Sums of Sequences of Negatively Dependent Random Variables

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Journal of Probability and Statistics Volume 2011, Article ID 202015,16pages doi:10.1155/2011/202015

Research Article

Complete Convergence for Weighted

Sums of Sequences of Negatively Dependent Random Variables

Qunying Wu

College of Science, Guilin University of Technology, Guilin 541004, China

Correspondence should be addressed to Qunying Wu,[email protected] Received 30 September 2010; Accepted 21 January 2011

Academic Editor: A. Thavaneswaran

Copyrightq2011 Qunying Wu. 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.

Applying to the moment inequality of negatively dependent random variables the complete convergence for weighted sums of sequences of negatively dependent random variables is discussed. As a result, complete convergence theorems for negatively dependent sequences of random variables are extended.

1. Introduction and Lemmas

Definition 1.1. Random variablesXandY are said to be negatively dependentNDif P

X≤x, Y ≤y

≤PX≤xP Y ≤y

1.1 for allx, y∈R. A collection of random variables is said to be pairwise negatively dependent PNDif every pair of random variables in the collection satisfies1.1.

It is important to note that1.1implies that P

X > x, Y > y

≤PX > xP Y > y

1.2 for allx, y ∈ R. Moreover, it follows that1.2implies1.1, and, hence,1.1and1.2are equivalent. However,1.1and1.2are not equivalent for a collection of 3 or more random variables. Consequently, the following definition is needed to define sequences of negatively dependent random variables.

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Definition 1.2. The random variablesX1, . . . , Xnare said to be negatively dependentNDif, for all realx1, . . . , x_n,

P

⎛

⎝ⁿ

j1

X_j≤x_j⎞

⎠≤ ⁿ

j1

P

X_j≤x_j ,

P

⎛

⎝ⁿ

j1

X_j> x_j⎞

⎠≤ ⁿ

j1

P

X_j> x_j .

1.3

An infinite sequence of random variables{Xn;n ≥ 1}is said to be ND if every finite subset X1, . . . , X_nis ND.

Definition 1.3. Random variablesX1, X2, . . . , Xn,n ≥ 2, are said to be negatively associated NAif, for every pair of disjoint subsetsA1andA2of{1,2, . . . , n},

cov

f1Xi;i∈A1, f2

Xj;j∈A2

≤0, 1.4

wheref1andf2are increasing in every variableor decreasing in every variable, provided this covariance exists. A random variables sequence{X_n;n ≥ 1}is said to be NA if every finite subfamily is NA.

The definition of PND is given by Lehmann 1, the concept of ND is given by Bozorgnia et al.2, and the definition of NA is introduced by Joag-Dev and Proschan 3.

These concepts of dependence random variables have been very useful in reliability theory and applications.

First, note that by lettingf1X1, X2, . . . , X_n−1 I_X₁_≤x₁_,X₂_≤x₂_,...,X_n−1_≤x_n−1,f2X_n I_X_n_≤x_n andf₁X1, X2, . . . , X_n−1 I_X₁_>x₁_,X₂_>x₂_,...,X_n−1_>x_n−1,f₂X_n I_X_n_>x_n, separately, it is easy to see that NA implies1.3. Hence, NA implies ND. But there are many examples which are ND but are not NA. We list the following two examples.

Example 1.4. LetX_i be a binary random variable such thatPX_i 1 PX_i 0 0.5 for i 1,2,3. LetX1, X2, X3take the values 0,0,1,0,1,0,1,0,0, and1,1,1, each with probability 1/4.

It can be verified that all the ND conditions hold. However,

PX1X3≤1, X2≤0 4

8 PX1X3≤1PX2≤0 3

8. 1.5

Hence,X1,X2, andX3are not NA.

In the next example X X1, X2, X3, X4 possesses ND, but does not possess NA obtained by Joag-Dev and Proschan3.

Example 1.5. LetXibe a binary random variable such thatPXi 1 .5 fori1,2,3,4. Let X1, X2andX3, X4have the same bivariate distributions, and letX X1, X2, X3, X4have joint distribution as shown inTable 1.

