We study the complete moment convergence for sequences of negatively dependent identically distributed random variables with EX=0, Eexp|X|α<∞, 0<α<1, and Eexp|X|ln-r|X|<∞, r>0. As a result, we establish the new complete moment convergence theorems.

National Natural Science Foundation of China11361019Support Program of the Guangxi China Science Foundation2015GXNSFAA1390081. Introduction and Main Results

Random variables X and Y are said to be negative quadrant dependent (NQD) if(1)PX≤x,Y≤y≤PX≤xPY≤yfor all x,y∈R. A collection of random variables is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the collection satisfies (1).

It is important to note that (1) implies(2)PX>x,Y>y≤PX>xPY>yfor all x,y∈R. Moreover, it follows that (2) implies (1) and, hence, (1) and (2) are equivalent. However, Ebrahimi and Ghosh [1] showed that (1) and (2) are not equivalent for a collection of 3 or more random variables. Accordingly, the following definition is needed to define sequences of negatively dependent random variables.

Definition 1.

Random variables X1,…,Xn are said to be negatively dependent (ND) if, for all real x1,…,xn, (3)P⋂j=1nXj≤xj≤∏j=1nPXj≤xj,P⋂j=1nXj>xj≤∏j=1nPXj>xj.An infinite sequence of random variables {Xn;n≥1} is said to be ND if every finite subset X1,…,Xn is ND.

Definition 2.

Random variables X1,X2,…,Xn,n≥2, are said to be negatively associated (NA) if, for every pair of disjoint subsets A1 and A2 of {1,2,…,n}, (4)covf1Xi;i∈A1,f2Xj;j∈A2≤0, where f1 and f2 are increasing for every variable (or decreasing for every variable) function such that this covariance exists. A sequence of random variables {Xi;i≥1} is said to be NA if its every finite subfamily is NA.

The definition of PNQD is given by Lehmann [2]. The definition of NA is introduced by Joag-Dev and Proschan [3], and the concept of ND is given by Bozorgnia et al. [4]. These concepts of dependent random variables are very useful for reliability theory and applications.

It is easy to see from the definitions that NA implies ND. But Example 1.5 in Wu and Jiang [5] shows that ND does not imply NA. Thus, it is shown that ND is much weaker than NA. In the articles listed earlier, a number of well-known multivariate distributions are shown to possess the ND properties. In many statistics and mechanic models, a ND assumption among the random variables in the models is more reasonable than an independent or NA assumption. Because of wide applications in multivariate statistical analysis and reliability theory, the notions of ND random variables have attracted more and more attention recently. A series of useful results have been established (cf. Bozorgnia et al. [4], Fakoor and Azarnoosh [6], Asadian et al. [7], Wu [5, 8], Wang et al. [9], and Liu et al. [10]). Hence, it is highly desirable and of considerable significance in the theory and application to study the limit properties of ND random variables theorems and applications.

Chow [11] first investigated the complete moment convergence, which is more exact than complete convergence. Thus, complete moment convergence is one of the most important problems in probability theory. The recent results can be found in Chen and Wang [12], Gut and Stadtmüller [13], Sung [14], Guo [15], and Qiu and Chen [16, 17]. In addition, Qiu and Chen [17] obtained complete moment convergence theorems for independent identically distributed sequences of random variables with EX=0, Eexplnα|X|<∞, α>1. A natural question is whether there is any type of complete moment convergence theorems for 0<α≤1. In this paper, we study the complete moment convergence for sequences of negatively dependent identically distributed random variables with EX=0, Eexp|X|α<∞, 0<α<1, and Eexp|X|ln-r|X|<∞, r>0. As a result, we establish the new complete moment convergence theorems.

In the following, the symbol c stands for a generic positive constant which may differ from one place to another. Let an≪bn denote that there exists a constant c>0 such that an≤cbn for sufficiently large n, lnx mean ln(max(x,e)), and I denotes an indicator function.

Theorem 3.

Let 0<α<1, {X,Xn;n≥1} be a sequence of ND identically distributed random variables with partial sums Sn=∑i=1nXi, n≥1. Suppose that(5)EX=0,EexpXα<∞,and then(6)∑n=1∞expnαnpEmax1≤k≤nSk-βn+q<∞∀β>1,q>0andallp∈R.Conversely, if (6) holds for p=α-2-q and some β>0, then(7)EexpX4βα<∞.

