# Some Types of Convergence for Negatively Dependent Random Variables under Sublinear Expectations

Authors: Ruixue Wang; Qunying Wu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In this paper, we research complete convergence and almost sure convergence under the sublinear expectations. As applications, we extend some complete and almost sure convergence theorems for weighted sums of negatively dependent random variables from the traditional probability space to the sublinear expectation space.


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## Body

## 1. Introduction

As is well known that limit theorems are important research topics in probability and statistics, classical limit theorems only hold in the case of model certainty. However, in financial and other field research, statistical model uncertainty, risk random fluctuation, and other problems are emerging, and the classical probabilistic space cannot meet the need of solving such problems. Academician Peng [1–3] of Shandong University creatively proposed the concept and system of the sublinear expectation. Since it was introduced, it has been widely concerned and studied by a large number of scholars, and many excellent results have been achieved. For instance, Chen [4] studied Kolmogorov’s strong law of larger numbers. Zhang and Chen [5] established a weighted central limit theorem for independent random variables under sublinear expectation. Zhang [6–8] got the expectation inequalities, Rosenthals inequalities, and strong law of larger numbers. Yu and Wu [9] obtained complete convergence for weighted sums of END random variables under sublinear expectations, and so on.Complete convergence and almost sure convergence are important problems in limit theorems. The initial definition of complete convergence was presented by Hsu and Robbins [10] in 1947. From then on, lots of results on complete convergence for different sequences have been found under classical probability space. For instance, Wu and Jiang [11] obtained complete convergence for negatively associated sequences of random variables. Wang et al. [12, 13] studied complete convergence for martingale difference sequence and complete convergence for a type of random variables satisfying Rosenthal-type inequality. Some results on almost sure convergence can be found in Sung et al. [14, 15], Wu [16], Chen and Gan [17], Xu and Yu [18], and so on. In this article, we establish the complete convergence and almost sure convergence for weighted sums of ND random variables under sublinear expectations. The results obtained by Sung [15] have been generalized to the sublinear expectation space.


## 2. Preliminaries

We also use the framework and notations of Peng [1–3]. Here, we do not discuss notations, concepts, and properties of the sublinear expectation space in detail. By the properties of sublinear expectations and capacity, we can get Markov inequality:(1)∀X∈H,VX≥x≤E^Xpxp,∀x>0,p>0.Definition 1 ([6]).
(i) (Negative dependence) [6] In a sublinear expectation space (Ω,H,E^), a random vector Y=(Y1,…,Yn),Yi∈H is said to be negatively dependent (ND) to another random vector X=(X1,…,Xm),Xi∈H under E^ if for each pair of test functions φ1∈Cl,Lip(Rm) and φ2∈Cl,Lip(Rn) we have E^(φ1(X)φ2(Y))≤E^(φ1(X))E^(φ2(Y)), whenever φ1, φ2 are coordinate-wise nondecreasing or φ1, φ2 are coordinate-wise nonincreasing with φ1(X)≥0, E^φ2(Y)≥0,  E^|φ1(X)φ2(Y)|<∞,  E^|φ1(X)|<∞,  E^|φ2(Y)|<∞.
(ii) (ND random variables) [6] Let {Xn;n≥1} be a sequence of random variables in the sublinear expectation space (Ω,H,E^). X1,X2,… are said to be negatively dependent if Xi+1 is negatively dependent to (X1,…,Xi) for each i≥1.Obviously, if{Xn;n≥1} is a sequence of ND random variables and functions f1(x),f2(x),…∈Cl,Lip(R) are all nondecreasing (resp., all nonincreasing), then {fn(Xn);n≥1} is also a sequence of ND random variables.The symbolc stands for a positive constant which may have different values in different places. Let an≪bn denote that there exists a constant c>0 such that an≤cbn for sufficiently large n.Lemma 2.
Let{Xni,1≤i≤n,n≥1} be an array of row-wise ND random variables with E^Xni=0. Suppose that
(a)max1≤i≤n⁡|Xni|=O(log⁡n)-1,
(b)∑i=1nE^Xni2=o(log⁡n)-1.
Then, for anyε>0,(2)∑n=1∞V∑i=1nXni>ε<∞.Proof.
Takingbn=(log⁡n)-1 in Lemma 2.2 of [9], we can get conditions (a) and (b). The condition (iii) of Lemma 2.2 of [9] is clearly satisfied for α>1 and bn=(log⁡n)-1. So, we can get Lemma 2.Lemma 3.
AssumeX∈H, p>1, c>0. Then(3)CVXp<∞⇔∑i=1∞VX>ci1/p<∞.(4)CVXp<∞⇔∑n=1∞nVX>cn2/p<∞.Proof.
C V X p < ∞ is equivalent to CVXp/cp<∞. Note that (5)CVXpcp=∫0∞VX>cx1/pdx<∞⇔∑i=1∞VX>ci1/p<∞.Therefore, (3) holds. Then, let x=y2(6)CVXpcp=∫0∞VX>cx1/pdx=∫0∞2yVX>cy2/pdy<∞⇔∑n=1∞nVX>cn2/p<∞. Thus, (4) holds.Lemma 4 (Borel-Cantelli Lemma, Zhang [6], Lemma 3.9).
LetAn;n≥1 be a sequence of events in F. Suppose that V is a countably subadditive capacity. If ∑n=1∞V(An)<∞, then V(An;i.o.)=0, where An;i.o.=⋂n=1∞⋃m=n∞Am.In the sublinear expectation space, because of the uncertainty of expectation and capacity, the study of complete convergence and almost sure convergence is much more complex and difficult. As we all know, in the probability space, there is an equality:EI(|X|≤a)=P(|X|≤a). However, the expression E^I(|X|≤a) does not exist in the sublinear expectation space. This needs to modify the indicator function by functions in Cl,Lip. To this end, for 0<μ<1, we define the even function g(x)∈Cl,Lip(R), 0≤g(x)≤1 for all x, and g(x)=1 if |x|≤μ, g(x)=0 if |x|>1. Then(7)Ix≤μ≤gx≤Ix≤1,Ix>1≤1-gx≤Ix>μ.