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Table 1 X1, X2

0,0 0,1 1,0 1,1 Marginal

0,0 .0577 .0623 .0623 .0577 .24

0,1 .0623 .0677 .0677 .0623 .26

X3, X4 1,0 .0623 .0677 .0677 .0623 .26

1,1 .0577 .0623 .0623 .0577 .24

marginal .24 .26 .26 .24

It can be verified that all the ND conditions hold. However,

PX_i1, i1,2,3,4> PX1X21PX3 X41, 1.6

violating NA.

From the above examples, it is shown that ND does not imply NA and ND is much weaker than NA. In the papers listed earlier, a number of well-known multivariate distributions are shown to possess the ND properties, such as a multinomial, b convolution of unlike multinomials, c multivariate hypergeometric, d dirichlet, e dirichlet compound multinomial, andfmultinomials having certain covariance matrices.

Because of the wide applications of ND random variables, the notions of ND random variables have received more and more attention recently. A series of useful results have been establishedcf. Bozorgnia et al.2, Amini4, Fakoor and Azarnoosh5, Nili Sani et al.

6, Klesov et al.7, and Wu and Jiang8. Hence, the extending of the limit properties of independent or NA random variables to the case of ND random variables is highly desirable and of considerable significance in the theory and application. In this paper we study and obtain some probability inequalities and some complete convergence theorems for weighted sums of sequences of negatively dependent random variables.

In the following, leta_n b_na_n b_ndenote that there exists a constantc >0 such thatan ≤cbnan ≥cbnfor suﬃciently largen, and letan ≈bn meanan bnandan bn. Also, let logxdenote lnmaxe, xandSn_n

j1Xj.

Lemma 1.6see2. LetX1, . . . , X_n be ND random variables and{f_n;n≥1}a sequence of Borel functions all of which are monotone increasing (or all are monotone decreasing). Then{fnXn;n≥1}

is still a sequence of ND r. v. ’s.

Lemma 1.7see2. LetX1, . . . , X_nbe nonnegative r. v. ’s which are ND. Then

E

⎛

⎝ ⁿ

j1

X_j

⎞

⎠≤ ⁿ

j1

EX_j. 1.7

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In particular, letX1, . . . , Xnbe ND, and lett1, . . . , tnbe all nonnegative (or non-positive) real numbers.

Then

E

⎛

⎝exp

⎛

⎝ⁿ

j1

t_jX_j

⎞

⎠

⎞

⎠≤ ⁿ

j1

E exp

tjX_j

. 1.8

Lemma 1.8. Let{X_n;n ≥1}be an ND sequence withEX_n 0 andE|X_n|^p <∞, p ≥2. Then for Bn_n

i1EX²_i,

E|Sn|^p≤cp

_n

i1

E|Xi|^pB_n^p/2

, 1.9

E

max1≤i≤n|Si|^p

≤c_plog^pn _n

i1

E|Xi|^pB^p/2_n

, 1.10

wherec_p>0 depends only onp.

Remark 1.9. If {X_n;n ≥ 1} is a sequence of independent random variables, then 1.9 is the classic Rosenthal inequality 9. Therefore, 1.9 is a generalization of the Rosenthal inequality.

Proof ofLemma 1.8. Let a > 0, X_i minXi, a, andS_n _n

i1X_i. It is easy to show that {X_i;i ≥ 1}is a negatively dependent sequence by Lemma 1.6. Noting thate^x−1−x/x² is a nondecreasing function ofxon R and thatEX_i≤EX_i0,tX_i ≤ta, we have

E e^tXⁱ

1tEX_iE

e^tXⁱ−1−tX_i t²X_i² t²X_i²

≤1

e^ta−1−ta

a⁻²EX_i²

≤1

e^ta−1−ta a⁻²EX²_i

≤exp

e^ta−1−ta

a⁻²EX_i² .

1.11

Here the last inequality follows from 1x≤e^x, for allx∈R.

Note thatB_n_n

i1EX_i²and{X_i;i≥1}is ND, we conclude from the above inequality andLemma 1.7that, for anyx >0 andh >0, we get

e^−hxE e^hSⁿ

e^−hxE _n

i1

e^hXⁱ

≤e^−hx

n i1

E e^hXⁱ

≤exp

−hx

e^ha−1−ha a⁻²B_n

.

