# Complete Convergence for Weighted Sums of Widely Acceptable Random Variables under Sublinear Expectations

Authors: Rong Hu; Qunying Wu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Using different methods than the probability space, under the condition that the Choquet integral exists, we study the complete convergence theorem for weighted sums of widely acceptable random variables under sublinear expectation space. We proved corresponding theorem which was extended to the sublinear expectations’ space from the probability space, and similar results were obtained.


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## Body

## 1. Introduction

In the study of probability theory and mathematical statistics, limit theory is an important research topic, which is widely used in the financial sector and other fields. However, the establishment of the classical limit theory requires strict conditions for the certainty model, especially in the practice of financial statistics and financial risk measurement, so its limitations are gradually highlighted by many uncertainties. Therefore, Peng [1–3] made an improvement, proposed a sublinear expectations’ space that can model the probability and distribution of uncertainty, and gave the corresponding theoretical system, which has aroused the attention of the majority of scholars. At present, the limit theory has obtained many excellent results under sublinear expectation. For example, in the early research on sublinear expectation, Peng [1–3] extended the central limit theory in the traditional probability space to the sublinear expectation space. Zhang [4–6] continued his research on extended negatively dependent random variables and obtained Kolmogorov’s strong law of large numbers (SLLN) and a series of inequalities under sublinear expectation. Zhang and Chen [7] obtain the central limit theorem for weighted sums in sublinear expectations’ space. Feng et al. [8] obtain a complete convergence of the weighted sum of negatively dependent (ND) sequences in the sublinear expectations’ space. Wang and Wu [9] study on complete convergence and almost sure convergence under the sublinear expectations. Chen [10] obtains a SLLN for an independent identically distributed sequence in the sublinear expectations space. Liang and Wu [11] research on complete convergence and complete integral convergence for extended negatively dependent (END) random variables under sublinear expectations.Complete convergence is one of the most important problems in limit theory research because of the extensive application of weighted sum in statistics, and its properties attract more scholars to study and discuss. In the study of complete convergence in probability space, statistician Hsu and Robbins [12] first propose the concept of complete convergence in 1947, which aroused the interest of many scholars. So far, complete convergence has been studied very deeply in probability space, for example, Liang and Su [13] obtain the complete convergence theorem for weighted sums of negatively associated (NA) sequences and discuss its necessity. Sung [14], based on the exponential inequality, obtains new complete convergence results for weighted sums of independent random variables. Wu [15, 16] proves the complete convergence theorems for ND sequences and arrays of row-wise ND random variables. Lita [17] explores the property of complete convergence for END random variables. In this article, we establish the complete convergence theorem for weighted sums of widely acceptable (WA) random variables under sublinear expectations. The results have been obtained by Lang et al. [18] and have been generalized to the sublinear expectation space.


## 2. Preliminaries

We use the framework and notations of Peng [1–3]. Let Ω,ℱ be a given measurable space and let ℋ be a linear space of real functions defined on Ω,ℱ such that if X1,X2,…,Xn∈ℋ, then φX1,…,Xn∈ℋ for each φ∈Cl,Lipℝn, where Cl,Lipℝn denotes the linear space of (local Lipschitz) functions φ satisfying(1)φx−φy≤c1+xm+ymx−y,∀x,y∈ℝn,for some c>0,m∈ℕ depending on φ.ℋ is considered as a space of random variables. In this case, we denote X∈ℋ.Definition 1.
(see [2]). A sublinear expectation E^ on ℋ is a function E^:ℋ⟶ℝ¯ satisfying the following properties: for all X,Y∈ℋ, we have(a)
Monotonicity: ifX≥Y, then E^X≥E^Y(b)
Constant preserving:E^c=c(c)
Subadditivity:E^X+Y≤E^X+E^Y, whenever E^X+E^Y is not of the form +∞−∞ or −∞+∞(d)
Positive homogeneity:E^λX=λE^X,λ≥0
Here,ℝ¯=−∞,∞. The triple Ω,ℋ,E^ is called a sublinear expectation space.
Given a sublinear expectationE^, let us denote the conjugate expectation ε^ of E^ by(2)ε^X≔−E^−X,∀X∈ℋ.
