A H ´ajek-R ´enyi-Type Maximal Inequality and Strong Laws of Large Numbers for

Volume 2010, Article ID 569759,14pages doi:10.1155/2010/569759

Research Article

A H ´ajek-R ´enyi-Type Maximal Inequality and Strong Laws of Large Numbers for

Multidimensional Arrays

Nguyen Van Quang

and Nguyen Van Huan

1Department of Mathematics, Vinh University, Nghe An 42000, Vietnam

2Department of Mathematics, Dong Thap University, Dong Thap 871000, Vietnam

Correspondence should be addressed to Nguyen Van Huan,[email protected] Received 1 July 2010; Accepted 27 October 2010

Academic Editor: Alexander I. Domoshnitsky

Copyrightq2010 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 Cwill 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 integerd-dimensional lattice points will be denoted by ^d₀, and the set of all positive integerd-dimensional lattice points will be denoted by ^d. We will write1, m, n, and n1 for points1,1, . . . ,1,m1, m2, . . . , md,n1, n2, . . . , nd, andn11, n21, . . . , nd1, respectively. The notationm norn mmeans thatmi ni for all i 1,2, . . . , d, the 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|_d

i1ni.

Let{b_n,n ∈ ^d}be ad-dimensional array of real numbers. We define Δb_n to be the dth-order finite difference of theb’s at the pointn. Thus,bn

1knΔbk for alln∈ ^d. For example, ifd2, then for alli, j∈ ²,Δbijbij−bi,j−1−bi−1,jbi−1,j−1with the convention thatb0,0 bi,0 b0,j 0. We say that{b_n,n∈ ^d}is a nondecreasing array ifbk bl for any pointskl.

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

moments, andbj, j 1is a nondecreasing sequence of positive real numbers, then for any ε >0 and for any positive integersn, n0 n0< n,

⎛

⎝max

n0 i n

1 bi

i j1

Xj

ε

⎞

⎠ 1 ε²

⎛

⎝ⁿ⁰

j1

X_j² b²n0

ⁿ

jn01

X_j² b²_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 Klesov2 gave a general method for obtaining the strong law of large numbers for sequences of random variables by using a H´ajek-R´enyi-type maximal inequality. Afterwards, Nosz´aly and T ´om´acs 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

Xk−→0 a.s. as n−→ ∞, 1.2

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

i1bⁱni, where{bnⁱi, ni 1}is a nondecreasing sequence of positive real numbers for each i1,2, . . . , d. Then, we have

bn

1kn

Δbk b¹n1bn²2 · · ·b^dnd, n∈ ^d. 1.3

This implies that Δbn

b¹n1 −b_n¹₁₋₁

b²n2 −b_n²₂₋₁

· · ·

b^dnd −b^d_nd−1

, n∈ ^d. 1.4

Therefore,

Δbn 0, n∈ ^d, 1.5 ΔbnΔbn1 Δbn1n2···nd−1,nd1Δbn11,n21,...,nd−11,nd, n∈ ^d. 1.6

On the other hand, we can show that under the assumption that{b_n,n ∈ ^d} is an array of positive real numbers satisfying1.5, it is not possible to guarantee that1.6holdsfor details, seeExample 2.8in the next section.

Thus, if{b_n,n∈ ^d}is a positive, nondecreasingd-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 satisfying1.5and 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.

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 ∈ ^d₀}of nondecreasing sub-σ- algebras ofFrelated to the partial orderon ^d₀ is said to be a stochastic basic.

Let{F_n,n ∈ ^d₀}be a stochastic basic such thatF_n {∅,Ω}if|n| 0, letE be a real separable Banach space, letBEbe theσ-algebra of all Borel sets inE, and let{Xn,n ∈ ^d} be an array of random elements such thatXn isFn/BE-measurable for alln ∈ ^d. Then {X_n,F_n,n∈ ^d}is said to be an adapted array.

For a given stochastic basic{Fn,n∈ ^d₀}, forn∈ ^d₀, we set

F¹_n

ki12 i d

Fn1k2k3···kd : ^∞

k21

∞ k31

· · ·^∞

kd1

Fn1k2k3···kd,

F_n^j

ki1^{1 i j−1}

ki1^{j1 i d}

Fk1···kj−1njkj1···kd if 1< j < d,

F^d_n

ki11 i d−1

Fk1k2···kd−1nd,

1.7

in the cased1, we setF_n¹Fn.

