Complete Convergence for Negatively Dependent Sequences of Random Variables

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Volume 2010, Article ID 507293,10pages doi:10.1155/2010/507293

Review Article

Complete Convergence for Negatively Dependent

Sequences of Random Variables

Qunying Wu

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

Correspondence should be addressed to Qunying Wu,[email protected]

Received 5 January 2010; Accepted 6 March 2010

Academic Editor: Jewgeni Dshalalow

Copyrightq2010 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.

We study the complete convergence for negatively dependent sequences of random variables. As a result, we extend some complete convergence theorems for independent random variables to the case of negatively dependent random variables without necessarily imposing any extra conditions.

1. Introduction and Lemmas

Definition 1.1. Random variablesXandY are said to negatively dependentNDif

PX≤x, Y ≤y≤PX≤xPY ≤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

PX > x, Y > y≤PX > xPY > y 1.2

for allx, y ∈ R. Moreover, it follows that 1.2implies1.1, and hence,1.1and 1.2are equivalent. Ebrahimi and Ghosh 1 showed that 1.1 and 1.2 are not equivalent for a collection of 3 or more random variables. They considered random variables X1, X2, and

X3 where X1, X2, X3 assumed the values 0,1,1,1,0,1,1,1,0, and 0,0,0 each with

probability 1/4. The random variablesX1, X2, andX3are pairwise independent, and hence,

they satisfy both1.1and1.2for all pairs. However,

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for allx1, x2, andx3, but

PX1≤0, X2≤0, X3≤0

1 4/≤

1

8 PX1≤0PX2≤0PX3≤0. 1.4

Placing probability 1/4 on each of the other vertices {1,0,0,0,1,0,0,0,1,1,1,1} provides the converse example of pairwise independent random variables which will not satisfy1.3withx1 0, x2 0, andx3 0 but where the desired “≤” inPX1 ≤ x1, X2 ≤

x2, X3 ≤ x3 ≤

₃

i1PXi ≤ xi hold for all x1, x2, and x3. Consequently, the following

definition is needed to define sequences of negatively dependent random variables.

Definition 1.2. Random variablesX1, . . . , Xn are said to be negatively dependentNDif for

all realx1, . . . , xn,

P

⎛ ⎝n

j1

Xj≤xj

⎞ ⎠_≤n

j1

PXj≤xj,

P

⎛ ⎝n

j1

Xj> xj

⎞ ⎠_≤n

j1

PXj> xj.

1.5

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

X1, . . . , Xnis 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},

covf1Xi; i∈A1, f2

Xj; j∈A2

≤0, 1.6

wheref1andf2are increasing for every variableor decreasing for every variable, such that

this covariance exists. An infinite sequence of random variables{X_n; n≥1}is said to be NA if every finite subfamily is NA.

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

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

It is easy to see that NA implies ND from the definitions. But in the following example, we will show that ND does not imply NA.

Example 1.4. LetXi be a binary random variable such thatPXi 0 PXi 1 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/≤

3

8 PX1X3≤1PX2≤0. 1.7

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From the above example, it is shown that ND is much weaker than NA. 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 established

cf. 3, 5–11. Hence, it is highly desirable and of considerable significance to extend the limit properties of independent or NA random variables to the case of ND random variables theorems and applications.

Complete convergence is one of the most important problems in probability theory. Recent results of the complete convergence can be found in Wu11,12and Sung13,14. 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.

Lemma 1.5see3. LetX1, . . . , Xn be ND random variables and let{fn; n ≥ 1}be a sequence

of Borel functions all of which are monotone increasing (or all are monotone decreasing). Then {fnXn; n≥1}is a sequence of ND r.v.’s.

Lemma 1.6see3. LetX1, . . . , Xnbe nonnegative r.v.’s which are ND. Then

E

⎛ ⎝n

j1

Xj

⎞ ⎠_≤n

j1

EXj. 1.8

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

Then

E

⎛ ⎝_exp

⎛ ⎝n

j1

tjXj

⎞ ⎠

⎞ ⎠_≤n

j1

Eexpt_jX_j. 1.9

Lemma 1.7. Let{X_n; n≥1}be a sequence of ND random variables withEX_i0, EX2_i <∞. Then

forn≥1,

ES2

n≤ n

k1

EX2

k, 1.10

whereSn nk1Xk.

Proof. Obviously, ND implies PND from their definitions. Thus, by Lemma 2 of Wu12,

Lemma 1.7holds.

2. Main Results and the Proof

In the following, letan bndenote that there exists a constantc >0 such thatan ≤ cbn for

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Theorem 2.1. Let{Xn; n≥1}be a sequence of ND identically distributed random variables. Let for

some0< α≤1

E|X1|2/α _<_∞_{, EX}

10, 2.1 |ank| ≤cn−α fork≤nand some0< c <∞, ank0 fork > n, 2.2

cn n

k1

a2

nko

log−1n. 2.3

Then

Tn n

k1

ankXk−→c 0, 2.4

where →c denotes complete convergence.