## 3. Main Results

Theorem 5.
Letp≥1, and {Xni,1≤i≤n,n≥1} be an array of row-wise negatively dependent random variables under sublinear expectations. There exist a r.v. X and a constant c satisfying(8)E^hXni≤cE^hX,foralln≥1,1≤i≤n,0≤h∈Cl,LipR,(9)E^Xp≤CVXp<∞.Let {ani,1≤i≤n,n≥1} be an array of positive constants such that(10)∑i=1nanip=O1nα,forsomeα>0.(11)∑i=1nani2E^X2ganiXlog⁡n=olog⁡n-1.Then, for any positive whole number N satisfying αN>1, and ∀ε>0,(12)∑n=1∞V∑i=1naniXni-E^XniIaniXni≤εN>ε<∞,and(13)∑n=1∞V∑i=1naniXni-ε^XniIaniXni≤εN<-ε<∞.In particular, if E^Xni=ε^Xni, then(14)∑n=1∞V∑i=1naniXni-E^XniIaniXni≤εN>ε<∞.Remark 6.
Condition (8) is similar to stochastic domination condition in Theorem 2.1 of [15]. Condition (9) corresponds to the moment condition of probability space. Condition (11) is similar to condition (ii) of Theorem 2.1 of [15]. Our results extend Theorem 2.1 of [15] from the traditional probability space to sublinear expectation space.Theorem 7.
Suppose thatp≥1, and V is countably subadditive. Let {Xn,n≥1} be a sequence of negatively dependent random variables. There exist a r.v. X and a constant c satisfying(15)E^hXn≤cE^hX,foralln≥1,0≤h∈Cl,LipR,(16)E^Xp≤CVXp<∞.Let {ani,1≤i≤n,n≥1} be an array of positive constants such that(17)max1≤i≤n⁡ani=O1n1/p,(18)∑i=1nanip=O1nα,forsomeα>0,(19)∑i=1nani2E^X2ganiXlog⁡n=olog⁡n-1.Then(20)limsupn→∞∑i=1naniXi-E^Xi≤0a.s.V,and(21)liminfn→∞∑i=1naniXi-ε^Xi≥0a.s.V.In particular, if E^Xi=ε^Xi, then(22)limn→∞∑i=1naniXi-E^Xi=0a.s.V.Theorem 8.
Suppose that1≤p≤2, and V is countably subadditive. Let {Xn,n≥1} be a sequence of negatively dependent random variables. There exist a r.v. X and a constant c satisfying (15), (16). Let {ani,1≤i≤n,n≥1} be an array of positive constants such that (17), (18) hold. Then we also can get (20), (21), and (22).Theorem 9.
Suppose thatp≥1, and V is countably subadditive. Let {Xni,1≤i≤n,n≥1} be an array of row-wise negatively dependent random variables. There exist a r.v. X and a constant c satisfying (8) and (9). Let {ani,1≤i≤n,n≥1} be an array of positive constants such that(23)max1≤i≤n⁡ani=O1n2/p,(24)∑i=1nani2E^X2ganiXlog⁡n=olog⁡n-1.Then(25)limsupn→∞∑i=1naniXni-E^Xni≤0a.s.V,and(26)liminfn→∞∑i=1naniXni-ε^Xni≥0a.s.V.In particular, if E^Xni=ε^Xni, then(27)limn→∞∑i=1naniXni-E^Xni=0a.s.V.Remark 10.
Theorems7–9 extend Corollary 2.1, Corollary 2.2, and Corollary 2.5 of Sung [15] from the probability space to sublinear expectation space.Proof of Theorem5.
To prove our main results, we just need to prove (12). Because of considering {-Xni,1≤i≤n,n≥1} instead of {Xni,1≤i≤n,n≥1} in (12), we can get (13). Without loss of generality, we can assume that E^Xni=0.