An adapted array {X_n,F_n,n ∈ ^d} is said to be a martingale difference array if

X_n|Fⁱ_n−1 0 for alln∈ ^d and for alli1,2, . . . , 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 bep-uniformly smooth1 p 2if

ρτ supxy x−y

2 −1, ∀x, y∈E, x 1, yτ

Oτ^p. 1.8

A Banach spaceE is said to bep-smoothable if there exists an equivalent norm under whichE isp-uniformly smooth.

Pisier6 proved that a real separable Banach spaceE isp-smoothable 1 p 2 if and only if there exists a positive constantCsuch that for everyL^pintegrableE-valued martingale difference sequence{Xj,1 j n},

n j1

Xj

p

C n

j1

Xj^p. 1.9

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 withY11 Y1−1 1/2. LetE^∞E×E×E× · · · and define

E

⎧⎨

⎩v1, v2, . . .∈E^∞: ∞ j1

Yjvj converges inprobability

⎫⎬

⎭. 1.10

Let 1 p 2. Then,E is said to be of Rademacher typepif there exists a positive constantC such that

∞ j1

Yjvj

p

C ∞

j1

vj^p ∀v1, v2, . . .∈ E. 1.11

It is well known that if a real separable Banach space is of Rademacher typep1 p 2, then it is of Rademacher typeqfor all 1 q p. Every real separable Banach space is of Rademacher type 1, while theL^p-spaces and^p-spaces are of Rademacher type 2∧pforp1.

The real lineis of Rademacher type 2. Furthermore, if a Banach space isp-smoothable, then it is of Rademacher typep. For more details, the reader may refer to Borovskikh and Korolyuk 7 , Pisier8 , and Woyczy ´nski9 .

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{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 thatbn → ∞asn → ∞. If

n1

xn <∞, 1.12

then

1 bn

1kn

b_kx_k−→0 as n−→ ∞. 1.13

Proof. For everyε >0, there exists a pointn0∈ ^dsuch that

k1

xk−

1kn0

xk ε. 1.14

Therefore, for allnn0,

0 1

bn

1kn

bkxk−

1kn0

bkxk

1kn

xk−

1kn0

xk

ε. 1.15

It means that

nlim→ ∞

1 bn

1kn

bkxk−

1kn0

bkxk

0. 1.16

On the other hand, sincebn → ∞asn → ∞,

nlim→ ∞

1 bn

1kn0

bkxk0. 1.17

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. LetΩ,F,be a probability space, and let{A_n,n∈ ^d}be an array of sets inFsuch thatAn⊂Amfor any pointsmn. Then,

n1An

lim

n→ ∞An. 1.18

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

nlim→ ∞

sup

kn Xk ε

0, 1.19

thenXn → 0 a.s.asn → ∞.

Proof. For eachi1, we have

n1

kn

X_k 1 i

lim

n→ ∞

kn

X_k 1

i by Lemma 1.2

nlim→ ∞

sup

kn Xk 1 i

0.

1.20

Set

A

i1

n1

kn

Xk 1 i

. 1.21

Then,A 0 and for allω /∈A, for anyi1, there exists a pointl∈ ^d such that Xkω <

1/ifor allkl. It means that

X_k−→0 a.s.ask−→ ∞. 1.22

The proof is completed.

Lemma 1.4Quang and Huan5 . Let 1 p 2, and letE be a real separable Banach space.

Then, the following two statements are equivalent.

iThe Banach spaceE isp-smoothable.

iiFor everyL^pintegrable martingale difference array{Xn,Fn,n∈ ^d}, there exists a positive constantCp,d(depending only onpandd) such that

1kn

Xk

p

Cp,d

1kn

X_k ^p, n∈ ^d. 1.23

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. Letp >0, let{b_n,n∈ ^d}be an array of positive real numbers satisfying1.5, and let{Xn,n∈ ^d}be an array of random elements in a real separable Banach space. Then, there exists a positive constantCp,dsuch that for anyε >0 and for any pointsmn,

mknmax 1 bk

1lk

Xl

ε

Cp,d

ε^p max

1kn

1lk

Xl

blbm

p

. 2.1

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

mknmax 1 bk

1lk

Xl

ε

mknmax 1 bkbm

1lk

Xl

ε

2

1knmax 1 bkbm

1lk

Xl

ε

2

.