1.12

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Lettinghlnxa/Bn1/a >0, we get

e^ha−1−ha

a⁻²B_n x a−B_n

a² ln xa

B_n 1

≤ x

a. 1.13

Putting this one into1.12, we get furthermore

e^−hxE e^hSⁿ

≤exp x

a−x aln

xa B_n 1

. 1.14

Puttingx/atinto the above inequality, we get

PSn≥x≤ⁿ

i1

PXi> a P S_n≥x

≤ⁿ

i1

PX_i> a e^−hxEe^hSⁿ

≤ⁿ

i1

P X_i> x

t

exp

t−tln x²

tBn 1

ⁿ

i1

P X_i> x

t e^t

1 x²

tB_n _−t

.

1.15

Letting−Xitake the place ofXiin the above inequality, we can get

P−S_n≥x PS_n ≤ −x≤ⁿ

i1

P

−X_i > x t

e^t

1 x² tBn

_−t

ⁿ

i1

P

X_i< −x t

e^t

1 x²

tB_n _−t

.

1.16

Thus

P|S_n| ≥x PS_n≥x PS_n≤ −x≤ⁿ

i1

P

|X_i|< x t

2e^t

1 x² tBn

_−t

. 1.17

Multiplying1.17bypx^p−1, lettingtp, and integrating over 0< x <∞, according to

E|X|^pp _∞

0

x^p−1P|X| ≥xdx, 1.18

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we obtain

E|Sn|^pp _∞

0

x^p−1P|Sn| ≥xdx

≤pⁿ

i1

_∞

0

x^p−1P

|X_i| ≥x p

dx2pe^p _∞

0

x^p−1

1 x² pB_n

_−p dx

p^p1ⁿ

i1

E|X_i|^ppe^p

pB_n_p/2_∞

0

u^p/2−1 1u^pdu

p^p1ⁿ

i1

E|Xi|^pp^p/21e^pBp 2,p

2 B^p/2n ,

1.19

whereBα, β ₁

0x^α−11−x^β−1dx_∞

0 x^α−11x^−αβdx, α, β >0 is Beta function. Letting cp maxp^p1, p^1p/2e^pBp/2, p/2,we can deduce 1.9 from1.19. From1.9, we can prove1.10by a similar way of Stout’s paper10, Theorem 2.3.1.

Lemma 1.10. Let{Xn;n ≥ 1}be a sequence of ND random variables. Then there exists a positive constantcsuch that, for anyx≥0 and alln≥1,

1−P

1≤k≤nmax|X_k|> x2n k1

P|X_k|> x≤cP

1≤k≤nmax|X_k|> x

. 1.20

Proof. LetA_k |X_k|> xandα_n1−P_n

k1A_k 1−Pmax_1≤k≤n|X_k|> x. Without loss of generality, assume thatα_n>0. Note that{I_X_k_>x−EI_X_k_>x; k≥1}and{I_X_k_<−x−EI_X_k_<−x; k≥ 1}are still ND byLemma 1.6. Using1.9, we get

E _n

k1

IAk−EIAk 2

E _n

k1

IXk>x−EIXk>x

IXk<−x−EIXk<−x2

≤2E _n

k1

IXk>x−EIXk>x₂ 2E

_n

k1

IXk<−x−EIXk<−x₂

≤cⁿ

k1

PA_k.

1.21

Combining with the Cauchy-Schwarz inequality, we obtain n

k1

PAk

n k1

P

⎛

⎝Ak,ⁿ

j1

Aj

⎞

⎠ⁿ

k1

E

IAkIⁿ_j1_A_j

E _n

k1

I_A_k −EI_A_k

Iⁿ_j1_A_jⁿ

k1

PA_kP

⎛

⎝ⁿ

j1

A_j

⎞

⎠

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≤

⎛

⎝E _n

k1

IAk −EIAk 2

EIⁿ_j1_A_j

⎞

⎠

1/2

1−αnⁿ

k1

PAk

≤

c1−αn αn αn

n k1

PAk _1/2

1−αnⁿ

k1

PAk

≤ 1 2

c1−α_n

α_n α_nⁿ

k1

PA_k

1−α_nⁿ

k1

PA_k.

