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Volume 2010, Article ID 569759, 14 pagesdoi:10.1155/2010/569759 Research Article A H ´ajek-R ´enyi-Type Maximal Inequality and Strong Laws of Large Numbers for Multidimensional Arrays 1

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Volume 2010, Article ID 569759, 14 pages

doi:10.1155/2010/569759

Research Article

A H ´ajek-R ´enyi-Type Maximal Inequality and

Strong Laws of Large Numbers for

Multidimensional Arrays

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,vanhuandhdt@yahoo.com

Received 1 July 2010; Accepted 27 October 2010

Academic Editor: Alexander I Domoshnitsky

Copyrightq 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 H´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 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 d

0, and the set of all

positive integer d-dimensional lattice points will be denoted by d We will write1, m, n, and

n 1 for points 1, 1, , 1, m1, m2, , m d , n1, n2, , n d , and n1 1, n2 1, , n d 1, respectively The notationm n or n m means that m i n i for all i 1, 2, , d, the

limit n → ∞ is interpreted as n i → ∞ for all i 1, 2, , d this limit is equivalent to

min{n1, n2, , n d} → ∞, and we define |n| d

i1 n i 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 1knΔbk for alln ∈ d For

example, if d 2, then for all i, j ∈ 2,Δb ij b ij − b i,j−1 − b i−1,j b i−1,j−1with the convention

that b 0,0 b i,0 b 0,j 0 We say that {bn, n ∈ d } is a nondecreasing array if bk bl for any pointsk l.

H´ajek and R´enyi 1 proved the following important inequality: If X j , j 1 is a sequence ofreal-valued independent random variables with zero means and finite second

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moments, andb j , j 1 is a nondecreasing sequence of positive real numbers, then for any

ε > 0 and for any positive integers n, n0 n0< n,

⎛

⎝ max

n0 i n

1

b i

i

j1

X j

ε

⎞

ε2

⎛

⎝n0

j1

X2

j

b2n0

n

jn0 1

X2

j

b2j

⎞

This inequality is a generalization of the Kolmogorov inequality and is a useful tool to prove the strong law of large numbers Fazekas and Klesov2 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 T ómács 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

bn

1kn

where {bn, n ∈ d } is a positive, nondecreasing d-sequence of product type, that is, bn

d

i1 b i n i, where{b n i i , n i 1} is a nondecreasing sequence of positive real numbers for each

i 1, 2, , d Then, we have

bn

1kn

Δbk  b1n1b n22 · · · b d n d , n ∈ d 1.3 This implies that

Δbn b1n1 − b n11−1 b2n2 − b n22−1· · · b d n d − b d n d−1, n ∈ d 1.4 Therefore,

ΔbnΔbn1 Δb n1n2···n d−1 ,n d1Δb n11,n21, ,n d−1 1,n d , 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 satisfying1.5, it is not possible to guarantee that 1.6 holds for details, seeExample 2.8in 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 satisfying1.5, but the reverse is not true

In this paper, we use the hypothesis that {bn, n ∈ d} is an array of positive real numbers satisfying1.5 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

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The paper is organized as follows In the rest of this section, we recall some definitions and present some lemmas.Section 2is devoted to our main results and their proofs

LetΩ, F, be a probability space A family {Fn, n ∈ d

0} of nondecreasing

sub-σ-algebras ofF related to the partial order on d

0 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, letBE be the σ-algebra of all Borel sets in E, and let {Xn, n ∈ d}

be an array of random elements such that Xn isFn/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 ∈ d

0}, for n ∈ d

0, we set

F1

n

k i 12 i d

Fn1k2k3···k d : ∞

k2 1

∞

k3 1

· · ·∞

k d1

Fn1k2k3···k d ,

Fj

n

k i 11 i j−1

k i 1j1 i d

Fk1···k j−1 n j k j1 ···k d if 1 < j < d,

Fd

n

k i 11 i d−1

Fk1k2···k d−1 n d ,

1.7

in the case d 1, we set F1

n Fn

An adapted array {Xn, Fn, n ∈ d } is said to be a martingale diﬀerence array if

Xn|Fi

n−1 0 for all n ∈ d and for all i 1, 2, , d.

In Quang and Huan5 , 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 spaceE is said to be p-uniformly smooth 1 p 2 if

ρ τ sup x y x − y

2 − 1, ∀x, y ∈ E, x 1, y τ Oτ p . 1.8

A Banach spaceE is said to be p-smoothable if there exists an equivalent norm under which E

is p-uniformly smooth.