2.2

Fork∈ ^d, set

rkbkbm, Dk

1lk

Xl

rl. 2.3

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

1lk

Xl

1lk

1tl

Δr_t Xl

rl

1tk

Δr_t

tlk

Xl

rl

. 2.4

Thus, sinceΔr_t0,

1knmax 1 rk

1lk

Xl

2^dmax

1ln Dl . 2.5

By2.2and2.5and the Markov inequality, we have

mknmax 1 b_k

1lk

Xl

ε

max1ln Dl ε 2^d1

2^pd1 ε^p max

1ln D_l ^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. Letp >0, let{an,n∈ ^d}be an array of nonnegative real numbers, let{bn,n∈ ^d} be an array of positive real numbers satisfying1.5andbn → ∞asn → ∞, and let{X_n,n∈ ^d} be an array of random elements in a real separable Banach space such that for any pointsmn,

max

1kn

1lk

Xl

blbm

p

C

1kn

ak

bkbm^p. 2.7

Then, the condition

n1

an

b^p_n <∞ 2.8

implies1.2.

Proof. By2.7andTheorem 2.1, for anyε >0 and for any pointsmn, we have

mknmax 1 bk

1lk

Xl

ε

C ε^p

1kn

ak

bkbm^p. 2.9

This implies, by lettingn → ∞, that

sup

km

1 bk

1lk

Xl

ε

C ε^p

k1

ak

bkbm^p C

ε^p

1km

ak

b^p_m

k1

ak

b^p_k −

1km

ak

b^p_k

.

2.10

Lettingm → ∞, by2.8andLemma 1.1, we obtain

mlim→ ∞

sup

km

1 bk

1lk

Xl

ε

0. 2.11

Lemma 1.3ensures that1.2holds. The proof is completed.

The next theorem provides three characterizations ofp-smoothable Banach spaces. The equivalence ofiand iiis an improvement of a result of Quang and Huan5 stated as Lemma 1.4above.

Theorem 2.3. Let 1 p 2, and letE be a real separable Banach space. Then, the following four statements are equivalent.

iThe Banach spaceE isp-smoothable.

iiFor everyL^pintegrable martingale difference array{Xn,Fn,n∈ ^d}, there exists a positive constantCp,dsuch that

max

1kn

1lk

Xl

p

Cp,d

1kn

Xk p

, n∈ ^d. 2.12

iiiFor everyL^pintegrable martingale difference array {X_n,F_n,n ∈ ^d}, for every array of positive real numbers {bn,n ∈ ^d} satisfying1.5, for anyε > 0, and for any points mn, there exists a positive constantCp,dsuch that

mknmax 1 bk

1lk

Xl

ε

Cp,d

ε^p

1kn

Xk

bkbm

^p. 2.13

ivFor every martingale difference array{X_n,F_n,n ∈ ^d}, for every array of positive real numbers{bn,n∈ ^d}satisfying1.5andbn → ∞asn → ∞, the condition

n1

Xn p

b^pn <∞ 2.14

implies1.2.

Proof. i⇒ii: We easily obtain2.12in the casep1. Now, we consider the case 1< p 2.

By virtue ofLemma 1.4, it suffices to show that

max

1kn

1lk

Xl

p

p p−1

pd

1kn

Xk

p

, n∈ ^d. 2.15

First, we remark that ford1,2.15follows from Doob’s inequality. We assume that 2.15holds fordD−11, we wish to show that it holds fordD.

Fork∈ ^D, we set

S_k

1lk

X_l, YkD max

1 ki ni1 i D−1 S_k . 2.16

Then,

Sk1k2···kD−1kD | F^D_{k1k2···kD−1,kD−1}

Sk1k2···kD−1,kD−1 | F_k^D_{1k2···kD−1,kD−1}

⎛

⎝

1 li ki1 i D−1

Xl1l2···lD−1kD | F^D_{k1k2···kD−1,kD−1}

⎞

⎠ Sk1k2···kD−1,kD−1.

2.17

Therefore,

YkD | F_k^D_{1k2···kD−1,kD−1}

1 ki nmaxi1 i D−1 Sk | F^D_{k1k2···kD−1,kD−1}

max

1 ki ni1 i D−1

Sk| F^D_{k1k2···kD−1,kD−1} Yk_D−1.

2.18

It means that{YkD,F^D_k

1k2···kD−1kD, kD 1}is a nonnegative submartingale. Applying Doob’s inequality, we obtain

max

1kn Sk p

1maxkD nD

YkD

_p p p−1

_p

Yn^pD

p

p−1 _p

max

1 ki ni1 i D−1 Sk1k2···kD−1nD

p.

2.19

We set

X_k^D−1

1k2···kD−1 ^nD

kD1

Xk1k2···kD−1kD, F^D−1_k

1k2···kD−1 ^∞

kD1

Fk1k2···kD−1kD. 2.20

Then we again have that {X_k^D−1

1k2···kD−1,F^D−1_k

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

max

1 ki ni1 i D−1 Sk1k2···kD−1nD

p

max

1 ki ni1 i D−1

1 li ki1 i D−1

X^D−1_l

1l2···lD−1

p

p p−1

_pD−1

1 li ni1 i D−1

X_l^D−1_{1l2···lD−1}

p

p

p−1 _pD−1

Sn1n2···nD

p.