From the definition, it is easily shown that, for allX,Y∈ℋ,(3)ε^X≤E^X,E^X+c=E^X+c,E^X−Y≤E^X−Y,E^X−Y≥E^X−E^Y.
IfE^Y=ε^Y, then E^X+aY=E^X+aE^Y for any a∈ℝ. Next, we consider the capacities corresponding to the sublinear expectations. Let G⊂ℱ. A function V:G⟶0,1 is called a capacity if(4)V∅=0,VΩ=1,VA≤VB,for∀A⊆B,A,B∈G.
It is called subadditive ifVA∪B≤VA+VB for all A,B∈G with A∪B∈G. In the sublinear space Ω,ℋ,E^, we denote a pair V,V of capacities by(5)VA≔infE^ξ;IA≤ξ,ξ∈ℋ,VA≔1−VAc,∀A∈ℱ,where Ac is the complement set of A. By definition of V and V, it is obvious that V is subadditive, and(6)VA≤VA,∀A∈ℱ.
IfIA∈ℋ, then(7)VA=E^IA,VA=ε^IA.
Iff≤IA≤g,f,g∈ℋ, then(8)E^f≤VA≤E^g,ε^f≤VA≤ε^g.
This implies Markov inequality:∀X∈ℋ,(9)VX≥x≤E^Xpxp,∀x>0,p>0,from IX≥x≤Xp/xp∈ℋ.Definition 2.
(see [2]). We define the Choquet integrals/expectations CV,CV by(10)CVX=∫0∞VX≥tdt+∫−∞0VX≥t−1dt,with V being replaced by V and V, respectively.Definition 3.
(see [2]).(i)
E^ countably subadditive: E^ is called to be countably subadditive if it satisfies(11)E^X≤∑n=1∞E^Xn,whereX≤∑n=1∞Xn,X,Xn∈ℋ,X≥0,Xn≥0,n≥1.(ii)
V is called to be countably subadditive if(12)V∪n=1∞An≤∑n=1∞VAn,∀An∈ℱ.Definition 4 (identical distribution, see [2]).
LetX1 and X2 be two n-dimensional random vectors defined, respectively, in a sublinear expectation spaces Ω1,ℋ1,E^1 and Ω2,ℋ2,E^2. They are called identically distributed if(13)E^φX1=E^φX2,∀φ∈Cl,Lipℝn,whenever the sublinear expectation is finite. A sequence Xn;n≥1 of random variables is said to be identically distributed if, for each i≥1,Xi and X1 are identically distributed.Definition 5.
(widely acceptable, see [19]). Let Yn,n≥1 be a sequence of random variables in a sublinear expectation spaceΩ,ℋ,E^. The sequence Yn,n≥1 is called widely acceptable (WA) if for t≥0, and for all n∈ℕ,(14)E^exp∑i=1ntYi≤gn∏i=1nE^exptYi,where 0<gn<∞.
It is obvious that ifXn;n≥1 is a sequence of widely acceptable random variables and functions f1x,f2x,…,∈Cl,Lipℝn are all nondecreasing (resp. all nonincreasing), then fnXn;n≥1 is also a sequence of widely acceptable random variables.
In the following, letXn;n≥1 be a sequence of random variables in Ω,ℋ,E^, and Sn=∑i=1nXi. The symbol c stands for a generic positive constant which may differ from one place to another. Let ax∼bx denote limx⟶∞ax/bx=1. ax≪bx denotes that there exists a constant c>0 such that ax≤cbx for sufficiently large x, and I· denotes an indicator function.
To prove our results, we need the following two lemmas.Lemma 1 (see [19]).
LetX1,…,Xn be a sequence of WA random variables in Ω,ℋ,E^, with E^Xk≤0 for 1<i≤n. Then, for all x>0,d>0, we have(15)VSn≥x≤Vmax1≤i≤nXi>d+gnexpxd−xdln1+xd∑i=1nE^Xi2.Proof.
We can easily get (15) from Theorem 1 of [19].Lemma 2.