Theorem 2.2. Let{X_n; n≥1}be a sequence of ND identically distributed random variables with

E|X1|2/α _<_∞_, _{for some}_{α >}₁_. _2.5

Then

Sn

nα −→c 0. 2.6

Remark 2.3. Theorems2.1and2.2extend corresponding results for independent r.v.s. to ND r.v.s. without necessarily adding any extra conditions.

Proof ofTheorem 2.1. By 2/α ≥ 2 and 2.1, we have EX2

1 < ∞. Let ε > 0 be given. By

Tnnk1ankXk−

_n

k1a−nkXk, where ank maxank,0 ≥ 0,and a−nk max−ank,0 ≥ 0,

without loss of generality, we can assume thatank >0 for alln≥1,k ≤n,andEX12 1. For

k≤n,let

X_nk1 −n−α/3_a−1

nkIXk<−n−α/3a−nk1XkI|Xk|≤n−α/3a−nk1n

−α/3_a−1

nkIXk>n−α/3a−nk1,

X_nk2 Xkn−α/3a−nk1

I_X_k_<₋_εa−1 nk/4

Xk−n−α/3a−nk1

I_X_k_>_εa−1 nk/4,

X_nk3 Xk−X_k1−X_k2

Xkn−α/3a−_nk1

I₋_εa−1

nk/4<Xk<−n−α/3a−nk1

Xk−n−α/3a−_nk1

I_n−α/3_a−1

nk<Xk<εa−nk1/4,

Tni n

k1

ankX_nki, i1,2,3.

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Thus,

TnTn1Tn2Tn3. 2.8

Note that

|Tn|>3ε⊂Tn1> ε

∪Tn2> ε

∪Tn3> ε

. 2.9

We shall prove that∞_n₁P|Tn|>3ε<∞by proving that

∞

n1

PTni> ε

<∞, i1,2,3. 2.10

ByEXk0 and2.2,

ETn1

n

k1

ankEX_nk1

n

k1

ankE

Xk−X1_nk

≤n

k1

ank

EXkn−α/3a−_nk1IXk<−n−α/3a−_nk1E

Xk−n−α/3a−_nk1IXk>n−α/3a−_nk1

≤n

k1

ankE|X1|I|X1|>n−α/3a−_nk1

n−α1_E_|_X_1|_I

|X1|>c−1n2α/3

≤n−1α/3_E_|_X_1|2/α_−→₀_{, n}_{−→ ∞}_.

2.11

So to prove∞_n₁P|T_n1|> ε<∞it suﬃces to show that

∞

n1

PTn1−ETn1> ₂ε

<∞. 2.12

LetX_nk1X_nk1−EX_nk1, T_n1T_n1−ET_n1. Fixn≥1. Letuminε/4c_n, nα/3_/₂_{. Since} |uankX1_nk| ≤1,EX_nk10,andEX1_nk2≤EX121, it follows that

EexpuankX1_nk

1

∞

j1

EuankX1_nk

_j

j!

1EuankX_nk1

E

uankX_nk1

2

⎛ ⎝₁₂∞

j3

1

j!

⎞ ⎠

≤1 u

2_a2

nkE

X_nk12

2 12e−2.5

≤1u2_a2

nk≤exp

u2_a2

nk

.

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SinceX1_nk and −X_nk1 are nondecreasing and nonincreasing functions of X_k, respec-tively, thus {uankX1_nk, n ≥ 1, k ≤ n} and {uank−X1_nk, n ≥ 1, k ≤ n} are also ND by

Lemma 1.5. It follows fromLemma 1.6and2.13that

EexpuTn1

E

_n

k1

expuankX1_nk

≤n

k1

Eexpua_nkX_nk1

≤n

k1

expu2a2_nkexpu2cn

.

2.14

By the Markov inequality,

PTn1> ε

≤exp−εuEexpuTn1

≤exp−εuu2cn

. 2.15

Since{−X_nk1}is also satisfying the conditions:|ua_nk−X1_nk| ≤1,E−X_nk1 0,andE−X2

nk≤

1, replacing theXkby−Xkin2.15the argument will then establish

PTn1 <−ε

≤exp−εuu2cn

. 2.16

Thus,

PTn1> ε₂

≤2 exp−εu

2 u

2_c

n

. 2.17

Ifε/2cn > nα/3,then by the definition ofu, we haveu nα/3/2,−εu/2u2cn ≤ −εnα/3/8.

Hence

∞

n1

PTn1> ε₂

≤2

∞

n1

exp

−εnα/3

8

<∞. 2.18

Ifε/2c_n ≤nα/3_{, then by the definition of}_u_{, we have}_u_ε/₄_c

n,and−εu/2u2cn−ε2/16cn.

