Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 569759, 14 pages
doi:10.1155/2010/569759
Research Article
A Hájek-Rényi-Type Maximal Inequality and
Strong Laws of Large Numbers for
Multidimensional Arrays
Nguyen Van Quang1 and Nguyen Van Huan2
1
2
Department of Mathematics, Vinh University, Nghe An 42000, Vietnam
Department of Mathematics, Dong Thap University, Dong Thap 871000, Vietnam
Correspondence should be addressed to Nguyen Van Huan, vanhuandhdt@yahoo.com
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.
A Hájek-Rényi-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
necessarily the same one in each appearance. Let d be a positive integer, the set of all
nonnegative integer d-dimensional lattice points will be denoted by d0 , and the set of all
positive integer d-dimensional lattice points will be denoted by d . We will write 1, m, n, and
n 1 for points 1, 1, . . . , 1, m1 , m2 , . . . , md , n1 , n2 , . . . , nd , and n1 1, n2 1, . . . , nd 1,
ni for all i 1, 2, . . . , d, the
respectively. The notation m n or n m means that mi
limit n → ∞ is interpreted as ni → ∞ for all i 1, 2, . . . , d this limit is equivalent to
min{n1 , n2 , . . . , nd } → ∞, and we define |n| di1 ni .
Let {bn , n ∈ d } be a d-dimensional array of real numbers. We define Δbn to be the
dth-order finite difference of the b’s at the point n. Thus, bn 1kn Δbk for all n ∈ d . For
example, if d 2, then for all i, j ∈ 2 , Δbij bij − bi,j−1 − bi−1,j bi−1,j−1 with the convention
bl for any
that b0,0 bi,0 b0,j 0. We say that {bn , n ∈ d } is a nondecreasing array if bk
points k l.
Hájek and Rényi 1 proved the following important inequality: If Xj , 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 bj , j 1 is a nondecreasing sequence of positive real numbers, then for any
ε > 0 and for any positive integers n, n0 n0 < n,
⎞
i
1
⎝ max Xj ε⎠
n0 i n bi j1
⎛
⎛
⎞
n0 X 2
n X2
1 ⎝
j
j
⎠.
2
ε2 j1 bn2 0
jn0 1 bj
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 large numbers for sequences of random variables by using a
Hájek-Rényi-type maximal inequality. Afterwards, Noszály and Tómács 3 extended this
result to multidimensional arrays see also Klesov et al. 4. They provided a sufficient
condition for d-dimensional arrays of random variables to satisfy the strong law of large
numbers
1 Xk −→ 0 a.s. as n −→ ∞,
bn 1kn
1.2
where {bn , n ∈ d } is a positive, nondecreasing d-sequence of product type, that is, bn d i
i
i1 bni , where {bni , ni 1} is a nondecreasing sequence of positive real numbers for each
i 1, 2, . . . , d. Then, we have
bn 1kn
1 2
d
Δbk bn1 bn2 · · · bnd ,
n∈
d
1.3
.
This implies that
1
1
2
2
d
d
Δbn bn1 − bn1 −1 bn2 − bn2 −1 · · · bnd − bnd −1 ,
n∈
d
.
1.4
Therefore,
Δbn
0,
n∈
d
1.5
,
Δbn Δbn1 Δbn1 n2 ···nd−1 ,nd 1 Δbn1 1,n2 1,...,nd−1 1,nd ,
n∈
d
.
1.6
On the other hand, we can show that under the assumption that {bn , n ∈ d } is an array
of positive real numbers satisfying 1.5, it is not possible to guarantee that 1.6 holds for
details, see Example 2.8 in the next section.
Thus, if {bn , n ∈ d } is a positive, nondecreasing d-sequence of product type, then it is
an array of positive real numbers satisfying 1.5, but the reverse is not true.
In this paper, we use the hypothesis that {bn , n ∈ 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 Hájek-Rényi-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 {Fn , n ∈ d0 } of nondecreasing sub-σalgebras of F related to the partial order on d0 is said to be a stochastic basic.
Let {Fn , n ∈ d0 } be a stochastic basic such that Fn {∅, Ω} if |n| 0, let E be a real
separable Banach space, let BE be the σ-algebra of all Borel sets in E, and let {Xn , n ∈ d }
be an array of random elements such that Xn is Fn /BE-measurable for all n ∈ d . Then
{Xn , Fn , n ∈ d } is said to be an adapted array.