SupposeX∈H, p>0, for any constant c>0:(i)
Then,(16)CVXp<∞⇔∑n=1∞np−1VX>cn<∞.Especially, whenp=1, we have(17)CVX<∞⇔∑n=1∞VX>cn<∞.(ii)
IfCVXp<∞, then, for any a>1,(18)∑k=1∞akVX>cak/p<∞.Proof.
For (i), similar to the proof of Lemma 3 in [9] and assuming x1/p=y, then (16) is obtained. For (ii), similar to the proof Theorem 6 in [20], we can get, for any a>1, (18) is satisfied.


## 3. The Main Results and Their Proofs

(A.1)
Letgx be a nondecreasing positive function on 0,∞, gx=gn, when x=n. And, gx/xτ↓ for some 0<τ<1(A.2)
There exists a nondecreasing positive functionhx on 0,∞, such that hx/x↓ and ∑n=1∞gn/nhγμn<∞ for some γ>0, 0<μ<1Theorem 1.
LetXn;n≥1 be a sequence of WA and identically distributed random variables under sublinear expectations. gx satisfies (A.1) or (A.2), and p≥1. When p=1, assume that (A.1) holds and(19)E^X1+δ≤CVX1+δ<∞,for some δ>0, or (A.2) holds and(20)E^XhX≤CVXhX<∞.
Whenp>1, assume that(21)E^Xp≤CVXp<∞,and (A.1) or (A.2) holds. Let ani;1≤i≤n,n≥1 be an array of real positive numbers satisfying(22)max1≤i≤nani=On−1.
Then, for allε>0,(23)∑n=1∞np−2V∑i=1naniXi−E^Xi>ε<∞,(24)∑n=1∞np−2V∑i=1naniXi−ε^Xi<−ε<∞.
In particular, ifE^Xi=ε^Xi, then(25)∑n=1∞np−2V∑i=1naniXi−ε^Xi>ε<∞.Proof.
To simplify our proof, we may assume thatmax1≤i≤nani≤n−1; thus,(26)∑i=1nani2≤max1≤i≤nani2∑i=1n1≤n−1.
Whenp=1, it is obvious that CVX<∞ by CVX1+δ<∞ or CVXhX<∞. Whenp>1, CVX<∞ follows from CVXp<∞. Therefore, there exists a positive integer N such that(27)E^XIX>N<ε6.
For widely acceptable random variablesXn;n≥1, in order to ensure that the truncated random variables are also widely acceptable, truncated functions should belong to Cl,Lip and should be nondecreasing. Denote for 1≤i≤n that(28)Xi′=NIXi>N+XiIXi≤N−NIXi<−N,Xi″=Xi−Xi′=Xi−NIXi>N+Xi+NIXi<−N.
It is easily checked that(29)∑n=1∞np−2V∑i=1naniXi−E^Xi>ε=∑n=1∞np−2V∑i=1naniXi′−E^Xi′+∑i=1naniXi′′+∑i=1naniE^Xi′−E^Xi>ε≤∑n=1∞np−2V∑i=1naniXi′−E^Xi′>ε3+∑n=1∞np−2V∑i=1naniXi″>ε3+∑n=1∞np−2V∑i=1naniE^Xi′−E^Xi>ε3≕I1+I2+I3.
Thus, to prove the desired result (23), we only need to show I1<∞, I2<∞, and I3<∞.
LetTi,1≤i≤n,n≥1 be a sequence of any random variable, and we all have(30)Vmax1≤i≤nTi>d≤∑i=1nVTi>d≤∑i=1n1dqE^Tiq.
For eachn≥1, Xi′,1≤i≤n is a sequence of WA, Xi′−E^Xi′ satisfies the conditions of Lemma 1. Let x=ε/3 and d=ε/6p+γ in Lemma 1; taking q>p and (30), we obtain(31)I1≤∑n=1∞np−2∑i=1n1dqE^aniXi′−E^Xi′q+gnexpε3d−ε3dln1+εd3∑i=1nE^aniXi′−E^Xi′2≕I11+I12.
ForI11, it follows from the definition of Xi′,i≥1 that E^Xi′≤N. By Cr inequality and (22), we obtain(32)I11≪∑n=1∞np−2∑i=1n1dqaniqE^Xi′q≤c∑n=1∞np−2∑i=1naniq≤c∑n=1∞np−2max1≤i≤naniq∑i=1n1≤c∑n=1∞np−q−1<∞.
