Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums of PNQD Random Variables

Abstract

The complete convergence for pairwise negative quadrant dependent (PNQD) random variables is studied. So far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. We establish a collection that contains relationship to overcome the difficulties that there is no general moment inequality. Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by Baum and Katz (1965) and Wu (2002).

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Qunying Wu 1,2

Recommended by Martin Weiser

1, College of Science, Guilin University of Technology, Guilin 541004, China
2, Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin University of Technology, Guilin 541004, China

Received 7 February 2012; Accepted 19 April 2012

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.

1. Introduction and Lemmas

Random variables X and Y are said to be negative quadrant dependent (NQD) if [figure omitted; refer to PDF] for all x,y∈R . A sequence of random variables {_Xn ;n...5;1} is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the sequence is NQD. This definition was introduced by Lehmann [1]. Obviously, PNQD sequence includes many negatively associated sequences, and pairwise independent random sequence is the most special case.

In many mathematics and mechanic models, a PNQD assumption among the random variables in the models is more reasonable than an independence assumption. PNQD series have received more and more attention recently because of their wide applications in mathematics and mechanic models, percolation theory, and reliability theory. Many statisticians have investigated PNQD series with interest and have established a series of useful results. For example, Matula [2], Li and Yang [3], and Wu and Jiang [4] obtained the strong law of large numbers, Wang et al. [5] obtained the Marcinkiewicz's weak law of large numbers, Wu [6] obtained the strong convergence properties of Jamison weighted sums, the three-series theorem, and complete convergence theorem, and Li and Wang [7] obtained the central limit theorem. It is interesting for us to extend the limit theorems to the case of PNQD series. However, so far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. In the above-mentioned conclusions, only the Kolmogorov-type strong law of large numbers obtained by Matula [2, Theorem 1] and Baum and Katz-type complete convergence theorem obtained by Wu [6, Theorem 4] achieve the corresponding conclusions of independent cases, and the rest did not achieve the optimal results of independent cases.

Complete convergence is one of the most important problems in probability theory. Recent results of the complete convergence can be found in Wu [6], Chen and Wang [8], and Li et al. [9]. In this paper, we establish a collection that contains relationship to overcome the difficulties that there is no the general moment inequality and obtain the complete convergence theorem for weighted sums of PNQD sequence, which extend and improve the corresponding results of Baum and Katz [10] and Wu [6].

Lemma 1.1 (see [1]).

Let X and Y be NQD random variables. Then

(i) cov (X, Y)...4;0 ,

(ii) P(X>x, Y>y)...4;P(X>x)P(Y>y) , for all x,y∈R ,

(iii): if f and g are Borel functions, both of which being monotone increasing (or both are monotone decreasing), then f(X) and g(Y) are NQD.

Lemma 1.2 (see [6, Lemma 2]).

Let {_Xn ;n...5;1} be a sequence of PNQD random variables with E_Xn =0, E^Xn2 <∞, _Tj (k)=^^∑i=j+1j+k_Xi , j...5;0 . Then [figure omitted; refer to PDF]

Lemma 1.3 (see [2, Lemma 1]).

(i) If ^∑n=1∞ P(_An )<∞ , then P(_An ;i.o.)=0 .

(ii) if P(_AkAm )...4;P(_Ak )P(_Am ), k...0;m , and ^∑n=1∞ P(_An )=∞ , then P(_An ;i.o.)=1 .

Lemma 1.4.

Let {_Xn ;n...5;1} be a sequence of PNQD random variables. Then for any x...5;0 , there exists a positive constant c such that for all n...5;1 , [figure omitted; refer to PDF]

Proof.

We can prove the Lemma by Lemma A.6 of Zhang and Wen [11].

2. Main Results and the Proof

In the following, the symbol c stands for a generic positive constant which may differ from one place to another. Let _an ...a;_bn ( _an ...b;_bn ) denote that there exists a constant c>0 such that _an ...4;c_bn ( _an ...5;c_bn ) for all sufficiently large n , and let _Xi [precedes]X ( _Xi [succeeds]X ) denote that there exists a constant c>0 such that P(|_Xi |>x)...4;cP(|X|>x) ( P(|_Xi |>x)...5;cP(|X|>x) ) for all i...5;1 and x>0 .

Theorem 2.1.

Let {_Xn ;n...5;1} be a sequence of PNQD random variables with _Xi [precedes]X . Let {_ank ;k...4;n,n...5;1} be a sequence of real numbers such that [figure omitted; refer to PDF] Let for αp>1, 0<p<2, α>0 , and E_Xi =0 , for α...4;1 . If [figure omitted; refer to PDF]

then [figure omitted; refer to PDF] where _Snk =^∑i=1k_aniXi .

Theorem 2.2.

Let {_Xn ;n...5;1} be a sequence of PNQD random variables with _Xi [succeeds]X . Let {_ank ;k...4;n, n...5;1} be a sequence of real numbers such that |_ank |...b;^n-α , for all k...4;n, n...5;1 . Let for α>0, αp>1, 0<p<2 . If (2.3) holds, then (2.2) holds.