For a given stochastic basic {Fn , n ∈ d0 }, for n ∈ d0 , we set
Fn1 Fn1 k2 k3 ···kd :
ki 1 2 i d
j
Fn ki 1 1 i j−1 ki 1 j1 i d
Fnd ∞ ∞
k2 1 k3 1
∞
···
Fk1 ···kj−1 nj kj1 ···kd
Fn1 k2 k3 ···kd ,
kd 1
if 1 < j < d,
1.7
Fk1 k2 ···kd−1 nd ,
ki 1 1 i d−1
in the case d 1, we set Fn1 Fn .
An adapted array {Xn , Fn , n ∈ d } is said to be a martingale difference array if
i
Xn |Fn−1
0 for all n ∈ d and for all i 1, 2, . . . , d.
In Quang and Huan 5, the authors showed that the set of all martingale difference
arrays is really larger than the set of all arrays of independent mean zero random elements.
A Banach space E is said to be p-uniformly smooth 1 p 2 if
x y x − y ρτ sup
− 1, ∀x, y ∈ E, x 1, y τ Oτ p .
2
1.8
A Banach space E is said to be p-smoothable if there exists an equivalent norm under which E
is p-uniformly smooth.
p
2
Pisier 6 proved that a real separable Banach space E is p-smoothable 1
if and only if there exists a positive constant C such that for every Lp integrable E-valued
martingale difference sequence {Xj , 1 j n},
p
n
Xj j1 C
n X j p .
1.9
j1
In Quang and Huan 5, this inequality was used to define p-uniformly smooth Banach
spaces.
Let {Yj , j 1} be a sequence of independent identically distributed random variables
with Y1 1 Y1 −1 1/2. Let E∞ E × E × E × · · · and define
⎫
⎧
∞
⎬
⎨
Yj vj converges inprobability .
E v1 , v2 , . . . ∈ E∞ :
⎭
⎩
j1
1.10
4
Let 1 p
such that
Journal of Inequalities and Applications
2. Then, E is said to be of Rademacher type p if there exists a positive constant C
p
∞
Yj vj j1
C
∞ vj p
∀v1 , v2 , . . . ∈ E.
1.11
j1
It is well known that if a real separable Banach space is of Rademacher type p1 p
2, then it is of Rademacher type q for all 1 q p. Every real separable Banach space is of
Rademacher type 1, while the Lp -spaces and p -spaces are of Rademacher type 2∧p for p 1.
The real line is of Rademacher type 2. Furthermore, if a Banach space is p-smoothable, then
it is of Rademacher type p. For more details, the reader may refer to Borovskikh and Korolyuk
7, Pisier 8, and Woyczyński 9.
Now, we present some lemmas which will be needed in what follows. The first lemma
is a variation of Lemma 2.6 of Fazekas and Tómács 10 and is a multidimensional version of
the Kronecker lemma.
Lemma 1.1. Let {xn , n ∈ d } be an array of nonnegative real numbers, and let {bn , n ∈
nondecreasing array of positive real numbers such that bn → ∞ as n → ∞. If
xn < ∞,
d
} be a
1.12
n1
then
1 bk xk −→ 0
bn 1kn
Proof. For every ε > 0, there exists a point n0 ∈
as n −→ ∞.
d
such that
xk −
xk
k1
1.13
1.14
ε.
1kn0
Therefore, for all n n0 ,
0
1
bn
bk xk −
1kn
bk xk
1kn0
xk −
1kn
xk
ε.
1.15
1kn0
It means that
1
lim
n → ∞ bn
1kn
bk xk −
1kn0
bk xk
0.
1.16
Journal of Inequalities and Applications
5
On the other hand, since bn → ∞ as n → ∞,
1 lim
n → ∞ bn
bk xk 0.
1.17
1kn0
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 {An , n ∈
that An ⊂ Am for any points m n. Then,
Lemma 1.3. Let {Xn , n ∈
d
d
} be an array of sets in F such
lim An .
An
1.18
n→∞
n1
} be an array of random elements. If for any ε > 0,
lim sup Xk ε
n→∞
0,
kn
1.19
then Xn → 0 a.s. as n → ∞.
Proof. For each i 1, we have
n1 kn
Xk 1
i
lim n→∞
1
Xk i
kn
lim supXk n→∞
kn
1
i
by Lemma 1.2
1.20
0.