1.22

Thus

α²_nⁿ

k1

PA_k≤c1−α_n, 1.23

that is,

1−P

1≤k≤nmax|Xk|> x 2n

k1

P|Xk|> x≤cP

1≤k≤nmax|Xk|> x

. 1.24

2. Main Results and the Proofs

The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins 11 as follows. A sequence {Yn;n ≥ 1} of random variables converges completely to the constantcif_∞

n1P|X_n−c| > ε< ∞, for allε >0. In view of the Borel- Cantelli lemma, this implies thatYn → 0 almost surely. Therefore, complete convergence is one of the most important problems in probability theory. Hsu and Robbins11proved that the sequence of arithmetic means of independent and identically distributedi.i.d.random variables converges completely to the expected value if the variance of the summands is finite. Baum and Katz12proved that if{X, Xn;n≥1}is a sequence of i.i.d. random variables with mean zero, thenE|X|^pt2 < ∞1 ≤ p < 2, t ≥ −1is equivalent to the condition that _∞

n1n^tP_n

11|Xi|/n^1/p > ε< ∞, for allε > 0. Recent results of the complete convergence can be found in Li et al.13, Liang and Su14, Wu15,16, and Sung17.

In this paper we study the complete convergence for negatively dependent random variables. As a result, we extend some complete convergence theorems for independent random variables to the negatively dependent random variables without necessarily imposing any extra conditions.

Theorem 2.1. Let{X, X_n;n≥1}be a sequence of identically distributed ND random variables and {ank; 1≤k≤n, n≥1}an array of real numbers, and letr >1,p >2. If, for some 2≤q < p,

Nn, m1

k≥1;|ank| ≥m1^−1/p

≈m^qr−1/p, n, m≥1, 2.1

EX0 for 1≤qr−1, 2.2

n k1

a²_nkn^δ for 2≤qr−1and some 0< δ < 2

p, 2.3

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then, forr ≥2,

E|X|^pr−1<∞ 2.4

if and only if

∞ n1

n^r−2P

1≤k≤nmax

k i1

a_niX_i > εn^1/p

<∞, ∀ε >0. 2.5

For 1 < r < 2, 2.4 implies 2.5, conversely, and 2.5 and n^r−2Pmax_1≤k≤n|a_nkX_k| > n^1/p decreasing onnimply2.4.

Forp2,q2, we have the following theorem.

Theorem 2.2. Let{X, X_n;n≥1}be a sequence of identically distributed ND random variables and {ank; 1≤k≤n, n≥1}an array of real numbers, and letr >1. If

Nn, m1

k;|ank| ≥m1^−1/2

≈m^r−1, n, m≥1, 2.6

EX 0, 1≤2r−1, n

k1

|a_nk|^2r−1O1, 2.7

then, forr ≥2,

E|X|^2r−1log|X|<∞ 2.8

if and only if

∞ n1

n^r−2P

1≤k≤nmax

k i1

a_niX_i > εn^1/2

<∞, ∀ε >0. 2.9

For 1 < r < 2, 2.8 implies 2.9, conversely, and2.9 and n^r−2Pmax_1≤k≤n|a_nkX_k| > n^1/2 decreasing onnimply2.8.

Remark 2.3. Since NA random variables are a special case of ND r. v. ’s, Theorems2.1and2.2 extend the work of Liang and Su14, Theorem 2.1.

Remark 2.4. Since, for some 2≤q≤p,

k∈N|ank|^qr−11 asn → ∞implies that

Nn, m1

k≥1;|a_nk| ≥m1^−1/p

m^qr−1/p asn−→ ∞, 2.10

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takingr 2, then conditions2.1and2.6are weaker than conditions2.13and2.9in Li et al.13. Therefore, Theorems 2.1and2.2not only promote and improve the work of Li et al.13, Theorem 2.2for i.i.d. random variables to an ND setting but also obtain their necessities and relax the range ofr.