Pisier6 proved that a real separable Banach space E is p-smoothable 1 p 2

if and only if there exists a positive constant C such that for every L pintegrableE-valued

martingale diﬀerence sequence {Xj , 1 j n},

n

j1

X j

p

C

n

j1

X jp

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 withY1 1 Y1 −1 1/2 Let E∞ E × E × E × · · · and define

E 

⎧

⎨

⎩v1, v2, ∈ E∞:

∞

j1

Y j v j converges inprobability

⎫

⎬

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Let 1 p 2 Then,E is said to be of Rademacher type p if there exists a positive constant C

such that

∞

j1

Y j v j

p

C

∞

j1

v jp ∀v1, v2, ∈ E. 1.11

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 theLp -spaces and p-spaces are of Rademacher type 2∧p for p1 The real lineis 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 Woyczy´nski 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 T ´om´acs10 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 ∈ d } be a

nondecreasing array of positive real numbers such that bn → ∞ as n → ∞ If

n1

then

1

bn

1kn

Proof For every ε > 0, there exists a point n0∈ dsuch that

k1

xk−

1kn0

Therefore, for alln n0,

bn

1kn

bkxk−

1kn0

bkxk

1kn

xk−

1kn0

xk

It means that

lim

n → ∞

1

bn

1kn

bkxk−

1kn

bkxk

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On the other hand, since bn → ∞ as n → ∞,

lim

n → ∞

1

bn

1kn0

Combining the above arguments, this completes the proof ofLemma 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 ∈ d } be an array of sets in F such

that An⊂ Amfor any points m n Then,

n1An

lim

n → ∞An. 1.18

Lemma 1.3 Let {Xn, n ∈ d } be an array of random elements If for any ε > 0,

lim

n → ∞

sup

kn Xk ε

then Xn → 0 a.s as n → ∞.

Proof For each i1, we have

n1

kn

Xk

1

i

lim

n → ∞

kn

Xk

1

i

by Lemma 1.2

lim

n → ∞

sup

kn Xk

1

i

 0.

1.20

Set

A 

i 1

n1

kn

Xk

1

i

Then,A 0 and for all ω /∈ A, for any i1, there exists a pointl ∈ d such that Xkω <

1/i for all k l It means that

Xk−→ 0 a.s as k −→ ∞. 1.22 The proof is completed

Lemma 1.4 Quang and Huan 5 Let 1 p 2, and let E be a real separable Banach space.

Then, the following two statements are equivalent.

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i The Banach space E is p-smoothable.

ii For every L p integrable martingale diﬀerence array {Xn, Fn, n ∈ d }, there exists a positive

constant C p,d (depending only on p and d) such that

1kn

Xk

p

C p,d

1kn

Xk p

2 Main Results

Theorem 2.1provides a H´ajek-R´enyi-type maximal inequality for multidimensional arrays of random elements This theorem is inspired by the work of Shorack and Smythe11

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 C p,d such that for any ε > 0 and for any points m n,

max

mkn

1

bk

1lk

Xl

ε

C p,d

ε p max

1kn

1lk

Xl

bl bm

p

Proof Since {bn, n ∈ d} is a nondecreasing array of positive real numbers,

max

mkn

1

bk

1lk

Xl

ε

max

mkn

1

bk bm

1lk

Xl

ε

2

max

1kn

1

bk bm

1lk

Xl

ε

2

.

2.2

Fork ∈ d, set

rk bk bm, Dk

1lk

Xl

Then, by interchanging the order of summation, we obtain the following

1lk

Xl

1lk

1tl

Δrt

Xl

rl

1tk

Δrt

tlk

Xl

rl

Thus, sinceΔrt0,

max

1kn

1

rk

1lk

Xl

2dmax1ln Dl 2.5

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By2.2 and 2.5 and the Markov inequality, we have

max

mkn

1

bk

1lk

Xl

ε

max

1ln Dl

ε

2d1

2pd1

ε p max

1ln Dl p

.

2.6

This completes the proof of the theorem

Now, we useTheorem 2.1to 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 ∈ d}

be an array of positive real numbers satisfying1.5 and bn → ∞ as n → ∞, and let {Xn, n ∈ d}

be an array of random elements in a real separable Banach space such that for any points m n,

max

1kn

1lk

Xl

bl bm

p

C

1kn

ak

Then, the condition

n1

an

implies1.2.

Proof By2.7 andTheorem 2.1, for any ε > 0 and for any points m n, we have

max

mkn

1

bk

1lk

Xl

ε

C

ε p

1kn

ak

This implies, by lettingn → ∞, that

sup

km

1

bk

1lk

Xl

ε

C

ε p

k1

ak

bk bmp

C

ε p

1km

ak

b pm

k1

ak

b pk −

1km

ak

b pk

.