Set
A
i 1 n1 kn
Xk 1
.
i
Then, A 0 and for all ω /
∈ A, for any i 1, there exists a point l ∈
1/i for all k l. It means that
Xk −→ 0
a.s. as k −→ ∞.
1.21
d
such that Xk ω <
1.22
The proof is completed.
Lemma 1.4 Quang and Huan 5. Let 1
p
Then, the following two statements are equivalent.
2, and let E be a real separable Banach space.
6
Journal of Inequalities and Applications
i The Banach space E is p-smoothable.
ii For every Lp integrable martingale difference array {Xn , Fn , n ∈
constant Cp,d (depending only on p and d) such that
p
Xk 1kn Cp,d
Xk p , n ∈
d
d
}, there exists a positive
1.23
.
1kn
2. Main Results
Theorem 2.1 provides a Hájek-Rényi-type maximal inequality for multidimensional arrays of
random elements. This theorem is inspired by the work of Shorack and Smythe 11.
Theorem 2.1. Let p > 0, let {bn , n ∈ d } be an array of positive real numbers satisfying 1.5, and
let {Xn , n ∈ d } be an array of random elements in a real separable Banach space. Then, there exists a
positive constant Cp,d such that for any ε > 0 and for any points m n,
1
Xl ε
max mkn bk 1lk
X p
Cp,d
l
max .
εp
1kn b
b
l
m
1lk
Proof. Since {bn , n ∈
d
} is a nondecreasing array of positive real numbers,
1
Xl ε
max mkn bk 1lk
For k ∈
d
2.1
1 ε
max
Xl 2
mkn bk bm 1lk
1 ε
.
max
Xl 2
1kn bk bm 1lk
2.2
, set
r k bk bm ,
Dk Xl
.
r
1lk l
2.3
Then, by interchanging the order of summation, we obtain the following
1lk
Thus, since Δrt
Xl 1lk
1tl
Δrt
Xl
Δrt
rl
1tk
Xl
r
tlk l
.
2.4
0,
1
Xl max 1kn rk 1lk
2d maxDl .
1ln
2.5
Journal of Inequalities and Applications
7
By 2.2 and 2.5 and the Markov inequality, we have
1
Xl ε
max mkn bk 1lk
maxDl 1ln
2
pd1
εp
ε
2d1
2.6
max Dl p .
1ln
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.
Theorem 2.2. Let p > 0, let {an , n ∈ d } be an array of nonnegative real numbers, let {bn , n ∈
be an array of positive real numbers satisfying 1.5 and bn → ∞ as n → ∞, and let {Xn , n ∈
be an array of random elements in a real separable Banach space such that for any points m n,
X p
l
max 1kn
b
b
m
1lk l
C
ak
.
bm p
1kn bk
d
d
}
}
2.7
Then, the condition
an
p
n1 bn
<∞
2.8
implies 1.2.
Proof. By 2.7 and Theorem 2.1, for any ε > 0 and for any points m n, we have
1
max Xl ε
mkn bk 1lk
C ak
.
εp 1kn bk bm p
2.9
This implies, by letting n → ∞, that
1
sup Xl ε
km bk 1lk
C
ak
εp k1 bk bm p
C
εp
ak
p
1km bm
ak
p
k1 bk
−
ak
p
1km bk
2.10
.
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Journal of Inequalities and Applications
Letting m → ∞, by 2.8 and Lemma 1.1, we obtain
1
Xl ε 0.
sup km bk 1lk
lim
m→∞
2.11
Lemma 1.3 ensures that 1.2 holds. The proof is completed.
The next theorem provides three characterizations of p-smoothable Banach spaces. The
equivalence of i and ii is an improvement of a result of Quang and Huan 5 stated as
Lemma 1.4 above.
Theorem 2.3. Let 1
p
statements are equivalent.
2, and let E be a real separable Banach space. Then, the following four
i The Banach space E is p-smoothable.
ii For every Lp integrable martingale difference array {Xn , Fn , n ∈
constant Cp,d such that
p
max Xl 1kn
1lk
Cp,d
Xk p , n ∈
d
d
}, there exists a positive
2.12
.