Proof ofTheorem 2.1. Equation2.4⇒2.5. To prove2.5it suﬃces to show that

∞ n1

n^r−2P

1≤k≤nmax

k i1

a^±_niXi

> εn^1/p

<∞, ∀ε >0, 2.11

where a_ni maxa_ni,0and a⁻_ni max−a_ni,0. Thus, without loss of generality, we can assume thata_ni >0 for alln≥1, i≤n. For 0< α <1/psmall enough and suﬃciently large integerK, which will be determined later, let

X¹_ni −n^αI_a_ni_X_i_<−n^αa_niX_iI_a_ni_|X_i_|≤n^αn^αI_a_ni_X_i_>n^α, X²_ni aniXi−n^αI_n^α_<a_ni_X_i_<εn^1/p_/K,

X³_ni aniX_in^αI_−εn^1/p_/K<a_ni_X_i_<−nα, X⁴_ni aniXni−X_ni¹−X_ni²−X_ni³

aniXin^αI_a_ni_X_i_≤−εn^1/p_/K aniXi−n^αI_a_ni_X_i_≥εn^1/p_/K,

S^j_nk ^k

i1

X_ni^j, j 1,2,3,4 ; 1≤k≤n, n≥1.

2.12

ThusS_nk_k

i1a_niX_i₄

j1S^j_nk. Note that

1≤k≤nmax|S_nk|>4εn^1/p

⊆⁴

j1

1≤k≤nmax

S^j_nk> εn^1/p

. 2.13

So, to prove2.5it suﬃces to show that

Ij^∞

n1

n^r−2P

1≤k≤nmax

S^j_nk> εn^1/p

<∞, j1,2,3,4. 2.14

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For anyq > q, n

i1

a^q_ni^r−1 ^∞

j1

^j1⁻¹^≤a^pni<j⁻¹

a^q_ni^r⁻¹≤^∞

j1

^j1⁻¹^≤a^pni<j⁻¹

j^−q^r−1/p

^∞

j1

N

n, j1

−N n, j

j^−q^r−1/p

^∞

j1

N n, j

j^−q^r−1/p−

j1_−q_r−1/p

^∞

j1

j^−1−q^−qr−1/p<∞.

2.15

Now, we prove that

n^−1/pmax

1≤k≤n

ES¹_nk−→0, n−→ ∞. 2.16

iFor 0< qr−1<1, takingq < q < psuch that 0< qr−1<1, by2.4and2.15, we get

n^−1/pmax

1≤k≤n

ES¹_nk

≤n^−1/pⁿ

i1

E|a_niX_i|I_|a_ni_X_i_|≤n^αn^αP|a_niX_i|> n^α

≤n^−1/p _n

i1

E|a_niX_i|^q^r−1|a_niX_i|^1−q^r−1I_|a_ni_X_i_|≤n^αn^α−αq^r−1ⁿ

i1

E|a_niX_i|^q^r−1

n^{−1/pα−αq}^r−1 −→0, n−→ ∞.

2.17

iiFor 1≤qr−1, lettingq < q < p, by2.2,2.4, and2.15, we get

n^−1/pmax

1≤k≤n

ES¹_nk

≤n^−1/pⁿ

i1

E|a_niX_i|I_|a_ni_X_i_|>n^αn^αP|a_niX_i|> n^α

≤n^−1/pⁿ

i1

E|a_niX_i|

|a_niX_i| n^α

_q_r−1−1

I_|a_ni_X_i_|≤n^αn^α−αq^r−1E|a_niX_i|^q^r⁻¹

n^{−1/pα−αq}^r−1−→0.

2.18

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Hence,2.16holds. Therefore, to proveI1<∞it suﬃces to prove that

I1^∞

n1

n^r−2P

1≤k≤nmax

S¹_nk −ES¹_nk> εn^1/p

<∞, ∀ε >0. 2.19

Note that{X¹_ni ; 1≤i≤n, n≥1}is still ND by the definition ofX¹_ni andLemma 1.6. Using the Markov inequality andLemma 1.8, we get for a suitably largeM, which will be determined later,

I1^∞

n1

n^r−2−M/pE

1≤k≤nmax

S¹_nk−ES¹_nk _M

^∞

n1

n^r−2−M/plog^Mn

⎡

⎣ⁿ

i1

EX_ni¹^M _n

i1

E

X¹_ni 2_M/2⎤

⎦

I11I12.

