Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 949608, 7 pages http://dx.doi.org/10.1155/2014/949608

Research Article

On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables

Aiting Shen,1 Ying Zhang,1 and Andrei Volodin2

1 School of Mathematical Science, Anhui University, Hefei 230601, China

2 Department of Mathematics and Statistics, University ofRegina, Regina, SK, Canada S4S 0A2

Correspondence should be addressed to Aiting Shen; shenaiting1114@126.com Received 15 December 2013; Accepted 7 April 2014; Published 8 May 2014 Academic Editor: Angelo Favini

Copyright © 2014 Aiting Shen et al. 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.

Let {an, n> 1} be a sequence of positive constants with an/n 1 and let {X, Xn, «> 1} be a sequence of pairwise negatively quadrant dependent random variables. The complete convergence for pairwise negatively quadrant dependent random variables is studied under mild condition. In addition, the strong laws of large numbers for identically distributed pairwise negatively quadrant dependent random variables are established, which are equivalent to the mild condition P(|X| > an) < m. Our results obtained in the paper generalize the corresponding ones for pairwise independent and identically distributed random variables.

1. Introduction

Throughout the paper, let {an,n > 1} be a sequence of positive constants with ajn 1, and let {X, Xn, n> 1} be a sequence of pairwise i.i.d. random variables. Denote Sn = Xt for each n > 1. Now, we consider the following assumptions:

(i) Z~1 P(IXI > an) < m;

(ii) SJan ^ 0 a.s.;

(iii) IX,Ilan ^ 0 a.s.

Recently, Sung [1] proved that the three assumptions above are equivalent for pairwise i.i.d. random variables. In addition, he presented some results on complete convergence for pairwise i.i.d. random variables. For more details about the strong law of large numbers and complete convergence for independent random variables or dependent random variables, one can refer to Etemadi [2], Wang et al. [3], Chen et al. [4], Tang [5], and so forth.

We point out that the keys to the proofs of the main results of Sung [1] are the Khintchine-Kolmogorov-type convergence theorem and the second Borel-Cantelli lemma for pairwise independent events (e.g., see Theorem 4.2.5 in [6] or Theorem 2.18.5 in [7]), while these are not proved for pairwise negatively quadrant dependent random variables

(pairwise NQD, in short; see Definition 1). If we want to generalize the main results of Sung [1] to the case of pairwise NQD random variables, we should propose new methods or prove the Khintchine-Kolmogorov-type convergence theorem and the second Borel-Cantelli lemma for pairwise NQD random variables. The answer is positive.

Firstly, let us recall the concept of pairwise negatively quadrant dependent random variables as follows.

Definition 1. The pair (X, Y) of random variables X and Y is said to be negatively quadrant dependent (NQD, in short), if, for all x,y e R,

P(X<x,Y <y)<P(X<x)P(Y < y). (1)

A sequence of random variables {Xn,n > 1} is said to be pairwise NQD, if (Xi,Xj) is NQD for every i = j, i,j = 1, 2,....

An array {Xni, i > 1,n > 1} of random variables is called rowwise pairwise NQD random variables if for every n > 1, {Xni, i > 1} is a sequence of pairwise NQD random variables.

The concept of pairwise NQD random variables was introduced by Lehmann [8], which includes pairwise independent random sequence and some negatively dependent

sequences, such as negatively associated sequences (see [913]), negatively orthant dependent sequences (see [9, 1418]), and linearly negative quadrant dependent sequences (see [19-21]). Hence, studying the probability limiting behavior of pairwise NQD random variables and its applications in probability theory and mathematical statistics are of great interest. Many authors have dedicated themselves to the study of it. Matula [10] gained the Kolmogorov-type strong law of large numbers for the identically distributed pair-wise NQD sequences; Wu [22] gave the generalized three-series theorem for pairwise NQD sequences and proved the Marcinkiewicz strong law of large numbers; Chen [23] discussed Kolmogorov-Chung strong law of large numbers for the nonidentically distributed pairwise NQD sequences under very mild conditions; Wan [24] and Huang et al. [25] obtained the complete convergence for pairwise NQD random sequences; Wang et al. [26], Li and Yang [27], Gan and Chen [28], Shi [29], Xu and Tang [30], and Tang [31] studied the strong convergence properties for pairwise NQD random variables; Sung [21] established the Lr convergence for weighted sums of arrays of rowwise pairwise NQD random variables under weaker uniformly integrable conditions; and so on. The main purpose of the paper is to establish the second Borel-Cantelli lemma for pairwise NQD random variables and generalize the main results of Sung [1] to thecaseofpairwiseNQD random variableswithout adding any extra conditions.