For fixedn≥1, denote for 1≤i≤n that (28)Xni1=XniIaniXni≤log⁡n-1+ani-1log⁡n-1IaniXni>log⁡n-1-ani-1log⁡n-1IaniXni<-log⁡n-1,Xni2=Xni-ani-1log⁡n-1Ilog⁡n-1<aniXni≤εN,Xni3=Xni+ani-1log⁡n-1I-εN≤aniXni<-log⁡n-1,Xni4=-ani-1log⁡n-1IaniXni>εN+ani-1log⁡n-1IaniXni<-εN.Then, we have (29)aniXniIaniXni≤εN=aniXni1+aniXni2+aniXni3+aniXni4. On account of the arbitrariness of ε, it follows that (30)∑n=1∞V∑i=1naniXniIaniXni≤εN>4ε≤∑n=1∞V∑i=1naniXni1>ε+∑n=1∞V∑i=1naniXni2>ε+∑n=1∞V∑i=1naniXni3>ε+∑n=1∞V∑i=1naniXni4>ε≔I1+I2+I3+I4.
Because ofani>0, by the definition of ND, {aniXni,1≤i≤n} is also a sequence of ND random variables. Obviously, aniXni(1) is nondecreasing by aniXni. Therefore, aniXni(1) is also a sequence of ND random variables. Firstly, we prove I1<∞.
For the array{ani(Xni(1)-E^Xni(1)),1≤i≤n,n≥1}, we can make use of Lemma 2, noting that max1≤i≤n⁡|ani(Xni(1)-E^Xni(1))|≤2(log⁡n)-1, and condition (a) of Lemma 2 holds.
By (7) and the Cr inequality, for any r>0,(31)Xni1r≪XnirIaniXni≤log⁡n-1+ani-rlog⁡n-rIaniXni>log⁡n-1≤XnirgμaniXnilog⁡n+ani-rlog⁡n-r1-ganiXnilog⁡n.E^Xni1r≪E^XrgμaniXlog⁡n+ani-rlog⁡n-rE^1-ganiXlog⁡n≤E^XrgμaniXlog⁡n+ani-rlog⁡n-rVaniX>μlog⁡n-1.Then, by (1), (8), (9), (10), (11), (31), we have (32)∑i=1nE^aniXni1-E^Xni12≤∑i=1nani2E^2Xni21+2E^Xni12≤c∑i=1nani2E^Xni12≤∑i=1nani2E^X2gμaniXlog⁡n+log⁡n-2∑i=1nVaniX>μlog⁡n-1≪olog⁡n-1+log⁡np-2∑i=1nanipE^Xp≤olog⁡n-1+log⁡np-2O1nαCVXp=olog⁡n-1. Now, conditions (a) and (b) of Lemma 2 hold. So, by Lemma 2, we can get that (33)∑n=1∞V∑i=1naniXni1-E^Xni1>ε2<∞.So to prove I1<∞, it suffices to prove that(34)∑i=1naniE^Xni1→0,n→∞.By E^Xni=0, (1), (7), (8), (9), (10), (31), we can get (35)∑i=1naniE^Xni1=∑i=1naniE^Xni1-E^Xni≤∑i=1naniE^Xni-Xni1≪∑i=1naniE^X1-ganiXlog⁡n≤∑i=1naniE^Xpanip-1log⁡np-1μ1-p1-ganiXlog⁡n≪log⁡np-1∑i=1nanipE^Xp≤clog⁡np-1O1nα→0. So, we get (34).
Now, we proveI2<∞. By the condition of 0≤aniXni(2)≤ε/N, ∑i=1naniXni(2)=∑i=1naniXni(2)>ε means that there is at least one N that makes Xni2≠0. By the definition of ND, (1), (7), (8), (9), (10), we can obtain that (36)V∑i=1naniXni2>ε≤∑1≤k1<⋯<kN≤nVXn,k12≠0,…,Xn,kN2≠0≤∑1≤k1<⋯<kN≤nVan,k1Xn,k1≥log⁡n-1,…,an,kNXn,kN≥log⁡n-1≪∑1≤k1<⋯<kN≤nE^1-gan,k1Xlog⁡n⋯1-gan,kNXlog⁡n≤∑1≤k1<⋯<kN≤nE^1-gan,k1Xlog⁡n⋯E^1-gan,kNXlog⁡n≤∑i=1nE^1-ganiXlog⁡nN≤∑i=1nVaniX>μlog⁡n-1N≪log⁡np∑i=1nanipE^XpN≪log⁡np1nαE^XpN≤clog⁡np1nαN. When αN>1, we can get I2<∞.