2.21

Combining2.19and2.21yields that2.15holds fordD.

ii ⇒ iii: let {X_n,F_n,n ∈ ^d} be an arbitraryL^p integrable martingale difference array. Then, for allm ∈ ^d, {X_n/b_n bm,F_n,n ∈ ^d}is also anL^p integrable martingale difference array. Therefore, the assertioniiandTheorem 2.1ensure that2.13holds.

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

iv⇒i: for a given positive integerd, assume thativholds. Let{Xj,Fj, j 1}be an arbitrary martingale difference sequence such that

∞ j1

Xj^p

j^p <∞. 2.22

Forn∈ ^d, set

XnXn1 if ni12 i d,

Xn0 if there exists a positive integeri2 i dsuch thatni>1, F_n Fn1, bnn1.

2.23

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

n1

Xn p

b^p_n ^∞

n11

Xn1

p

n^p₁ <∞, 2.24

and so1.2holds. It means that 1 n1

n1

j1

Xj−→0 a.s.asn1−→ ∞. 2.25

Then, by Theorem 2.2 of Hoffmann-Jørgensen and Pisier12 ,E isp-smoothable.

Remark 2.4. The inequality2.15holds for everyp > 1 and for every martingale difference array without imposing any geometric condition on the Banach space.

In the cased1,Theorem 2.3reduces to the following corollary which was proved by Gan13 and Gan and Qiu14 .

Corollary 2.5. Let 1 p 2, and letE be a real separable Banach space. Then, the following three statements are equivalent.

iThe Banach spaceE isp-smoothable.

iiFor every L^p 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 integersn, n0 n0< n, there exists a positive constantCsuch that

⎛

⎝max

n0 i n

1 bi

i j1

Xj

ε

⎞

⎠ C ε^p

⎛

⎝ⁿ⁰

j1

Xj^p b^pn0

ⁿ

jn01

Xj^p b_j^p

⎞

⎠. 2.26

iiiFor every martingale difference sequence {Xj,Fj, j 1} and for every nondecreasing sequence of positive real numbers{bj, j1}such thatbj → ∞asj → ∞, the condition

∞ j1

Xj^p

b^p_j <∞ 2.27

implies

1 bi

i j1

Xj−→0 a.s.asi−→ ∞. 2.28

Remark 2.4ensures that the inequality2.15holds for everyp >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⇒2.1.2of Theorem 2.1of Hoffmann-Jørgensen and Pisier 12 and the same arguments as in the proof ofTheorem 2.3, we get the following theorem which generalizes some results given by Christofides and Serfling15 and Gan and Qiu14 . We omit its proof.

Theorem 2.6. Let 1 p 2, and letE be a real separable Banach space. Then, the following four statements are equivalent.

iThe Banach spaceE is of Rademacher typep.

iiFor every array ofL^pintegrable independent mean zero random elements{X_n,n ∈ ^d}, there exists a positive constantCp,dsuch that2.12holds.

iiiFor every array ofL^pintegrable independent mean zero random elements{X_n,n∈ ^d}, for every array of positive real numbers{b_n,n∈ ^d}satisfying1.5, for anyε > 0, and for any pointsmn, there exists a positive constantCp,dsuch that2.13holds.

ivFor every array of independent mean zero random elements{X_n,n∈ ^d}, for every array of positive real numbers{bn,n∈ ^d}satisfying1.5andbn → ∞asn → ∞, the condition 2.14implies1.2.

We close this paper by giving a remark on Theorem 2.6 and an example which illustrates Theorems2.2,2.3, 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 ofTheorem 2.3, we can extendTheorem 2.6toM-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

Xn−|n|^1/4

Xn |n|^1/4 1

2. 2.29

Then,{X_n,n∈ ^d}is an array of independent mean zero random variables taking values in the 2-smoothable Banach spaceusing the absolute value as norm.

Letbn |n|min{n1, n2, . . . , nd} n∈ ^d. Then,

Δb_n

⎧⎨

⎩

2 ifn1n2· · ·nd,

1 otherwise. 2.30

It means that{b_n,n∈ ^d}is an array of positive real numbers satisfying1.5andbn → ∞ asn → ∞. Moreover, by virtue of1.6, we can show that{bn,n ∈ ^d} is not a positive, nondecreasingd-sequence of product type. Therefore,1.2does not follow from Theorem 3.2 of Klesov et al.4 . But for every array of positive real numbers{r_n,n∈ ^d},{X_n/r_n,F_n σXk,1kn,n∈ ^d}is a martingale difference array such that

max

1kn

1lk

Xl

rl

2

C

1kn

|Xk|²

r_k² , n∈ ^d

by Theorem 2.3

n1

|Xk|² b²_n

n1

|n|^1/2

|n|² <∞,

2.31

and so2.7and2.8are satisfied,Theorem 2.2ensures that1.2holds.