ForI12, we get, by (26), Cr inequality, τ<1<1+γ, and (A.1) or (A.2) holds, such that(33)I12≤c∑n=1∞np−2gn∑i=1nE^aniXi′−E^Xi′2ε/3d≤c∑n=1∞np−2gn∑i=1nani2E^Xi′2ε/3d≤c∑n=1∞np−2gn∑i=1nani2ε/3d≤c∑n=1∞np−2+τ∑i=1nani2ε/3d,ifA.1holds,c∑n=1∞np−2−ε/3dgnhγnnγ+1nhγnnγ,ifA.2holds,≤c∑n=1∞np−2+τ−ε/3d,ifA.1holds,c∑n=1∞np−1+γ−ε/3d,ifA.2holds,=c∑i=1∞np−2+τ−ε/3ε/6p+γ,ifA.1holds,c∑i=1∞np−1+γ−ε/3ε/6p+γ,ifA.2holds,≤∑n=1∞n−1−p−γ<∞.
Thus, we have provedI1<∞.
Now, we deal withI2. For n>N, denote for 1≤i≤n that(34)Yni=Xi−NIN<Xi≤n+n−NIXi>n+Xi+NI−n<Xi≤−N+−n+NIXi<−n.
Then,(35)I2=∑n=1∞np−2V∑i=1naniXi″>ε3≤∑n=1∞np−2∑i=1nVXi>n+∑n=1∞np−2∑i=1nV∑i=1naniYni>ε3≕I21+I22.
For theμ of condition (A.2), let zx∈Cl,Lipℝ, 0≤zx≤1, for all x, and zx=1 if x≤μ, zx=0 if x>1, and zx are nonincreasing for any x>0. Then,(36)Ix≤μ≤zx≤Ix≤1,Ix>1≤1−zx≤Ix>μ.
ForI21, we can get by (36); then,(37)I21≤∑n=1∞np−2∑i=1nE^1−zXin=∑n=1∞np−2∑i=1nE^1−zXn≤∑n=1∞np−1VX>μn.
Whenp=1, we can get I21≤∑n=1∞VX>μn<∞ by (17). When p>1, we can get I21<∞ by (16). Next, we prove I22<∞. Obviously,(38)I22=∑n=1∞np−2V∑i=1naniYni>ε3≤∑n=1∞np−2V∑i=1naniYni−E^Yni>ε6+∑n=1∞np−2V∑i=1naniE^Yni>ε6.
And, it follows from (22) and (27); when n>N, we have(39)∑i=1naniE^Yni≤∑i=1naniE^XiIXi>N≤max1≤i≤nani∑i=1nE^XIX>N≤E^XIX>N<ε6.
So, we obtainV∑i=1naniE^Yni>ε/6=0, when n>N. It follows that ∑n=1∞np−2V∑i=1naniE^Yni>ε/6<∞. Thus, to estimate I22, it suffices to show ∑n=1∞np−2V∑i=1naniYni−E^Yni>ε/6<∞. Next, we consider the following two cases.Case 1.
p>1.
Yni,i≥1,n>N also is a sequence of WA random variables, andYni−E^Yni satisfies the conditions of Lemma 1. Applying Lemma 1 with x=ε/6, d>0, and (30), taking q>p, we have(40)I22≪∑n=1∞np−2V∑i=1naniYni−E^Yni>ε6≤∑n=1∞np−2∑i=1n1dqE^aniYni−E^Yniq+gnexpε6d−ε6dln1+εd6∑i=1nE^aniYni−E^Yni2≕I221+I222.
Firstly, we proveI221<∞. For any r>0, by the Cr inequality and (36), we have(41)Ynir≤XirIXi≤n+nrIXi>n≤XirzμXin+nr1−zXin.
Thus,(42)E^Ynir≤E^XirzμXin+nrE^1−zXin=E^XrzμXn+nrE^1−zXn≤E^XrzμXn+nrVX>μn.
