kn
X
k
1
i
by Lemma 1.2
lim
n →∞
sup
kn
X
k
ω <
1/i for all k l.Itmeansthat
X
k
−→ 0 a.s. as k −→ ∞ . 1.22
The proof is completed.
Lemma 1.4 Quang and Huan 5. Let 1 p 2,andletE be a real separable Banach space.
Then, the following two statements are equivalent.
6 Journal of Inequalities and Applications
i The Banach space E is p-smoothable.
ii For every L
p
integrable martingale difference array {X
n
, F
n
, n ∈
d
}, there exists a positive
constant C
p,d
(depending only on p and d)suchthat
1kn
d
} be an array of positive real numbers satisfying 1.5,and
let {X
n
, n ∈
d
} be an array of random elements in a real separable Banach space. Then, there exists a
positive constant C
p,d
such that for any ε>0 and for any points m n,
max
mkn
1
b
k
1lk
X
l
, n ∈
d
} is a nondecreasing array of positive real numbers,
max
mkn
1
b
k
1lk
X
l
ε
max
mkn
1
b
1lk
X
l
ε
2
.
2.2
For k ∈
d
,set
r
k
b
k
b
m
,D
k
tlk
X
l
r
l
. 2.4
Thus, since Δr
t
0,
max
1kn
1
r
k
1lk
X
l
ε
max
1ln
D
l
ε
2
d1
2
pd1
ε
p
max
1ln
D
l
p
.
2.6
This completes the proof of the theorem.
Now, we use Theorem 2.1 to prove a strong law of large numbers for multidimensional
arrays of random elements. This result is inspired by Theorem 3.2 of Klesov et al. 4.
b
m
p
C
1kn
a
k
b
k
b
m
p
. 2.7
Then, the condition
n1
a
n
b
p
n
< ∞ 2.8
k
b
k
b
m
p
. 2.9
This implies, by letting n →∞,that
sup
km
1
b
k
1lk
X
l
k1
a
k
b
p
k
−
1km
a
k
b
p
k
.
2.10
8 Journal of Inequalities and Applications
Letting m →∞,by2.8 and Lemma 1.1,weobtain
lim
m →∞
sup
km
1
b
k
}, there exists a positive
constant C
p,d
such that
max
1kn
1lk
X
l
p
C
p,d
1kn
X
k
p
1lk
X
l
ε
C
p,d
ε
p
1kn
X
k
b
k
b
m
Proof. i⇒ii: We easily obtain 2.12 in the case p 1. Now, we consider the case 1 <p
2.
By virtue of Lemma 1.4,itsuffices to show that
max
1kn
1lk
X
l
p
p
p − 1
pd
max
1 k
i
n
i
1 i D−1
S
k
. 2.16
Then,
S
k
1
k
2
···k
D−1
k
D
|F
D
k
1
k
2
···k
D−1
⎝
1 l
i
k
i
1 i D−1
X
l
1
l
2
···l
D−1
k
D
|F
D
k
1
k
2
···k
D−1
,k
D
−1
⎞
⎠
S
max
1 k
i
n
i
1 i D−1
S
k
|F
D
k
1
k
2
···k
D−1
,k
D
−1
max
1 k
i
n
i
1 i D−1
1
k
2
···k
D−1
k
D
,k
D
1} is a nonnegative submartingale. Applying Doob’s
inequality, we obtain
max
1kn
S
k
p
max
1 k
D
n
D
Y
k
D
p
D
p
.
2.19
We set
X
D−1
k
1
k
2
···k
D−1
n
D
k
D
1
X
k
1
k
2
···k
D−1
k
D
2
···k
D−1
, F
D−1
k
1
k
2
···k
D−1
, k
1
,k
2
, ,k
D−1
∈
D−1
} is a martingale
difference array. Therefore, by the inductive assumption, we obtain
max
1 k
i
n
i
1 i D−1
S
k
1
l
2
···l
D−1
p
p
p − 1
pD−1
1 l
i
n
i
1 i D−1
X
.
