# On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

Authors: Xiaochen Ma; Qunying Wu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.


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## Body

## 1. Introduction

There are many uncertainties in the fields of economy, statistics, engineering, etc., which cannot be accurately predicted or described by traditional additive probability measures. In reality, many uncertain phenomena do not satisfy the linear additivity condition, so the application of classical limit theory is limited to some extent; a growing number of people abnegate the traditional tool of additivity of probability and instead use the new tool of nonadditive probability measure to portray problems with uncertainty. Taking financial derivatives as an example, the measures of risk are generally nonlinear and do not satisfy the linear additivity; the wrong assessment and management of its risk will lead to serious consequences. Recently, Peng [1–3] proposed a generalized sublinear expectation space theory. The sublinear expectation is not dependent on the corresponding probability and can be defined directly by the real functions satisfying monotonicity, constant preserving, subadditivity, and positive homogeneity. And combining the theory of partial differential equations, Peng has given the concepts of the identical distribution, independence, and maximum distribution of the random variables under sublinear expectation framework and has established a relatively complete axiom system. About other properties of the sublinear expectations theory, we can refer to Peng [4–6].Under sublinear expectation space, a suite of serviceable consequences have been established. For example, Zhang [7–9] acquired the exponential inequalities, Rosenthal’s inequalities, Kolmogorovs strong law of larger numbers (SLLNs) and Hartman-Wintners law of iterated logarithm, Yu and Wu [10] obtained complete convergence for weighted sums of extended negatively dependent random variables under a specific moment condition, Cheng [11] established strong law of large numbers (SLLNs) under a general moment condition and so on. And many strong laws of large numbers for weighted sums of some sequences have been obtained; one can refer to Chen [12], Wu and Jiang [13], Zhang [14], Chen et al. [15], and Chen and Liu [16]. These authors only researched Kolmogorov’s SLLNs or Marcinkiewicz’s SLLNs for some sequences of random variables. However, the strong law of large numbers for the weighted sums of extended negatively dependent (END) random variables under sublinear expectation has less related results.In this article, we will give some conditions for strong law of large numbers for weighted sums of END random variables. By the truncated random variable and some moment inequalities, we testify that the SLLNs for the weighted sums hold for END random variables. In addition, our results extend the corresponding results of Shen et al. [17] relative to the classical probability space. In the next part of this article, we provide some definitions under the sublinear expectations containing identical distribution, extended negatively dependent (END). In part 3, we give the main results. The proof of these theorems is given in the last part.