In the same way, we know that-ε/N≤aniXni(3)≤0, so, ∑i=1naniXni(3)=-∑i=1naniXni(3)>ε also means that there is at least one N that makes Xni(3)≠0. Using the same method of proving I2<∞, we can have I3<∞. At last, we prove I4<∞. Using the similar method to I2. By the value of Xni(4), we can get aniXni(4)≤(log⁡n)-1I(|aniXni|>ε/N)≤ε/N. Hence, by (1), (7), (8), (9), (10), we have (37)∑n=1∞V∑i=1naniXni4>ε≤∑n=1∞VthereareatleastNsuchthataniXni>εN≪∑n=1∞∑i=1nE^1-ganiXε/NN≤∑n=1∞∑i=1nVaniX>εμNN≤∑n=1∞∑i=1nanipE^XpεμN-pN≪∑n=1∞E^XpO1nαN<∞. Because of αN>1, we can get I4<∞. Thus, this completes the proof of Theorem 5.Proof of Theorem7.
Firstly, we prove (20). Without loss of generality, we assume that E^Xi=0.
By Theorem5, ∀ε>0, (38)∑n=1∞V∑i=1naniXiIaniXi≤εN>ε<∞. Thus, we can get(39)V∑i=1naniXiIaniXi≤εN>ε;i.o.=0.Equation (39) follows from Lemma 4, and V is countably subadditive. ∀ε>0, on account of the arbitrariness of ε, it implies(40)limsupn→∞∑i=1naniXiIaniXi≤εN≤εa.s.V.Thus, to prove (20), we just need to check that(41)∑i=1naniXiIaniXi>εN→0a.s.V.By (7) and (3) of Lemma 3, we can get(42)∑i=1∞VXi≥ci1/p≤∑i=1∞E^1-gXici1/p≪∑i=1∞E^1-gXci1/p≤∑i=1∞VX>μci1/p<∞.Thus, V(|Xi|≥ci1/p;i.o.)=0 follows from Lemma 4 and V being countably subadditive. By (17), it is easily checked that (43)∑i=1naniXiIaniXi>εN≪1n1/p∑i=1nXiI1n1/pXi>εN≪1n1/p∑i=1nXiIXi>i1/pεN→0a.s.V.
Hence, (41) holds; because of the arbitrariness of ε, we can get (20). Considering {-Xn;n≥1} instead of {Xn;n≥1} in (20), we can obtain(44)limsupn→∞⁡∑i=1nani-Xi-E^-Xi≤0.
We know thatε^X:=-E^(-X), ∀X∈H. Replacing -E^(-Xi) with ε^Xi in (44), we can get (21). When E^Xi=ε^Xi, combine (20) with (21), and we will have (22). Thus, we obtain the result of Theorem 7.Proof of Theorem8.
For1≤p≤2, by (16), (18), we obtain that (45)∑i=1nani2E^X2ganiXlog⁡n≤log⁡np-2∑i=1nanipE^XpganiXlog⁡n≤log⁡np-2∑i=1nanipE^Xp≤log⁡np-2O1nαCVXp=olog⁡n-1. We get condition (19) of Theorem 7. So the result of Theorem 8 can be proved.Proof of Theorem9.
Without loss of generality, we still assume thatE^Xni=0. By (23), obviously (46)∑i=1nanip≤nmax1≤i≤n⁡anip≤O1n. Then, by Theorem 5, to prove (25), we just need to prove that(47)∑n=1∞V∑i=1naniXniIaniXni>εN>ε<∞.By (4) of Lemma 3, (7), (8), (23), it is easy to check that (48)∑n=1∞V∑i=1naniXniIaniXni>εN>ε≤∑n=1∞∑i=1nVaniXni>εN≤∑n=1∞∑i=1nE^1-ganiXniε/N≪∑n=1∞∑i=1nE^1-ganiXε/N≤∑n=1∞∑i=1nVaniX>μεN≤∑n=1∞∑i=1nVc·1n2/pX>μεN≤∑n=1∞nVX>μεn2/pcN<∞.
Therefore, (47) holds, then, we obtain (25). Considering {-Xni;n≥1} instead of {Xni;n≥1} in (25), we have(49)limsupn→∞⁡∑i=1nani-Xni-E^-Xi≤0.
Replacing-E^(-Xni) with ε^Xni in (49), we can get (26). When E^Xni=ε^Xni, combine (25) with (26), and we will have (27). Therefore, we obtain the result of Theorem 9.



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