1kn
iii For every Lp integrable martingale difference array {Xn , Fn , n ∈ d }, for every array of
positive real numbers {bn , n ∈ d } satisfying 1.5, for any ε > 0, and for any points
m n, there exists a positive constant Cp,d such that
1
max Xl ε
mkn bk 1lk
Cp,d Xk p
.
εp 1kn bk bm 2.13
iv For every martingale difference array {Xn , Fn , n ∈ d }, for every array of positive real
numbers {bn , n ∈ d } satisfying 1.5 and bn → ∞ as n → ∞, the condition
Xn p
n1
p
bn
<∞
2.14
implies 1.2.
Proof. i⇒ii: We easily obtain 2.12 in the case p 1. Now, we consider the case 1 < p
By virtue of Lemma 1.4, it suffices to show that
p
max Xl 1kn 1lk
p
p−1
p
pd Xk ,
1kn n∈
d
.
2.
2.15
First, we remark that for d 1, 2.15 follows from Doob’s inequality. We assume that
2.15 holds for d D − 1 1, we wish to show that it holds for d D.
Journal of Inequalities and Applications
For k ∈
D
9
, we set
Sk YkD Xl ,
1lk
2.16
Sk .
max
1 ki ni 1 i D−1
Then,
Sk1 k2 ···kD−1 kD | FkD1 k2 ···kD−1 ,kD −1
Sk1 k2 ···kD−1 ,kD −1 | FkD1 k2 ···kD−1 ,kD −1
⎛
⎞
⎝
Xl1 l2 ···lD−1 kD |
1 li ki 1 i D−1
2.17
FkD1 k2 ···kD−1 ,kD −1 ⎠
Sk1 k2 ···kD−1 ,kD −1 .
Therefore,
YkD | FkD1 k2 ···kD−1 ,kD −1 1
Sk | FkD1 k2 ···kD−1 ,kD −1
max
1 ki ni 1 i D−1
Sk | FkD1 k2 ···kD−1 ,kD −1 max
2.18
ki ni 1 i D−1
YkD−1 .
It means that {YkD , FkD1 k2 ···kD−1 kD , kD
inequality, we obtain
max Sk p
1kn
1} is a nonnegative submartingale. Applying Doob’s
p
max YkD
1 kD nD
p
p−1
p
1 ki
max
p
p−1
ni 1 i D−1
p
YnpD
2.19
p
Sk1 k2 ···kD−1 nD .
We set
D−1
Xk1 k2 ···kD−1 nD
Xk1 k2 ···kD−1 kD ,
kD 1
D−1
Fk1 k2 ···kD−1 ∞
kD 1
Fk1 k2 ···kD−1 kD .
2.20
10
Journal of Inequalities and Applications
D−1
D−1
Then we again have that {Xk1 k2 ···kD−1 , Fk1 k2 ···kD−1 , k1 , k2 , . . . , kD−1 ∈
difference array. Therefore, by the inductive assumption, we obtain
1 ki
D−1
} is a martingale
Sk1 k2 ···kD−1 nD p
max
ni 1 i D−1
1 ki
max
ni 1 i D−1
1
pD−1 p
p−1
1
p
p−1
pD−1
p
li ki 1 i D−1
li ni 1 i
D−1 Xl1 l2 ···lD−1 p
D−1 Xl1 l2 ···lD−1 D−1
2.21
Sn1 n2 ···nD p .
Combining 2.19 and 2.21 yields that 2.15 holds for d D.
ii ⇒ iii: let {Xn , Fn , n ∈ d } be an arbitrary Lp integrable martingale difference
array. Then, for all m ∈ d , {Xn /bn bm , Fn , n ∈ d } is also an Lp integrable martingale
difference array. Therefore, the assertion ii and Theorem 2.1 ensure that 2.13 holds.
iii ⇒ iv: the proof of this implication is similar to the proof of Theorem 2.2 and is
therefore omitted.
iv ⇒ i: for a given positive integer d, assume that iv holds. Let {Xj , Fj , j 1} be
an arbitrary martingale difference sequence such that
∞ X p
j
jp
j1
For n ∈
d
< ∞.
2.22
, set
Xn Xn1
Xn 0
if ni 1 2
i
if there exists a positive integer i 2
Fn Fn1 ,
d,
i
d such that ni > 1,
2.23
b n n1 .