## 2. Preliminaries

The research of this article uses the framework of Peng [1–3]. For brevity, we omit these definitions. By definition of V and V, we have(1)E^f≤VA≤E^g,ε^f≤VA≤ε^g,iff≤IA≤g,f,g∈H.Under the sublinear expectation space, Chen [12] and Zhang [7, 8] researched the almost sure convergence of sequence. Hence, we define the almost sure as follows.Definition 1 (Wu and Jiang 2017 [13]).
A sequence of random variables{Xn;n≥1} is called to converge to X almost surely Va.s.V, showed by Xn→X a.s. V as n→∞, if VXn↛X=0.
V can be substituted by V and V severally. By VA≤VA and VA+VAc=1 for any A∈F, it is quite clear that Xn→X a.s. V signifies Xn→X a.s. V, but Xn→X a.s. V does not signify Xn→X a.s. V. Further, (2)Xn→Xa.s.V⇔VXn→X=1⇔VXn-X≥ε,i.o.=0for∀ε>0,Xn→Xa.s.V⇔VXn↛X=0⇔VXn→X=1.Definition 2 (Peng 2006 [1], 2008 [6]).
LetX1 and X2 be two random vectors defined severally in sublinear expectation spaces Ω,H,E^. They are named identically distributed, showed by X1=dX2 if (3)E^1φX1=E^2φX2,∀φ∈Cl,LipR,whenever the sublinear expectations are finite. A sequence {Xn,n≥1} of random variables is called to be identically distributed if Xi=dX1 for each i≥1.Definition 3 (Zhang 2016 [7]).
A sequence of random variables{Xn,n≥1} is called to be upper (resp., lower) extended negatively dependent (END) if there is some dominating constant K≥1 such that (4)E^∏i=1nφiXi≤K∏i=1nE^φiXi,∀n≥2, whenever the nonnegative functions φi(Xi)∈CL,lip(R),i=1,2,…, are all nondecreasing (resp., all nonincreasing). They are named extended negatively dependent if they are both upper extended negatively dependent and lower extended negatively dependent.In the following passage, let{Xn;n≥1} be a random variable sequence in (Ω,H,E^). c will signify a positive constant that may have different effects in different places. Let an≪bn indicate that there exists a constant c>0 such that an≤cbn for sufficiently large n, and I(·) indicates an indicator function.Lemma 4 (Zhang 2016, Lemma 3.9 [8]).
Let{An;n≥1} be a sequence of events in F. Assume that V is a countably subadditive capacity. If ∑n=1∞V(An)<∞, then (5)VAn;i.o.=0,whereAn;i.o.=⋂n=1∞⋃i=n∞Ai.Lemma 5 (Zhang 2016, Theorem 3.1 [9]).
Let{Xk;k≥1} be a sequence of upper extended negatively dependent (END) random variables in (Ω,H,E^) with E^Xk≤0. Then, (6)VSn≥x≤Cpδ-2pK∑k=1nE^Xkpxp+Kexp-x22Bn1+δ,where p≥2; there exists a constant Cp≥1 such that for all x>0 and 0<δ≤1.(7)VSn≥x≤1+KeBnx2,∀x>0.where Bn=∑k=1nE^Xk2.


## 3. Main Results

Theorem 6.
Suppose thatq>max⁡(β,2),β>1,1≤p<2,1/p=1/q+1/β and V is countably subadditive. Let Xn;n≥1 be a sequence of upper extended negatively dependent (END) random variables and identically distributed. Assume that(8)E^Xβ≤CVXβ<∞.Let {an,n≥1} and {bn,n≥1} be sequences of positive numbers with an↑, bn↑∞, such that(9)∑i=1nai=Obn,(10)anbn=On-1/p,Then,(11)limsupn→∞1bn∑i=1naiXi-E^Xi≤0a.s.V,
Further, ifXn;n≥1 is extended negatively dependent random variables, then(12)liminfn→∞1bn∑i=1naiXi-ε^Xi≥0a.s.V.
In particular, ifXn;n≥1 is extended negatively dependent random variables and E^Xi=ε^Xi, then(13)limn→∞1bn∑i=1naiXi-E^Xi=0a.s.V.Theorem 7.
Suppose that1<r<2, and V is countably subadditive. Let {Xn;n≥1} be a sequence of extended negatively dependent (END) random variables and identically distributed. Assume that(14)CVXr<∞.
Let{an,n≥1} and {bn,n≥1} be sequences of positive numbers with ∑n=i∞bn-2≪bi-2 for sufficiently i, bn↑∞. Put cn=bn/an, for n≥1, such that(15)Nn=cardi;ci≤n≪nr.Then,(16)limsupn→∞1bn∑i=1naiXi-E^Xi≤0a.s.V,
Further, ifXn;n≥1 is extended negatively dependent random variables, then(17)liminfn→∞1bn∑i=1naiXi-ε^Xi≥0a.s.V.
In particular, ifXn;n≥1 is extended negatively dependent random variables and E^Xi=ε^Xi, then(18)limn→∞1bn∑i=1naiXi-E^Xi=0a.s.V.