2.10

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Lettingm → ∞, by 2.8 andLemma 1.1, we obtain

lim

m → ∞

sup

km

1

bk

1lk

Xl

ε

Lemma 1.3ensures that1.2 holds The proof is completed

The next theorem provides three characterizations of p-smoothable Banach spaces The

equivalence ofi and ii is an improvement of a result of Quang and Huan 5 stated as Lemma 1.4above

Theorem 2.3 Let 1 p 2, and let E be a real separable Banach space Then, the following four

statements are equivalent.

ii For every L p integrable martingale diﬀerence array {Xn, Fn, n ∈ d }, there exists a positive

constant C p,d such that

max

1kn

1lk

Xl

p

C p,d

1kn

Xk p

iii For every L p 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 C p,d such that

max

mkn

1

bk

1lk

Xl

ε

C p,d

ε p

1kn

Xk

bk bm

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

n1

Xn p

implies1.2.

Proof. i⇒ii: We easily obtain 2.12 in the case p 1 Now, we consider the case 1 < p 2

By virtue ofLemma 1.4, it suﬃces to show that

max

1kn

1lk

Xl

p

p − 1

pd

1kn

Xk

p

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.

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Fork ∈ D, we set

Sk

1lk

Xl, Y k D max

Then,

S k1k2···k D−1 k D | FD

k1k2···k D−1 ,k D−1

S k1k2···k D−1 ,k D−1| FD

k1k2···k D−1 ,k D−1

⎛

1 l i k i 1 i D−1

X l1l2···l D−1 k D | FD

k1k2···k D−1 ,k D−1

⎞

⎠

 S k1k2···k D−1 ,k D−1.

2.17

Therefore,

Y k D | FD

k1k2···k D−1 ,k D−1

max

1 k i n i 1 i D−1 Sk | FD

k1k2···k D−1 ,k D−1

1 k i n i 1 i D−1

Sk| FD

k1k2···k D−1 ,k D−1

 Y k D−1

2.18

It means that{Y k D , F D

k1k2···k D−1 k D , k D 1} is a nonnegative submartingale Applying Doob’s inequality, we obtain

max

1kn Sk p

max

1 k D n D

Y k D

p

p − 1

p

Y n p D

p

p − 1

p

1 k i n i 1 i D−1 S k1k2···k D−1 n D p

.

2.19

We set

X k D−1

1k2···k D−1 n D

k 1

X k1k2···k D−1 k D , FD−1 k

1k2···k D−1 ∞

k 1

Fk1k2···k D−1 k D 2.20

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Then we again have that {X k D−1

1k2···k D−1 , F D−1 k

1k2···k D−1 , k1, k2, , k D−1 ∈ D−1} is a martingale diﬀerence array Therefore, by the inductive assumption, we obtain

1 k i n i 1 i D−1 S k1k2···k D−1 n D p

1 k i n i 1 i D−1

1 l i k i1 i D−1

X D−1 l

1l2···l D−1

p

p − 1

pD−1

1 l i n i1 i D−1

X l D−11l2···l D−1

p

p − 1

pD−1

S n1n2···n D p

2.21

Combining2.19 and 2.21 yields that 2.15 holds for d D.

ii ⇒ iii: let {Xn, Fn, n ∈ d } be an arbitrary L p integrable martingale diﬀerence array Then, for allm ∈ d , {Xn/bn bm, Fn, n ∈ d } is also an L p integrable martingale diﬀerence array Therefore, the assertion ii andTheorem 2.1ensure that2.13 holds

iii ⇒ iv: the proof of this implication is similar to the proof ofTheorem 2.2and is therefore omitted

iv ⇒ i: for a given positive integer d, assume that iv holds Let {X j , F j , j 1} be

an arbitrary martingale diﬀerence sequence such that

∞

j1

X jp

Forn ∈ d, set

Xn X n1 if n i 1 2 i d ,

Xn 0 if there exists a positive integer i2 i d such that n i > 1,

Fn Fn1, bn n1.

2.23

Then,{Xn, Fn, n ∈ d } is a martingale diﬀerence array, and {bn, n ∈ d} is an array of positive real numbers satisfying1.5 and bn → ∞ as n → ∞ Moreover, we see that

n1

Xn p

b pn ∞

n1

X n1 p

n p1 < ∞, 2.24

equivalence of i and ii is an improvement of a result of Quang and Huan 5 stated as Lemma 1.4above

Theorem 2.3 Let 1 p 2, and let E be a real separable Banach space Then,... class="text_page_counter">Trang 6

ii For every L p integrable martingale... for all q p Every real separable Banach space is of< /i>

Rademacher type 1, while theLp -spaces and p-spaces are of Rademacher type 2∧p for

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