2.20

TakingM >max2, pr−11−αq/1−αp, thenr−2−M/pαM−αqr−1<−1, and, by2.15, we get

I11≤^∞

n1

n^r−2−M/plog^Mnⁿ

i1

E|aniXi|^MI|aniXi|≤n^αn^MαP|aniXi|> n^α

≤^∞

n1

n^r−2−M/plog^Mnⁿ

i1

E|a_niX_i|^q^r−1n^αM−q^r−1n^αM−q^r−1E|a_niX_i|^q^r−1

^∞

n1

nr−2−M/pαM−αqr−1log^Mn

<∞.

2.21

iForqr−1<2, takingq < q < psuch thatqr−1<2 and takingM >max2, 2pr− 1/2−2αpαpqr−1, from2.15andr−2−M/pαM−Mαqr−1/2<−1, we have

I12 ≤^∞

n1

n^r−2−M/plog^Mn

#_n

i1

E|aniXi|^q^r−1n^α2−q^r−1I|aniXi|≤n^α

n^2α−αq^r−1E|aniXi|^q^r−1

$_M/2

^∞

n1

n−r−2−M/pαM−Mαqr−1/2log^Mn

<∞.

2.22

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iiForqr−1≥2, takingq < q < pandM >max2, 2pr−1/2−pδ, whereδis defined by2.3, we get, from2.3,2.4,2.15, andr−2−M/pδM/2<−1,

I12^∞

n1

n^r−2−M/plog^Mn

#_n

i1

a²_nin^2α−αq^r−1E|aniXi|^q^r−1

$_M/2

^∞

n1

nr−2−M/pδM/2log^Mn

<∞.

2.23

Since _n

i1

X_ni²> εn^1/p

_n

i1

aniXi−n^αI_n^α_<a_ni_X_i_<εn^1/p_/K> εn^1/p

⊆there at least existK indicesksuch that ankXk> n^α,

2.24

we have

P _n

i1

X²_ni > εn^1/p

≤

1≤i1<i2<···<iK≤n

Pani1Xi1> n^α, ani2Xi2> n^α, . . . , aniKXiK > n^α.

2.25

ByLemma 1.6,{a_niX_i; 1≤i≤n, n≥1}is still ND. Hence, forq < q < pwe conclude that

P _n

i1

X_ni²> εn^1/p

≤

1≤i1<i2<···<iK≤n K j1

P

a_ni_jX_i_j > n^α

≤ _n

i1

P|a_niX_i|> n^α _K

≤ _n

i1

n^−αq^r−1E|aniXi|^q^r−1 _K

n^−αq^r−1K,

2.26

via2.4and2.15.X_ni² >0 from the definition ofX_ni². Hence by2.26and by takingα >0 andKsuch thatr−2−αKqr−1<−1, we have

I2 ^∞

n1

n^r−2P _n

i1

X²_ni > εn^1/p

^∞

n1

n^r−2−αq^r−1K<∞. 2.27

Similarly, we haveX_ni³<0 andI3<∞.

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Last, we prove thatI4<∞. LetY KX/ε. By the definition ofX_ni⁴and2.1, we have

P

1≤k≤nmax

S⁴_nk> εn^1/p

≤P _n

i1

X_ni⁴> εn^1/p

≤P _n

i1

ani|Xi|> εn^1/p K

≤ⁿ

i1

P

ani|Xi|> εn^1/p K

^∞

j1

^j1⁻¹^≤a^pni<j⁻¹

P

|Y|>

nj_1/p

^∞

j1

N

n, j1

−N

n, j^∞

lnj

P

l≤ |Y|^p< l1

^∞

ln l/n

j1

N

n, j1

−N n, j

P

l≤ |Y|^p< l1

≈^∞

ln

l n

_qr−1/p P

l≤ |Y|^p< l1 .