## 4. Proof

Proof of Theorem6.
When we replace{-Xn;n≥1} with {Xn;n≥1} in (11), we can acquire (12). So, we just need to prove (11). And we suppose that E^Xi=0. It can be easily proved that (8) is isovalent for any c>0,(19)∑n=1∞VX>cn1/β<∞,∀c>0.Note that (20)∑n=1∞VX>cn1/β≥∑i=1∞∑2i-1≤n<2iVX>cn1/β≥∑k=1∞∑2i-1≤n<2iVX>c2i/β=2-1∑i=1∞2iVX>c2i/β. Hence, (19) implies(21)∑i=1∞2iVX>c2i/β<∞,∀c>0.
For upper extended negatively dependent random variablesXn;n≥1, there are truncated functions belonging to Cl,Lip and are nondecreasing, such that the sequences of truncated random variables are also upper extended negatively dependent. Let fc(x)=-cI(x<-c)+xI(x≤c)+cI(x>c) for any c>0, for any i≥1, (22)Yi≔fi1/βXi=-i1/βIXi<-i1/β+XiIXi≤i1/β+i1/βIXi>i1/β,Zi≔Xi-Yi=Xi+i1/βIXi<-i1/β+Xi-i1/βIXi>i1/β,Y≔fi1/βX,Z≔X-Y.
Then,Yi;i≥1 is also a sequence of upper extended negatively dependent random variables by fc(x)∈Cl,Lip and fc(x) being nondecreasing.
Note that(23)1bn∑i=1naiXi=1bn∑i=1naiZi+1bn∑i=1naiXi-E^Yi+1bn∑i=1naiE^Yi≔I1+I2+I3.
Thus, to testify (11), it suffices to verify that(24)limsupn→∞Ii≤0a.s.Vi=1,2,limn→∞I3=0.
It shall be pointed out that, in the classical linear expectation space, we know that the equality ofEI(X≤a)=PI(X≤a) is set up. Nevertheless, in the sublinear expectation space, E^ is defined through continuous functions in Cl,Lip and the indicator function I(X≤a) does not satisfy continuity. Hence, this expression E^I(X≤a) is no longer effective and we need to amend the indicator function by functions by substituting the original indicator function into the sublinear expectation space. Hence, we define the function g(x)∈Cl,Lip(R) as follows.
For0<μ<1, let g(x) be an even function and g(x)∈Cl.Lip(R) such that 0≤g(x)≤1 for all x, g(x)=1 if x≤μ,g(x)=0 if x≥1 and g(x) are nonincreasing for any x>0. Then,(25)Ix≤μ≤gx≤Ix≤1,Ix>1≤1-gx≤Ix>μ.By (1), (19), and (24), (26)∑i=1∞VZi≠0≤∑i=1∞VXi>i1/β≤∑i=1∞E^1-gXii1/β=∑i=1∞E^1-gXi1/β≤∑i=1∞VX>μi1/β<∞. Thus, V(Zi≠0,i.o.)=0 follows from the Borel-Cantelli lemma (Lemma 4) and V being countably subadditive. It follows that from (10)(27)I1=1bn∑i=1naiZi≪n-1/p∑i=1nZi→0,a.s.V.