Andcn< ε2/32 lognfor suﬃciently largenfromcnolog−1n. Hence

∞

n1

PTn1> ε₂

≤2

∞

n1

exp

− ε2

16c_n

∞

n1

1

n2 <∞. 2.19

Now we prove that∞_n₁P|Tn2|> ε<∞.Since

Tn2

n

k1

ankX_nk2

≤

n

k1

ankX2_nk≤ n

k1

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therefore,

_T2

n > ε

⊂n

k1

|Xk|> εa

−1

nk

4

⊂n

k1

|Xk|> εnα4c−1

. 2.21

Since 0< ank≤cn−αfork≤n,thus

PTn2> ε

≤n

k1

P|Xk|> εnα4c−1

nP|X1|> εnα₄_c−1 _Bnα_,

2.22

whereBε1/α4c−1/α>0. Therefore

∞

n1

PTn2> ε

≤∞

n1

nP|X1|1/α _{> Bn}

∞

n1

∞

jn

nPBj <|X1|1/α_≤_j₁_B

∞

j1

j

n1

nPBj <|X1|1/α_≤_j₁_B

≤∞

j1

j2_P_{Bj <}_|_X_1|1/α _≤_j₁_B

≤∞

j1

B−2_E_|_X_1|2/α_I

Bj<|X1|1/α≤j1B

E|X1|2/α_<_∞_.

2.23

Lastly, we prove that∞_n₁P|T_n3|> ε<∞. Since

_T3

n > ε

⊂there exist at least 4 indicesk such that|Xk|> a−_nk1n−α/3

, 2.24

we have

PTn3> ε

≤Pthere exist at least 4 indicesk such thatank|Xk|> n−α/3

≤

1≤i1<···<i4≤n

Pani1|Xi1|> n

−α/3_{, a}

ni2|Xi2|> n

−α/3_{, a}

ni3|Xi3|> n

−α/3_{, a}

ni4|Xi4|> n

−α/3_.

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By the definition of ND, and the fact that 0< ank≤cn−αfork≤n,we conclude that

PTn3> ε

≤

1≤i1<···<i4≤n

4

j1

PXij> c−1n2α/3

C4_nP4|X1|> c−1n2α/3≤n4P4|X1|> c−1n2α/3.

2.26

Thus, by2.1

∞

n1

PTn3> ε

∞

n1

n4_n−2α/32/α_E_|_X_1|2/α4

≤E|X1|2/α4∞

n1

n−4/3_<_∞_.

2.27

Together with2.19–2.27, we get

∞

n1

P|Tn|> ε<∞, 2.28

for allε >0 as desired. This completes the proof ofTheorem 2.1.

Proof ofTheorem 2.2. Let

Xkn−nαIXk<−nαXkI|Xk|≤nαnαIXk>nα, ∀n≥1, k≤n,

Skn n

k1

Xkn.

2.29

If 2/α≥1, thenE|X1|<∞andnP|X1|> nα → 0 from2.5. Thus

n−α_|_ES

kn| ≤n−α n

k1

E|Xk|I|Xk|≤nαnαP|Xk|> nα

≤E|X1|n1−α_nP_|_X_1|_{> n}α_−→₀_.

2.30

If 2/α <1, from above proof, we have

n−α_|_ES

kn| ≤n1−α n

k1

E|X1|Ik−1α_<_|_X

1|≤kαnP|X1|> n

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Since

∞

k1

k1−α_E_|_X_1|_I

k−1α_<_|_X 1|≤kα

∞

k1

k1−α_E_|_X_1|2/α_kα1−2/α_I

k−1<|X1|1/α≤k

∞

k1

k−1_E_|_X_1|2/α_I

k−1<|X1|1/α≤k

≤E|X1|2/α_<_∞_,

2.32

byα >1, and the Kronecker lemma, combining with2.31, we getn−α|ES_kn| → 0. Thus, for suﬃciently largen,

n−α_|_ES

kn|< ε₂. 2.33

Since Xnk is nondecreasing function ofXk, thus {Xnk, n ≥ 1, k ≤ n} is also ND

by Lemma 1.5. It follows from Lemma 1.7, 2.13, 2/α < 2, the Markov inequality, and

_∞

n1nP|X1|> nαE|X1|2/α<∞,that

∞

n1

P|Sn|> εnα≤

∞

n1

P

_n

k1

|Xk|> nα

∞

n1

P|Skn−ESkn|> εnα− |ESkn|

≤∞

n1

nP|X1|> nα ∞ n1

P

|Skn−ESkn|> εn α

2

E|X1|2/α∞ n1

1

n2αESkn−ESkn2

∞

n1

n−2αn

k1

EX2

kn

≤∞

n1

n−2α1_EX2

1I|X1|<nαn

2α_P_|_X_1|_{> n}α

∞

n1

n−2α1n

k1

EX2

1Ik−1α≤|X1|<kα

∞

n1

nP|X1|> nα

∞

k1

∞

nk

n−2α1_E_|_X_1|2/α_kα2−2/α_I

k−1α_≤|_X

1|<kαE|X1|

2/α

∞

k1

k−2α2_E_|_X_1|2/α_kα2−2/α_I

k−1α_≤|_X 1|<kα

∞

k1

E|X1|2/α_I

k−1α_≤|_X 1|<kα

<∞.

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Hence,

Sn

nα c

−→0. 2.35

This completes the proof ofTheorem 2.2.

Acknowledgments

The author is 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 China10661006, the Support Program of the New Century Guangxi China Ten-hundred-thousand Talents Project

2005214, and the Guangxi China Science Foundation0832262.

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