2.21
Combining 2.19 and 2.21 yields that 2.15 holds for d D.
ii ⇒ iii:let{X
n
, F
n
, n ∈
d
} be an arbitrary L
p
integrable martingale difference
array. Then, for all m ∈
d
, {X
n
/b
n
b
m
, F
n
, n ∈
d
} is also an L
p
integrable martingale
difference array. Therefore, the assertion ii and Theorem 2.1 ensure that 2.13 holds.
iii ⇒ iv: t he proof of this implication is similar to the proof of Theorem 2.2 and is
therefore omitted.
2 i d
,
X
n
0 if there exists a positive integer i
2 i d
such that n
i
> 1,
F
n
F
n
1
,b
n
n
1
.
2.23
Then, {X
n
, F
n
, n ∈
d
n
p
1
< ∞, 2.24
Journal of Inequalities and Applications 11
and so 1.2 holds. It means that
1
n
1
n
1
j1
X
j
−→ 0 a.s. as n
1
−→ ∞ . 2.25
Then, by Theorem 2.2 of Hoffmann-Jørgensen and Pisier 12, E is p-smoothable.
Remark 2.4. The inequality 2.15 holds for every p>1 and for every martingale difference
array without imposing any geometric condition on the Banach space.
In the case d 1, Theorem 2.3 reduces to the following corollary which was proved by
Gan 13 and Gan and Qiu 14.
Corollary 2.5. Let 1
p 2,andletE be a real separable Banach space. Then, the following three
statements are equivalent.
i The Banach space E is p-smoothable.
ii For every L
p
integrable martingale difference sequence {X
X
j
ε
⎞
⎠
C
ε
p
⎛
⎝
n
0
j1
X
j
p
b
p
n
∞
j1
X
j
p
b
p
j
< ∞ 2.27
implies
1
b
i
i
j1
X
j
−→ 0 a.s. as i −→ ∞ . 2.28
Remark 2.4 ensures that the inequality 2.15 holds for every p>1 and for every array
of independent mean zero random elements in a real separable Banach space. Therefore, by
using the implication 2.1.1 ⇒ 2.1.2 of Theorem 2.1 of Hoffmann-Jørgensen and Pisier
12 and the same arguments as in the proof of Theorem 2.3,wegetthefollowingtheorem
which generalizes some results given by Christofides and Serfling 15 and Gan and Qiu 14.
We omit its proof.
iv For every array of independent mean zero random elements {X
n
, n ∈
d
}, for every array of
positive real numbers {b
n
, n ∈
d
} satisfying 1.5 and b
n
→∞as n →∞, the condition
2.14 implies 1.2.
We close this paper by giving a remark on Theorem 2.6 and an example which
illustrates Theorems 2.2, 2.3,and2.6.
Remark 2.7. By the same method as in the proof of Lemma 3 of M
´
oricz et al. 16 and the
same arguments as in the proof of Theorem 2.3,wecanextendTheorem 2.6 to M-dependent
random fields.
Example 2.8. Let d be a positive integer d
2, and let {X
n
, n ∈
d
} be an array of
independent random variables with
X
n
1
,n
2
, ,n
d
} n ∈
d
. Then,
Δb
n
⎧
⎨
⎩
2ifn
1
n
2
··· n
d
,
1otherwise.
2.30
It means that {b
n
, n ∈
d
} is an array of positive real numbers satisfying 1.5 and b
n
→∞
1lk
X
l
r
l
2
C
1kn
|
X
k
|
2
r
2
k
, n ∈
d
by Theorem 2.3
n1
|
−1
n
1kn
X
k
−→ 0 a.s. as
|
n
|
−→ ∞ , 2.32
where {b
n
, n ∈
d
} is one of the special kinds of positive, nondecreasing d-sequences of
product type. For more details, the reader may refer to 17–19. Therefore, this example also
shows that the implications i ⇒ iv of Theorem 2.3 and i ⇒ iv of Theorem 2.6 are
independent of results obtained in 17–19.
Acknowledgments
The authors are grateful to the referee for carefully reading the paper and for offering some
comments which helped to improve the paper. This research was supported by the National
Foundation for Science Technology Development, Vietnam NAFOSTED, no. 101.02.32.09.
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´
ajek and A. R
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enyi, “Generalization of an inequality of Kolmogorov,” Acta Mathematica Academiae
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om
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acs, “Strong laws of large numbers for pairwise independent random variables
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`
ajek-R
`
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