Remark 2.3.

Taking _ani =^n-α , for all i...4;n, n...5;1 in Theorem 2.1, then [figure omitted; refer to PDF] Hence, Theorem 4 in Wu [6] is a particular case of our Theorem 2.1.

Remark 2.4.

When {_Xn ;n...5;1} is i.i.d. and _ani =^n-α , for all i...4;n, n...5;1 , then Theorems 2.1 and 2.2 become Baum and Katz [10] complete convergence theorem. Hence, our Theorems 2.1 and 2.2 improve and extend the well-known Baum and Katz theorem.

Proof of Theorem 2.1.

Without loss of generality, assume that _ank >0 for k...4;n, n...5;1 . Let q>0 such that (1+(1/αp))/2<q<1 . For all i...4;n , let [figure omitted; refer to PDF] Write [figure omitted; refer to PDF]

Firstly, we prove that [figure omitted; refer to PDF]

For any ω∈_Dn , we have [figure omitted; refer to PDF] and for any 1...4;i<j...4;n , [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] where the symbol #A denotes the number of elements in the set A .

When a=b=0 , then |_aniXi (ω)|...4;^nα(q-1) for any 1...4;i...4;n ; thus, _Yni (ω)=_Xi (ω) , and therefore by (2.8), [figure omitted; refer to PDF]

When a=1, b=0 (or a=0, b=1 ), then there exists only an _i0 : 1...4;_i0 ...4;n such that _ani0Xi0 (ω)>^nα(q-1) (or _ani0Xi0 (ω)<-^nα(q-1) ), the remaining j, |_anjXnj (ω)|...4;^nα(q-1) ; thus, _Xj (ω)=_Ynj (ω) . If 1...4;k...4;_i0 -1 , then _Snk (ω)=_Unk (ω) . If _i0 ...4;k...4;n , then by (2.8), [figure omitted; refer to PDF]

When a=b=1 , then there exist 1...4;_i1 , _i2 ...4;n such that _ani1Xi1 (ω)>^nα(q-1) , _ani2Xi2 (ω)<-^nα(q-1) , the remaining j, |_anjXj (ω)|...4;^nα(q-1) ; thus, _Xj (ω)=_Ynj (ω) . Without loss of generality, assume that _i1 ...4;_i2 . If 1...4;k...4;_i1 -1 , then _Snk (ω)=_Unk (ω) ; if _i1 ...4;k<_i2 , then by (2.8), [figure omitted; refer to PDF] If k...5;_i2 , then by (2.8), [figure omitted; refer to PDF] Hence, (2.7) holds, that is: [figure omitted; refer to PDF] Therefore, in order to prove (2.3), we only need to prove that [figure omitted; refer to PDF] By (2.1), (2.2), _Xi [precedes]X , and αp>1 , [figure omitted; refer to PDF] That is, (2.16) holds.

By Lemma 1.1(ii), _Xi [precedes]X , and the definition of q, αp(1-2q)<-1 , [figure omitted; refer to PDF] That is, (2.17) holds.

In order to prove (2.18), firstly, we prove that [figure omitted; refer to PDF]

(i) When α...4;1 , then p>1/α...5;1 ; from E_Xi =0 and the definition of q , we have q<1, αpq>αp+1-αpq=1+αp(1-q)>1: [figure omitted; refer to PDF]

(ii) When α>1 , and p...5;1 , then E|X|<∞ from (2.2), thus, [figure omitted; refer to PDF]

(iii) When α>1 , and p<1 , by -(αp-1)-α(1-q)(1-p)<0 , and -α(1-q)-(αpq-1)<0 , we get [figure omitted; refer to PDF] Hence, (2.21) holds; that is, for any [varepsilon]>0 , we have _{max 1...4;k...4;n} |E_Unk |<[varepsilon] for all sufficiently large n . Thus, [figure omitted; refer to PDF]

Let _Y~ni =_Yni -E_Yni . Obviously, _Yni is monotonic on _Xi . By Lemma 1.1(iii), {_Yni ;n...5;1,i...4;n} is also a sequence of PNQD random variables with E_Y~ni =0 , by Lemma 1.2 and -1-α(1-q)(2-p)<-1 : [figure omitted; refer to PDF] This completes the proof of Theorem 2.1.

Proof of Theorem 2.2.

Noting that _{max 1...4;k...4;n} |_ankXk |...4;2 _{max 1...4;k...4;n} |_Snk | and |_ank |...b;^n-α , from (2.3), [figure omitted; refer to PDF] Thus, by αp-2>-1 , we get [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF] Hence, for all sufficiently large n , [figure omitted; refer to PDF] By Lemma 1.4, [figure omitted; refer to PDF] which together with (2.27), [figure omitted; refer to PDF] By _Xk [succeeds]X , we obtain [figure omitted; refer to PDF] This completes the proof of Theorem 2.2.

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. Supported by the National Natural Science Foundation of China (11061012), and project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ( [2011] 47), and the support program of Key Laboratory of Spatial Information and Geomatics (1103108-08).

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