Our main results are as follows. The first two results are the complete convergence for pairwise NQD random variables.

Theorem 2. Let |an, n > 1} be a sequence of positive constants with ajn Let |X, Xn, n > 1} be a sequence of pairwise NQD random variables with identical distribution. If ^^ P(|X| > fln) < to, then

£[X¡ -EX,/(|X,|<aJ]

> a„e ) < to

Ve > 0.

Theorem 3. Let |an, n > 1} be a sequence of positive constants with ajn | to. Let |X, Xn, n > 1} be a sequence of pairwise NQD random variables with identical distribution. If P(|X| > flj < TO,then

I n-1P| i * \ i

n P | max |SJ

Vlsfcsn ' fcl

> a„e ) < to Ve > 0.

The following two theorems are the results on strong convergence for pairwise NQD random variables.

Theorem 4. Let |an, n > 1} be a sequence of positive constants with ajn Let |X, Xn, n > 1} be a sequence of pairwise NQD random variables with identical distribution. Then, the following statements are equivalent:

(i) Z~1 №1 > «J < to,

(ii) (1/flJ - W(|X,| < flj] ^ 0 a.s.

Theorem 5. Let |an, n > 1} be a sequence of positive constants with ajn | to. Let |X, Xn, n > 1} be a sequence of pairwise NQD random variables with identical distribution. Then, the following statements are equivalent:

(i) Z~1 > «J < to,

(ii) S„/a„ ^ 0 a.s.,

(iii) 27=1 |^i|/flB ^ 0 a.s.

With Theorem 5 and the second Borel-Cantelli lemma for pairwise NQD random variables (see Corollary 16) in hand, we can get the following result for pairwise NQD random variables.

Corollary 6. Let |a„, n > 1} be a sequence of positive constants with ajn Let |X, Xn, n > 1} be a sequence of pairwise NQD random variables with identical distribution and £|X| = to. Then,

, n TO

nlimorIlx¡l = 0 «-S. i// Ip(jxj>«n)

lim sup = to a.s. I -P(j-^l > fln) = to.

Remark 7. Theorems 2 and 3 deal with the complete convergence for pairwise NQD random variables. Theorems 4 and 5 deal with the strong laws of large numbers for pairwise NQD random variables, which are equivalent to the mild condition -P( | ^| > a„) < to. Pairwise NQD is a very wide dependence structure, which includes independent sequence as a special case. Hence, Theorems 2-5 generalize the corresponding ones for pairwise i.i.d. random variables to the case of pairwise NQD random variables.

Remark 8. Under the conditions of Theorem 3 and a2n < Can, we can get the Marcinkiewicz-Zygmund-type strong law of large numbers for pairwise NQD random variables as follows:

0 as n

n ¡=1

Remark 9. For a sequence |X, Xn, n > 1} of pairwise i.i.d. random variables with £|X| < to, Etemadi [2] proved that Zf=1(Xi - £X;)/n ^ 0 a.s. Note that £|X| < to is equivalent to -P(|-X] > n) < to and £|X| < to implies Zi=1 > 0M ^ 0. Hence, Etemadi's strong law of

large numbers follows from Theorem 4 with a„ = n.

Remark 10. Note that limsupn^TO(|Sn|/an) = to a.s. is equivalent to P(|SJ > aan, i.o.) = 1 for any a > 0. Hence, Corollary 6 improves the corresponding result of Kruglov [32].

Throughout the paper, let 7(A) be the indicator function of the set A. C denotes a positive constant not depending on n, which may be different in various places. Denote a0 = 0, = (x > 0), and x- = -xT (x < 0).

n ¡=1

2. Preliminaries

In this section, we will present some important lemmas which will be used to prove the main results of the paper.

The first three lemmas come from Sung [1].

Lemma 11 (cf.[1]). Let {an,n > 1} be a sequence of positive constants with ajn Then the following properties hold.

(i) {an, n> 1} isa strictly increasing sequence with an an \ x.

(ii) Zzi p(x >an )<x if and only if p(X > 2an) < x.

(iii) P(x > an )< (XI if and only if P(X > aan) < x for any a > 0.