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

b⁻¹_n

1kn

Xk−→0 a.s.as|n| −→ ∞, 2.32

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 to17–19 . Therefore, this example also shows that the implicationsi⇒ivofTheorem 2.3andi⇒ivofTheorem 2.6are independent of results obtained in17–19 .

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, VietnamNAFOSTED, no. 101.02.32.09.

References

1 J. H´ajek and A. R´enyi, “Generalization of an inequality of Kolmogorov,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 6, pp. 281–283, 1955.

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.

3 C. Nosz´aly and T. T ´om´acs, “A general approach to strong laws of large numbers for fields of random variables,” Annales Universitatis Scientiarum Budapestinensis de Rolando E¨otv¨os Nominatae.

Sectio Mathematica, vol. 43, pp. 61–78, 2001.

4 O. Klesov, I. Fazekas, C. Nosz´aly, and T. T ´om´acs, “Strong laws of large numbers for sequences and fields,” Theory of Stochastic Processes, vol. 5, no. 3-4, pp. 91–104, 2008.

5 N. V. Quang and N. V. Huan, “A characterization of p-uniformly smooth Banach spaces and weak laws of large numbers for d-dimensional adapted arrays,” The Indian Journal of Statistics, vol. 72, pp.

344–358, 2010.

6 G. Pisier, “Martingales with values in uniformly convex spaces,” Israel Journal of Mathematics, vol. 20, no. 3-4, pp. 326–350, 1975.

7 Y. V. Borovskikh and V. S. Korolyuk, Martingale Approximation, VSP, Utrecht, The Netherlands, 1997.

8 G. Pisier, “Probabilistic methods in the geometry of Banach spaces,” in Probability and Analysis (Varenna, 1985), vol. 1206 of Lecture Notes in Mathematics, pp. 167–241, Springer, Berlin, Germany, 1986.

9 W. A. Woyczy ´nski, “On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence,” Probability and Mathematical Statistics, vol. 1, no. 2, pp. 117–131, 1981.

10 I. Fazekas and T. T ´om´acs, “Strong laws of large numbers for pairwise independent random variables with multidimensional indices,” Publicationes Mathematicae Debrecen, vol. 53, no. 1-2, pp. 149–161, 1998.

11 G. R. Shorack and R. T. Smythe, “Inequalities for|Sk|/bkwherek∈N^r,” Proceedings of the American Mathematical Society, vol. 54, pp. 331–336, 1976.

12 J. Hoffmann-Jørgensen and G. Pisier, “The law of large numbers and the central limit theorem in Banach spaces,” Annals of Probability, vol. 4, no. 4, pp. 587–599, 1976.

13 S. Gan, “The H`ajek-R`enyi inequality for Banach space valued martingales and the p smoothness of Banach spaces,” Statistics & Probability Letters, vol. 32, no. 3, pp. 245–248, 1997.

14 S. Gan and D. Qiu, “On the H´ajek-R´enyi inequality,” Wuhan University Journal of Natural Sciences, vol.

12, no. 6, pp. 971–974, 2007.

15 T. C. Christofides and R. J. Serfling, “Maximal inequalities for multidimensionally indexed submartingale arrays,” The Annals of Probability, vol. 18, no. 2, pp. 630–641, 1990.

16 F. M ´oricz, U. Stadtm ¨uller, and M. Thalmaier, “Strong laws for blockwise M-dependent random fields,” Journal of Theoretical Probability, vol. 21, no. 3, pp. 660–671, 2008.

17 L. V. Dung, T. Ngamkham, N. D. Tien, and A. I. Volodin, “Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces,” Lobachevskii Journal of Mathematics, vol. 30, no. 4, pp. 337–346, 2009.

18 N. V. Quang and N. V. Huan, “On the strong law of large numbers andLp-convergence for double arrays of random elements in p-uniformly smooth Banach spaces,” Statistics & Probability Letters, vol.

79, no. 18, pp. 1891–1899, 2009.

19 L. V. Thanh, “On the strong law of large numbers for d-dimensional arrays of random variables,”

Electronic Communications in Probability, vol. 12, pp. 434–441, 2007.