Now, we proveI2. For any r>0, by the Cr inequality and (25), Yir≪XirIXi≤i1/β+ir/βIXi>i1/β≤XirgμXi/i1/β+ir/β1-gXi/i1/β,(28)E^Yir≤E^XirgμXii1/β+ir/βE^1-gXii1/β=E^XrgμXi1/β+ir/βE^1-gXi1/β≤E^XrgμXi1/β+ir/βVX>μi1/β.
BecauseaiYi-E^Yi;i≥1 is a sequence of upper extended negatively dependent random variables with E^aiYi-E^Yi=0, using (6) in Lemma 5, let δ=1, for aiYi-E^Yi;i≥1, by (10), we obtain for ϵ>0,(29)∑n=1∞V1bn∑i=1naiYi-E^Yi>ϵ≤∑n=1∞ϵ-qbn-q∑i=1nE^aiYi-E^Yiq+exp-ϵ2bn24∑i=1nE^aiYi-E^Yi2≤∑n=1∞n-q/p∑i=1nE^Yi-E^Yiq+∑n=1∞exp-ϵ2n2/p4∑i=1nE^Yi-E^Yi2=^I21+I22.
Letgj(x)∈Cl,LipR,j≥1 such that 0≤gj(x)≤1 for all x and gjx/2j/β=1 if 2(j-1)/β<x≤2j/β,gjx/2j/β=0 if x≤2(j-1)/β or x>(1+μ)2j/β. Then,(30)gjx2j/β≤Iμ2j-1/β<x≤1+μ2j/β,XrgX2i/β≤1+∑j=1iXrgjX2j/β,∀r>0.By the Cr inequality, (28) and (29), g(x)↓ for all x>0, we have (31)I21≪∑n=1∞n-q/p∑i=1nE^Yiq≤∑n=1∞n-q/p∑i=1nE^XqgμXi1/β+iq/βVX>μi1/β≤∑n=1∞n1-q/pE^XqgμXn1/β+∑n=1∞n-q/p∑i=1niq/βVX>μi1/β≔I211+I212. By the Markov inequality, (19), and 1/p=1/q+1/β, we get (32)I212=∑i=1∞iq/βVX>μi1/β∑n=i∞n-q/p≤∑i=1∞iq/βi-q/p+1VX>μi1/β=∑i=1∞VX>μi1/β<∞.
For everyn, there exists an i such that 2i-1≤n<2i; thus, by (1), (21), (25), (30), and g(x)↓ for all x>0, we get (33)I211=∑n=1∞n-q/βE^XqgμXn1/β≤∑i=1∞∑2i-1≤n<2i2-i-1q/βE^XqgμX2i/β≪∑i=1∞21-q/βiE^XqgμX2i/β≤∑i=1∞21-q/βi∑j=1iE^XqgjμX2j/β=∑j=1∞E^XqgjμX2j/β∑i=j∞21-q/βi≤∑j=1∞21-q/βj1+μq2qj/βVX>2j-1/β=c∑j=1∞2jVX>2-1/β2j/β<∞. So, we have(34)I21<∞.Next, we prove I22<∞. Since gμX/n1/β≤IX≤cn1/β. So, by (8), (28), and Markov inequality, (35)∑i=1nE^Yi-E^Yi2≪∑i=1nE^Yi2≤∑i=1nE^X2gμXi1/β+∑i=1ni2/βVX>μi1/β≤nE^X2gμXn1/β+c∑i=1ni2/β-1≤n1+2-β/β+n2/β=2n2/β. Since 2/q>0, so cn2/q≥ln⁡n2, for ∀c>0. Hence,(36)I22≪∑n=1∞exp-cn2/pn2/β=∑n=1∞exp-cn2/q≤∑n=1∞exp-2ln⁡n=∑n=1∞n-2<∞.Combining (29), (34), the Borel-Cantelli lemma (Lemma 4), and arbitrariness of ε, we obtain(37)limsupn→∞I2≤0a.s.V.
Finally, we proveI3→0asn→∞. By E^Xi=0, β>1. So, by (8), (25), (38)E^Yi=E^Xi-E^Yi≤E^Xi-Yi=E^Xi+i1/βIXi<-i1/β+Xi-i1/βIXi>i1/β≪E^Xi1-gXii1/β=E^X1-gXi1/β≤i-β-1/βE^Xβ→0,i→∞. Once again, using (8) and (9), by Toeplitz lemma, we get (39)I3≤1bn∑i=1naiE^Yi→0. Hence, we get (40)I3→0,n→∞. Together with (27) and (37), (24) holds. The proof of Theorem 6 is completed.Proof of Theorem7..