For α=1, we have the following.

Theorem 4.

Let {X,Xn;n≥1} be a sequence of ND identically distributed random variables with partial sums Sn=∑i=1nXi, n≥1. Suppose that(8)EX=0,EexpXln-rX<∞forsomer>0,and then(9)∑n=1∞expnln-rnnpEmax1≤k≤nSk-βn+q<∞∀β>1,q>0andallp∈R.Conversely, if ∑n=1∞expnln-rnn-q-1ln-rnEmax1≤k≤n|Sk|-βn+q<∞ for some β>0, then(10)EexpX4βln-rX4β<∞.

Remark 5.

By mimicking the analogous part in the proof of Theorem 2.1 in Qiu and Chen [17], (6) and (9) imply, respectively, (11)∑n=1∞expnαnpEsup1≤k≤nSkk-β+q<∞∀β>1,q>0 and all p∈R,(12)∑n=1∞expnln-rnnpEsup1≤k≤nSkk-β+q<∞∀β>1,q>0 and all p∈R.

Remark 6.

Because (6) and (9) hold for all q>0, p∈R, and ND random sequences, (6) and (9) are very broad conclusions.

2. Proofs

The following four lemmas play important roles in the proof of our theorems.

Lemma 7 (Bozorgnia et al. [<xref ref-type="bibr" rid="B2">4</xref>]).

Let {Xn;n≥1} be a sequence of ND random variables.

Let {fn;n≥1} be a sequence of Borel functions; all of them are monotone increasing (or all are monotone decreasing). Then, {fn(Xn);n≥1} is a sequence of ND r.v.’s.

Let X1,…,Xn be nonnegative. Then, (13)E∏j=1nXj≤∏j=1nEXj.

In particular, let t1,…,tn be all nonnegative (or nonpositive) real numbers. Then, (14)Eexp∑j=1ntjXj≤∏j=1nEexptjXj.

Lemma 8.

Let {X,Xi;i≥1} be a sequence of ND identically distributed random variables with Eexp(λX)<∞ for any λ>0. Assume that {ai;i≥1} is a sequence of positive real numbers such that an↑∞ as n→∞. Then, for Sk=∑i=1kXi, λ>0, and a positive integer n, (15)Pmax1≤k≤nSk>an≤exp-λan∑k=1nEexpλX-EXk.

Proof.

Obviously, δ0=^sup{λ≥0;Eexp(λ(X-EX))<∞}=∞ from condition Eexp(λX)<∞ for any λ>0. By Lemma 7(ii), (1.1) in Wang et al. [18] holds for any λ∈R. Together with condition 0<an↑∞, we know that the conditions of Theorem 2.2 in Wang et al. [18] are satisfied. Therefore, by Theorem 2.2 in Wang et al. [18], for any λ>0, (16)Pmax1≤k≤nSk>an≤minλ∈0,∞exp-λan∑k=1nEexpλX-EXk≤exp-λan∑k=1nEexpλX-EXk.

Lemma 9.

For any random variable X,(17)EexpXα<∞⟺∑n=1∞expnαn1-αPX>n<∞foranyα>0,(18)EexpXln-rX<∞⟺∑n=2∞expnln-rnln-rnPX>n<∞foranyr>0.

Proof.

Let an≈bn denote that there exist constants c1>0 and c2>0 such that c1an≤bn≤c2an for sufficiently large n. We have (19)∑n=1∞expnαn1-αPX>n=∑n=1∞expnαn1-α∑j=n∞Pj<X≤j+1=∑j=1∞Pj<X≤j+1∑n=1jexpnαn1-α≈∑j=1∞expjαEIj<X≤j+1≈∑j=1∞EexpXαIj<X≤j+1≈EexpXα, and it follows that (17) holds.

Note that (20)∑n=2jexpnln-rnln-rn≈∫2jexpxln-rxln-rxdx~∫2jexpxln-rxln-rx-rln-r-1xdx=∫2jexpxln-rx′dx≈expjln-rj, and hence, using similar methods used to prove (17), we can prove that (18) holds.