Then, {Xn , Fn , n ∈ d } is a martingale difference array, and {bn , n ∈ d } is an array of positive
real numbers satisfying 1.5 and bn → ∞ as n → ∞. Moreover, we see that
Xn p
n1
p
bn
∞
Xn1 p
n1 1
p
n1
< ∞,
2.24
Journal of Inequalities and Applications
11
and so 1.2 holds. It means that
n1
1
Xj −→ 0
n1 j1
a.s. as n1 −→ ∞.
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
statements are equivalent.
2, and let E be a real separable Banach space. Then, the following three
i The Banach space E is p-smoothable.
ii For every Lp integrable martingale difference sequence {Xj , Fj , j 1}, for every
nondecreasing sequence of positive real numbers {bj , j 1}, for any ε > 0, and for any
positive integers n, n0 n0 < n, there exists a positive constant C such that
⎞
i
1
⎝ max Xj ε⎠
n0 i n bi j1
⎛
⎛
⎞
n0 X p
n X p
C ⎝
j
j
⎠.
p
p
p
ε
b
b
n0
j1
jn0 1
j
2.26
iii For every martingale difference sequence {Xj , Fj , j 1} and for every nondecreasing
sequence of positive real numbers {bj , j 1} such that bj → ∞ as j → ∞, the condition
∞ X p
j
p
j1
bj
<∞
2.27
implies
i
1
Xj −→ 0
bi j1
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, we get the following theorem
which generalizes some results given by Christofides and Serfling 15 and Gan and Qiu 14.
We omit its proof.
p
Theorem 2.6. Let 1
statements are equivalent.
2, and let E be a real separable Banach space. Then, the following four
12
Journal of Inequalities and Applications
i The Banach space E is of Rademacher type p.
ii For every array of Lp integrable independent mean zero random elements {Xn , n ∈
there exists a positive constant Cp,d such that 2.12 holds.
d
},
iii For every array of Lp integrable independent mean zero random elements {Xn , n ∈ d }, for
every array of positive real numbers {bn , n ∈ d } satisfying 1.5, for any ε > 0, and for
any points m n, there exists a positive constant Cp,d such that 2.13 holds.
iv For every array of independent mean zero random elements {Xn , n ∈ d }, for every array of
positive real numbers {bn , n ∈ d } satisfying 1.5 and bn → ∞ 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, and 2.6.
Remark 2.7. By the same method as in the proof of Lemma 3 of Móricz et al. 16 and the
same arguments as in the proof of Theorem 2.3, we can extend Theorem 2.6 to M-dependent
random fields.
Example 2.8. Let d be a positive integer d
independent random variables with
2, and let {Xn , n ∈
d
} be an array of
1
2
Xn −|n|1/4 Xn |n|1/4 .
2.29
Then, {Xn , n ∈ d } is an array of independent mean zero random variables taking values in
the 2-smoothable Banach space using the absolute value as norm.
Let bn |n| min{n1 , n2 , . . . , nd } n ∈ d . Then,
Δbn ⎧
⎨2 if n1 n2 · · · nd ,
2.30
⎩1 otherwise.
It means that {bn , n ∈ d } is an array of positive real numbers satisfying 1.5 and bn → ∞
as n → ∞. Moreover, by virtue of 1.6, we can show that {bn , n ∈ d } is not a positive,
nondecreasing d-sequence of product type. Therefore, 1.2 does not follow from Theorem
3.2 of Klesov et al. 4. But for every array of positive real numbers {rn , n ∈ d }, {Xn /rn , Fn σXk , 1 k n, n ∈ d } is a martingale difference array such that
X 2
l
max 1kn
r 1lk l
C
|Xk |2
1kn
rk2
|Xk |2
n1
bn2
,
n∈
|n|1/2
n1
|n|2
d
by Theorem 2.3
2.31
< ∞,
and so 2.7 and 2.8 are satisfied, Theorem 2.2 ensures that 1.2 holds.
Journal of Inequalities and Applications
13
As we know, the limit |n| → ∞ is equivalent to max{n1 , n2 , . . . , nd } → ∞. Recently,
some authors have derived the sufficient conditions for the strong law of large numbers
bn−1
Xk −→ 0 a.s. as |n| −→ ∞,
2.32
1kn
where {bn , 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.
References
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2 I. Fazekas and O. Klesov, “A general approach to the strong laws of large numbers,” Theory of
Probability and Its Applications, vol. 45, no. 3, pp. 436–449, 2002.
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