Lemma 12 (cf. [1]). If {an,n > 1} is a sequence of positive constants with ajn \ and X is a random variable, then

-E\X\l(\X\<an)<^P(\X\>an).

Lemma 13 (cf. [1]). Let {an,n > 1} be a sequence of positive constants with ajn \ m and X is a random variable. If YZi p(\x\ > an) < m, then (n/an)E\X\I(\X\ < an) ^ 0.

The next one is the basic property for pairwise NQD random variables, which was given by Lehmann [8] as follows.

Lemma 14 (cf. [8]). Let X and Y be NQD; then

(i) EXY < EXEY;

(ii) P(X > x, Y > y) < P(X > x)P(Y > y), for any x,y e R;

(iii) if f and д are both nondecreasing (or nonincreasing) functions, then f(X) and g(Y) are NQD.

The following one is the generalized Borel-Cantelli lemma, which was obtained by Matula [10].

Lemma 15 (cf. [10]). Let {An, n> 1} be a sequence of events.

(i) If !Zi p(An) < m, then P(An, i.o.) = 0.

(ii) If P(AkAm) < P(Ak)P(Am) for k = m and ÏZi p(An) = m, then P(An, i.o.) = 1.

With the generalized Borel-Cantelli lemma accounted for, we can establish the second Borel-Cantelli lemma for pairwise NQD random variables as follows.

Corollary 16 (second Borel-Cantelli lemma for pairwise NQD random variables). Let {an,n > 1} be a sequence of positive constants with ajn Let {Xn, n > 1}bea sequence of pairwise NQD random variables. Then

TP(\Xn\>an)<m.

Proof. By Lemma 11, P(\Xn\ > an) < m is equivalent to P(\Xn\ > ane) < m for all e > 0, which yields that XJan ^ 0 a.s. by Borel-Cantelli lemma.

Let XJan ^ 0 a.s., which implies that X++/an a.s. and Xn/an ^ 0 a.s.

For any e > 0, denote

An (1) =

Hence,

a„ 2

An (2) =

a„ 2

P[An (j), i.o.} = 0, j=1,2.

By Lemma 14(iii), we can see that {X+, n > 1} and [Xn, n > 1} are both sequences of pairwise NQD random variables. It follows by Lemma 14(ii) that, for any k = m,

P A (j) Am (j)) < P (Ak (j)) P (Am (j)), j=1,2.

By Lemma 15(ii) and (9)-(10), we can see that ITO=1 P(A„(j)) < ot for j = 1,2. Hence,

TO TOTO

X P (\Xn\ > ane )<^P(An (1)) + lP A (2)) < ot

for any e > 0,

which is equivalent to ^^ P(\Xn\ > an) < m by Lemma 11. This completes the proof of the corollary. □

The last one is the Kolmogorov-type strong law of large numbers for pairwise NQD random variables obtained by Chen [23], which plays an important role in proving the main results of the paper.

Lemma 17 (cf. [23]). Let {Xn, n> 1}bea sequence of pairwise NQD random variables with Var (Xn) < m for each n >1. Let {an, n > 1} be a sequence of real numbers satisfying 0 < an | m. Suppose that

(i) supn^n1 Zn=iE\Xt -EXt\<m;

(ii) !Zi Var (Xn )/a2n < m. Then a-1 Z'n=1(Xi - EX) ^ 0 a.s.

3. Proofs of Theorems 2-5

Proof of Theorem 2. Note that the condition ajn | implies that

Z1 ^ I I 1 I 2 21

n2 ~ L—' n2vt2 ~ n2 и2 _ n2 ; n2

2 œ 1 i2 2 2i

, , , 2 2 2 2 2

n=i n п=г i

For fixed n >1, denote for 1 < i < n that

Yt = -aj (Xt < -an) + XtI (|Х,| < an) + aj (Xt > an).

It is easily checked that

I[X, -EX,I(\X,\ <an)]

> a„e

<ln-1p( U(I I >Un)

n=1 \i=1

I(Yt -EX,l(\ Xi\ <an))

I(Yt -EY,)

> a„e

IP(\X\>an) + In1p(

+ In1P(

= 11 +h + I3

I(EYt -EXtl(\Xt\<an))

To prove the desired result (2), it suffices to show ij < to for j = 1,2, 3. Note that i1 < to; we only need to prove i2 < to

and I3 < to.