Lemma 10.

Let {Xn;n≥1} be a sequence of ND random variables. Then, for any x≥0, there exists a positive constant c such that, for all n≥1, (21)1-Pmax1≤k≤nXk>x2∑k=1nPXk>x≤cPmax1≤k≤nXk>x.

Further, if P(max1≤k≤n|Xk|>x)→0 as n→∞, then there exists a positive constant c such that, for all n≥1, (22)∑k=1nPXk>x≤cPmax1≤k≤nXk>x.

Proof.

Obviously, ND implies pairwise negative quadrant dependent (PNQD) from the definitions of ND and PNQD. Thus, Lemma 10 holds from Lemma 1.4 of Wu [19].

Proof of Theorem <xref ref-type="statement" rid="thm1.3">3</xref>.

Note that(23)Emax1≤k≤nSk-βn+q=βq∫0nqxq-1Pmax1≤k≤nSk-βn>βxdx+βq∫n∞qxq-1Pmax1≤k≤nSk-βn>βxdx≪nqPmax1≤k≤nSk>βn+∫n∞xq-1Pmax1≤k≤nSk>βxdx.Hence, in order to establish (6), it suffices to prove, for any p∈R,(24)∑n=1∞expnαnpPmax1≤k≤nSk>βn<∞,(25)∑n=1∞expnαnp∫n∞xq-1Pmax1≤k≤nSk>βxdx<∞.Firstly, we prove (24). Let β>1 be arbitrary; define, for 1≤k≤n, (26)X′=-βnIX<-βn+XIX≤βn+βnIX>βn,Xk′=-βnIXk<-βn+XkIXk≤βn+βnIXk>βn,Sn′=∑k=1nXk′.It is easy to get(27)Pmax1≤k≤nSk>nβ≤Pmax1≤k≤nSk′>nβ+nPX>nβ.From (5) and the Markov inequality,(28)PX>nβ≤exp-βαnαEexpXα≪exp-βαnα.

Set X~=X′-EX′; then EX~=0; using the obvious inequality ey≤1+y+(1+eyI(y>0))y2/2, it follows that, for any λ>0,(29)EexpλX~≤1+λ22EX~21+expλX~IX~>0≤1+cλ2EX2+EX2expλX~.Let β-α<δ<δ1<1 and λ=δ(βn)α-1; then by cr inequality and E|X′|≤E|X|<∞,(30)λX~≤λX′+EX′αX′+EX′1-α≤λX′α+cX′1-α+c≤δβnα-1Xα+cβn1-α+c=δXα+c1+o1≤δ1Xα+c.

By (5), EX2≤Eexp(|X|α)<∞, and EX2exp(δ1|X|α)≤Eexp(|X|α)<∞, therefore, by combination with (29) and (30), we get(31)EexpλX~≤1+cλ2EX2+cEX2expδ1Xα≤1+cλ2=1+cn2α-1≤expcn2α-1.

Obviously, Xk′ is increasing on Xk; thus, by Lemma 7(i), {Xk′;k≥1} is also a sequence of ND random variables. Taking an=βn and λ=δ(βn)α-1 in Lemma 8, for 1<β1<δβα, we obtain(32)Pmax1≤k≤nSk′>nβ≤exp-λβn∑k=1nEexpλX~k≤nexp-λβn+cn2α-1=nexp-δβα+cn-1-αnα≤nexp-β1nα.

Replacing Xk′ by -Xk′ and by the same argument as above, (33)Pmax1≤k≤n-Sk′>nβ≤nexp-β1nα also holds. Hence,(34)Pmax1≤k≤nSk′>nβ≤2nexp-β1nα.

Thus, by combination with (27) and (28) and the fact that ∑n=1∞(np/expc1nα)<∞ for all c1>0p∈R, we obtain that, for β2=^min(βα,β1)>1, (35)∑n=1∞expnαnpPmax1≤k≤nSk>nβ≪∑n=1∞np+1expβ2-1nα<∞.

Hence, (24) holds. Next, we prove (25).