2.28

Combining with2.15,

I4≈^∞

n1

n^r−2^∞

ln

l n

_qr−1/p P

l≤ |Y|^p< l1

^∞

l1

l n1

n^{r−2−qr−1/p}l^qr−1/pP

l≤ |Y|^p< l1

≈^∞

l1

l^r−1P

l≤ |Y|^p< l1

≈E|Y|^pr−1≈E|X|^pr−1<∞.

2.29

Now we prove2.5⇒2.4. Since

max1≤j≤nanjXj≤max

1≤j≤n

j i1

aniXi

max

1≤j≤n

j−1 i1

aniXi

, 2.30

then from2.5we have ∞ n1

n^r−2P

max1≤j≤nanjXj> n^1/p

<∞. 2.31

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Combining with the hypotheses ofTheorem 2.1,

P

max1≤j≤na_njX_j> n^1/p

−→0, n−→ ∞. 2.32

Thus, for suﬃciently largen,

P

max1≤j≤nanjXj> n^1/p

< 1

2. 2.33

ByLemma 1.6,{a_njX_j; 1≤ j ≤ n, n ≥ 1}is still ND. By applyingLemma 1.10and2.1, we obtain

n k1

P

|a_nkX_k|> n^1/p

≤4CP

1≤k≤nmax|a_nkX_k|> n^1/p

. 2.34

Substituting the above inequality in2.5, we get ∞

n1

n^r−2ⁿ

k1

P

|a_nkX_k|> n^1/p

<∞. 2.35

So, by the process of proof ofI4<∞,

E|X|^pr−1≈^∞

n1

n^r−2ⁿ

k1

P

|a_nkX_k|> n^1/p

<∞. 2.36

Proof ofTheorem 2.2. Letp2,α <1/p1/2, andK >1/2α. Using the same notations and method ofTheorem 2.1, we need only to give the diﬀerent parts.

Letting 2.7 take the place of2.15, similarly to the proof of2.19and 2.26, we obtain

n^−1/2max

1≤k≤n

ES¹_nkn−1/2α−2αr−1−→0, n−→ ∞. 2.37

TakingM >max2,2r−1, we have

I11^∞

n1

n−1−1−2αM/2−r−1log^Mn <∞. 2.38

Forr−1≤1, takingM >max2,2r−1/1−2α2αr−1, we get

I12 ^∞

n1

n^{−1−1−2αr}−1−2αM/2r−1log^Mn <∞. 2.39

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Forr−1>1,EX_ni² <∞from2.8. LettingM >2r−1², by the H ¨older inequality,

I12 ^∞

n1

n^r−2−M/2log^Mn

# _n

i1

a²_nin^{2α−2αr−1}Ea_niX_i^2r−1

$_M/2

^∞

n1

n^r−2−M/2log^Mn

⎡

⎣ _n

i1

a^2r−1_ni

_1/r−1 _n

i1

1

_r−2/r−1⎤

⎦

M/2

^∞

n1

n−1−M/2r−1r−1log^Mn <∞.

2.40

By the definition ofK,

I2^∞

n1

n−1−r−12αK−1<∞. 2.41

Similarly to the proof2.31, we have

I4^∞

l1

l n1

n⁻¹l^r−1P

l≤ |Y|²< l1

^∞

l1

l^r−1loglP

l≤ |Y|²< l1

≈E

|Y|^2r−1log|Y|

≈E

|X|^2r−1log|X|

<∞.

2.42

Equation2.9⇒2.8Using the same method of the necessary part ofTheorem 2.1, we can easily get

E

|X|^2r−1log|X|

≈^∞

n1

n^r−2ⁿ

k1

P

|ankXk|> n^1/2

<∞. 2.43

Acknowledgments

The author is very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. This work was supported by the National Natural Science Foundation of China11061012, the Support Program the New Century Guangxi China Ten-Hundred-Thousand Talents Project2005214, and the Guangxi China Science Foundation2010GXNSFA013120. Professor Dr. Qunying Wu engages in probability and statistics.

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Sums of Sequences of Negatively Dependent Random Variables

Research Article

Complete Convergence for Weighted

Sums of Sequences of Negatively Dependent Random Variables

Qunying Wu

1. Introduction and Lemmas

2. Main Results and the Proofs

Acknowledgments

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

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The Scientific World Journal

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