When we replace{-Xn;n≥1} with {Xn;n≥1} in (16), we can obtain (17). So, we just need to prove (16). And we assume that E^Xn=0. Obviously, by the same methods as (19) and (21), (14) implies(41)∑k=1∞2krVX>c2k<∞,∀c>0.Denote (42)Yn≔-cnIXn<-cn+XnIXn≤cn+cnIXn>cn. By the definition of g,gj in Theorem 6, we have (43)X2gX2k≤∑j=1kX2gjX2j,gjX2j≤Iμ2j-1<X<1+μ2j.
LetN(1)=1. By (14), (15), (25), (41), g(x)↓ for all x>0, and E^ being countably subadditive, we have (44)∑n=1∞VXn≠Yn≤∑n=1∞VXn>cn≪∑n=1∞VX>cn≤∑n=1∞E^1-gXcn≤∑k=1∞∑2k-1<cn≤2kE^1-gX2k-1≤∑k=1∞N2k-N2k-1∑j=k-1∞E^gjX2j≤∑k=1∞N2k-N2k-1∑j=k-1∞VX>μ2j-1=∑j=1∞VX>μ2j-1∑k=1j+1N2k-N2k-1≤∑j=1∞N2j+1VX>μ2j-1≪∑j=1∞2jrVX>μ2-12j<∞which implies VXi≠Yi,i.o.=0 from the Borel-Cantelli lemma (Lemma 4). Hence, in order to testify (16), we just need to testify(45)limsupn→∞1bn∑i=1naiYi-E^Yi≤0a.s.V,(46)limn→∞1bn∑i=1naiE^Yi=0.By the proof of (44), for any c>0, we get(47)∑n=1∞VX>ccn<∞.
Firstly, we estimate (45). By fc(x)∈Cl,LipR, aiYi-E^Yi;i≥1 is also a sequence of upper extended negatively dependent random variables with E^aiYi-E^Yi=0. Using (7) in Lemma 5 for aiYi-E^Yi;i≥1, for every ϵ>0, we obtain (48)∑n=1∞V1bn∑i=1naiYi-E^Yi>ϵ≤∑n=1∞bn-2∑i=1nE^aiYi-E^Yi2≪∑n=1∞bn-2∑i=1nai2E^Yi2=∑i=1∞ai2E^Yi2∑n=i∞bn-2≪∑i=1∞ci-2E^Yi2.
Observe that, by (1), (15), the same methods as (28), (41), (47), g(x)↓ for all x>0, and E^ being countably subadditive, (49)∑i=1∞ci-2E^Yi2≤∑i=1∞ci-2E^X2gμXci+ci2VX>μci=∑i=1∞ci-2E^X2gμXci+∑i=1∞VX>μci≤∑k=1∞∑2k-1<ci≤2k2k-1-2E^X2gμX2k+c≪∑k=1∞N2k-N2k-12k-1-2∑j=1kE^X2gjμX2j≤∑k=1∞N2k-N2k-12k-1-2∑j=1k1+μ222jVX>2j-1=c∑j=1∞22jVX>2j-1∑k=j∞N2k-N2k-12k-1-2≤∑j=1∞22jVX>2j-1∑k=j∞N2k2k-1-2-2k-2≪∑j=1∞22jVX>2j-1∑k=j∞2kr-2≪∑j=1∞2jr-222jVX>2j-1≤∑j=1∞2jrVX>2-12j<∞. By the Borel-Cantelli lemma (Lemma 4) and arbitrariness of ε, we obtain (50)limsupn→∞1bn∑i=1naiYi-E^Yi≤0a.s.V.
Secondly, we estimate (46). Since IXn>cn≤1-gXn/cn and E^Xn=0, it follows that(51)∑n=1∞anbnE^Yn≤∑n=1∞cn-1E^Xn1-gXncn+cnE^1-gXncn=∑n=1∞cn-1E^Xn1-gXncn+∑n=1∞E^1-gXncn≤∑n=1∞cn-1E^Xn1-gXncn+∑n=1∞VX>μcn.
Observe that, byg(x)↓ for all x>0, and E^ being countably subadditive,(52)
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*Source: 7945431-2019-04-11.xml*