Let x≥n. Replace n by x in X′ and Xk′. Using similar methods to those used in the proof of (27)–(34), there is β2>1 such that(36)Pmax1≤k≤nSk>xβ≪nexp-β2xα.Thus,(37)∫n∞xq-1Pmax1≤k≤nSk>xβdx≪n∫n∞xq-1exp-β2xαdx~n∫n∞xq-1-q-αβ2α-1xq-1-αexp-β2xαdx=n∫n∞xq-αexp-β2xα-β2α′dx≪nq-α+1exp-β2nα, and it implies that (38)∑n=1∞expnαnp∫n∞xq-1Pmax1≤k≤nSk>xβdx≪∑n=1∞np+q+1-αexpβ2-1nα<∞.

That is, (25) holds.

Conversely, if (6) holds for p=α-q-2 and some β>0, then, by Emax1≤k≤n|Sk|-βn+q≥βq∫0nqxq-1Pmax1≤k≤n|Sk|-βn>βxdx≥βqnqPmax1≤k≤n|Sk|>2βn, we have (39)∑n=1∞expnαnα-2Pmax1≤k≤nSk>2βn<∞;by combination with max1≤k≤n|Xk|≤2max1≤k≤nSk, it follows that(40)∑n=1∞expnαn2-αPmax1≤k≤nXk>4βn<∞,and it implies that P(max1≤k≤n|Xk|>4βn)→0, n→∞. Thus, by Lemma 10, there is c>0 such that (41)nPX>4βn≤cPmax1≤k≤nXk>4βn. Consequently, by (40), (42)∑n=1∞expnαn1-αPX4β>n<∞, and, hence, Eexp|X/(4β)|α<∞ from Lemma 9. This completes the proof of Theorem 3.

Proof of Theorem <xref ref-type="statement" rid="thm1.4">4</xref>.

Let X′ and X~ be defined as Theorem 3. Let β-1<δ<1, and λ=δln-r(βn); then, (43)λX~≤δln-rβnX′+c≤δln-rX′X′+c≤δXln-rX+c from |X′|≤βn and xln-rx being monotonically increasing on x.

Thus, by (8), EX2exp(δ|X|ln-r|X|)≪Eexp(|X|ln-r|X|)<∞; therefore, similar to (29), we have (44)EexpλX~≤1+cln-2rn≤expcln-2rn.

By Lemma 8, for 1<β1<δβ<β, we get(45)Pmax1≤k≤nSk′>nβ≤nexp-λβn+cnln-2rn=nexp-δβnln-rβn+cnln-2rn≤nexp-β1nln-rn.

Replacing Xk′ by -Xk′ and by the same argument as above, (46)Pmax1≤k≤n-Sk′>nβ≤nexp-β1nln-rn also holds. Hence, (47)Pmax1≤k≤nSk′>nβ≤2nexp-β1nln-rn.

On the other hand, from (8) and the Markov inequality, (48)PX>nβ≪exp-βnln-rβn≤exp-β1nln-rn.

Hence, together with (27),(49)∑n=1∞expnln-rnnpPmax1≤k≤nSk>nβ≪∑n=1∞np+1expβ1-1nln-rn<∞.

Similar to (36) and the above discussion, for x≥n, (50)Pmax1≤k≤nSk>xβ≪nexp-β1xln-rx. Thus,(51)∫n∞xq-1Pmax1≤k≤nSk>xβdx≪n∫n∞xq-1exp-β1xln-rxdx~n∫n∞xq-1-rxq-1ln-1x-q-1β1-1xq-2lnrx-rβ1-1xq-2lnr-1xexp-β1xln-rxdx=n∫n∞xq-1lnrxexp-β1xln-rx-β1′dx≪nqlnrnexp-β1nln-rn, and it implies that (52)∑n=1∞expnln-rnnp∫n∞xq-1Pmax1≤k≤nSk>xβdx≪∑n=1∞np+qlnrnexpβ1-1nln-rn<∞. Thus, by combination with (23) and (49), (9) holds.

Conversely, by (18), using similar methods to those used in the proof of (7), we can get (10). This completes the proof of Theorem 4.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

Qunying Wu conceived the study and drafted, completed, read, and approved the final paper. Yuanying Jiang conceived the study and completed, read, and approved the final paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11361019) and the Support Program of the Guangxi China Science Foundation (2015GXNSFAA139008).

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