Note that {Y, - EYt, 1 < i < n} are pairwise NQD random variables by Lemma 14(iii); we have by Markov's inequality, Lemma 14(i), and the assumption ^^^ -P(\-X] > an) < to that

h = In1P

I(Yt -EY,)

-1 -2 a,

n=1 i=1

<Cl n-1a-21 E(Yi - EYi)2 <C% a-2 EY1

<C£ an2EX2I (\X\ <an)+ClP (\X\ > an)

n=1 n=1

<Clan2EX2l( \ X\<an)+C.

Combining with (12) and (15), we have

h <CI an21 EX21 (a-1 < \ X\ <a)+C

n=1 i=1

mm = c£ EX21 (a-1 <\ X\<at)I an2 + C

i=1 n=i

<CI EX21 (ai-1 < \ X\ < a) ia-2 + C i=1

<CliP(ai-1 < \X\ <at)+C i=1

<ClP(\X| > at)+C< to.

Finally, we will prove i3 < to. It is easily seen that

I3 = In-1P

I(EY, -EX,l(\X,\<an))

n=1 \i=l

<Xn1p(XP(\X1\>an )>£-

= In1p(nP(\X\>an)>2).

In the following we prove nP(\X\ > an) ^ 0. Note that Tzi p(\x\ > an) <to and 0 < P(\X\ > an) | as n we have P(\X\ > an) = o(1/n), which implies that nP(\X\ > an) ^ 0. Hence, I3 < to. This completes the proof of the theorem. □

Proof of Theorem 3. We use the same notations as those in Theorem 2. It is easy to see that

Yn 1p{max ^| >an£

<In-1 Ip(\xt\>*n)

Ixtl( \ Xi \<an)

+ I n-1P ( max

^-J \ 1<k<n

> a„e

<IP(\X\>an)

+ Zn1p(Z\Xtl(\Xt\<an)\>ane

n=1 \i=1

= IP(\X\>an)

+ I n-1P ( £ \XtI (|Xi| < an) -Yt +Y, - EX,I

x( \ X, \<an)+EXil( | Xi \<an) \>ane

n=1 \i=1

<C+I n-1P ( 1\ aj (X, < -an) - aj (X, > an) |

n=1 \i=1

n=1 \i=1

+ I«-1P(H Yt - EX,I (\X,\ < an)\ > ^

m ( n n

+ In-1p(I\ EX,l(\X,\<an)\>^

n=1 \i=1 3 )

œ / n

n=1 \i=1

<C+Xn-1P( \ Xi \ >an )>

œ ¡n

n=1 \i=1

+ ln-1p(l\Y, -EXil(\X,\<an)\>^f

œ / \

+ Yn1P(nE\X\l(\X\<an)> )

= C + h +J2 + J3.

To prove the desired result (3), it remains to show Ji < x for i = l,2,3.

By Markov's inequality and the assumption ^^ PCX > an) < x, we have

n=1 i=1

h <C^n-1 (N >an) = C^P (\X\ > an) < x.

By the assumptions of Theorem 3 and Lemma 13, we have (n/an)E\X\I(\X\ < an) ^ 0, which implies that J3 < x. In the following, we will prove J2 < x.Itis easily checked

n=1 \i=1

J2 <ln-1p(l\Y,-EYt\>a-f

n=1 \i=1

I"-1P( I\-P(Xt <-On)

+P(Xt >-dn)\>6

n=1 \i=1

<ln-1p(l(Y, -EY)

n=1 \i=1

In-1P(1(Y, -EYt)- > ^

œ / e\

+ Yn1P(nP(\X\>an)> 6)

=21 + J22 + J23•

Similar to the proof of I3 < x in Theorem 2, we can get that J23 < x.

Note that, for fixed n > l, {(Yt-EYi)+, l<i< n} and {(Yt -EYi1 < i < n} are both pairwise NQD random variables. Hence, similar to the proof of I2 < x in Theorem 2, we have

J21 <c£n1a-2 YE[(Yi -EY,)+ ]

Similarly, we have J22 < x. Therefore, J2 < x follows by the

statements above. This completes the proof of the theorem.

Proof of Theorem 4. Firstly, we will prove that (i) ^ (ii). For fixed n> 1, denote

Yn = - aj (Xn <-an) + Xni (\Xn\ < an) + aj (Xn > aH) .

Similar to the proof of l2 < x in Theorem 2, we have

X«n2 Var (Yn )<Y*n2EY2n

n=1 n=1

< X an2EX2I (\X\ <an) + £p (M > *n)

n=1 n=1

<cxp(\x\ >an) + C<x.

It follows by Lemma 12 that

sup«n1 XE\Y> -EY>\

m1 i=i

<2 sup anl XE\Yi\

m1 i=i

<2XP(\Xi\ >a) + 2 sup na^E^I^l < an)

i=1 n^1

<cxp(\x\ >an) < x.

Since Var(Yn) < a2 < x for each n> l,we have by (23) and (24) and Lemma 17 that

-1(Y, - BY) > 0

Note that

= -Y[Xil(\Xi \<ai)-EXil(\Xi\<ai)]

an i=1 1 n

+ Tt [aiI (Xi > ai) - a'1 (Xi < -ai) - aiP (Xi > ai)

<cZa-2EY?

+a,P(X, <-a,)]

and the assumption P(|X| > an) < x implies that 1(IXnl > an) < x a.s.; we can get that

£ - {anI (X„ > an) - anI (X„ < -an)

n=l an

- (xn > an) + an? (Xn < -aj))

<Tl(I Xn I >an) + ^P{\Xn\ >aH)<x a.s.,

n=1 n=1

which together with Kronecker's lemma yield that 1 n

- I W,I > a,) - a,I (X, < -a,) - a,P (X, > a,)

+ aiP(Xi < -a) —> 0 a.s. By (26) and (28), we have

~I(X,I(\X,\ < a,)-EX,I(\X,\ < a,)) > 0 a.s. (29)

an ,= 1

It follows by the assumption P(|X| > an) < x again that

~IX,I(\ X, | >a,)-^0 a.s. (30)

an i=l

Therefore, the desired result (ii) follows by (29) and (30) immediately.

Next, we will prove that (ii) ^ (i). Assume that

£[Xt -EX,I( I X, I <a,)]-^0 a.s. i=i

Then, we have

Xn -EXJ( \ Xn | <an)

= -£[Xi -EX,I(\X,\ <a,)]

n 1 n-1

^ —£[xt-EXi l( \ X, I <a,)]

an an-\ i=l

0 a.s.

Note that E\Xn\ I(\Xn\ <an)

E\X\(l(\X\<aN) + l(aN < \X\ < an))

< P(\X\<aN) + P(aN <\X\<an)

+P(\X\>aN):l^P(\X\>aN)-^0,

as N ^ x, which implies that

EXj(\Xn\ <an) 0 -1-!--> 0 as n —> x.

It follows by (32) and (34) that Xn/an ^ 0 a.s., which is equivalent to (i) by Corollary 16. The proof is completed. □

Proof of Theorem 5. Firstly, we will prove that (ii) ^ (i). It follows by (ii) that

in = !f x,x,

i n t-* ' a a , t-*

0 a.s..

which together with Corollary 16 imply that (i) holds.

On the other hand, assume that YTO=i P(|X| > an) < x; it follows by Lemma 13 that

£EX,I(\X,\ <an)

<—E\X\l(\X\<an)-^0 a.s.

The desired result (ii) follows by Theorem 4 and (36) immediately.

We have proved that (i) & (ii); next we prove (i) & (iii). It follows by Lemma 11(iii) that, for any e > 0,

£p(\X\>an)<x

£P(\X\ > ane) < x

£p(x+

> a„e) < x.

> a„e) < x

£p(x+

> a„) < x

> aj < x.

On the other hand, we have proved that (i) & (ii); hence,

Ip(X+ >an)<x,IP(X- >an)<x

n=1 n=1

n V"+ n V"-

ZA X-' ^ i

-i-->0 a.s., -->0 a.s. (38)

Z\ Ai \ n

0 a.s.

Therefore, (i) & (iii) follows by the statements above immediately. This completes the proof of the theorem. □

Proof of Corollary 6. The techniques used here are the second Borel-Cantelli lemma for pairwise NQD random variables (see Corollary 16) and Theorem 5. The proof is similar to that of Corollary 2.1 of Sung [1], so the details of the proof are omitted. □

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are most grateful to the Editor Angelo Favini and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001), the Natural Science Foundation of Anhui Province (1308085QA03), and the Students Innovative Training Project of Anhui University (201410357118).

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