ForI221, it follows from (42) and (22) that(43)I221≤c∑n=1∞np−2∑i=1naniqE^Yniq≤c∑n=1∞np−2∑i=1naniqE^XqzμXn+nqVX>μn≤c∑n=1∞np−2max1≤i≤naniq∑i=1n1E^XqzμXn+nqVX>μn≤c∑n=1∞np−1−qE^XqzμXn+nqVX>μn≪∑n=1∞np−1−qE^XqzμXn+∑n=1∞np−1VX>μn≕I221′+I221″.
Whenp>1, we can get I221″<∞ by (16). Next, we prove I221′<∞.
Letzjx be an even function and zjx∈Cl,Lipℝ, j≥1 such that 0≤zjx≤1 or all x and zjx/2j=1 if 2j−1<x≤2j and zjx/2j=0 if x≤μ2j−1 or x>1+μ2j. Then,(44)zjX2j≤Iμ2j−1<X<1+μ2j,XrzX2k≤1+∑j=1kXrzjX2j,for∀r>0.
For everyn, there exists k such that 2k−1≤n<2k; thus, by (8) and (44) and from zx which is nonincreasing for any x>0, we obtain(45)I221′=∑k=1∞∑2k−1≤n<2k2kp−1−qE^XqzμX2k≪∑k=1∞2kp−qE^XqzμX2k≪∑k=1∞2kp−q∑j=1kE^XqzjμX2j≪∑k=1∞2kp−q∑j=1k2qjE^zjμX2j=∑j=1∞2qjE^zjμX2j∑k=j∞2kp−q≪∑j=1∞2pjE^zjμX2j≤∑j=1∞2pjVX>μ2j−1=∑j=1∞2pjVX>μ2pj/p<∞.
It follows thatI221<∞. For I222, we consider the following two cases.Case 2.
p≥2.
By (8), we have E^Yni2≤CVYni2≤CVX2<∞, τ<1<1+γ, and (26), and taking d=ε/12p+γ and (A.1) or (A.2) hold that(46)I222≤c∑n=1∞np−2gnexp−ε6dlnεdc∑i=1nani2≤c∑n=1∞np−2gnexp−ε6dlnεdcn−1≤c∑n=1∞np−2gnn−ε/6d≤c∑n=1∞np−2+τ−ε/6d,ifA.1holds,c∑n=1∞np−2−ε/6dgnhγnnγ+1nhγnnγ,ifA.2holds,≤c∑n=1∞np−2+τ−ε/6d,ifA.1holds,c∑n=1∞np−1−ε/6d+γ,ifA.2holds,=c∑n=1∞np−2+τ−ε/6ε/12p+γ,ifA.1holds,c∑n=1∞np−1+γ−ε/6ε/12p+γ,ifA.2holds,≪∑n=1∞n−1−p−γ<∞.Case 3.
1<p<2.
Similar to Case2, taking d=ε/121+γ+1/p−1 and (A.1) or (A.2) hold, we have(47)I222≤c∑n=1∞np−2gnexp−ε6dlnεd6∑i=1nani2E^YnipYni2−p≤c∑n=1∞np−2gnexp−ε6dlnεdcn2−p∑i=1nani2≤c∑n=1∞np−2gnexp−ε6dlnεdcn1−p≤c∑n=1∞np−2gnn−ε/6dp−1≤c∑n=1∞np−2+τ−ε/6dp−1,ifA.1holds,c∑n=1∞np−2−ε/6dp−1gnhγnnγ+1nhγnnγ,ifA.2holds,≤c∑n=1∞np−2+τ−ε/6dp−1,ifA.1holds,c∑n=1∞np−1−ε/6dp−1+γ,ifA.2holds,=c∑n=1∞np−2+τ−ε/6ε/121+γ+1/p−1p−1,ifA.1holds,c∑n=1∞np−1+γ−ε/6ε/121+γ+1/p−1p−1,ifA.2holds,≪∑n=1∞n−1−p−γ<∞.
So, whenp>1, we have I22<∞.Case 4.
p=1.
Letx=ε/6 and d=εδ/121+γ in Lemma 1; taking q>p and (30), we still can obtain (40). And, similar to Case 1, we also get I221<∞. Next, we discuss the situation of (A.1) or (A.2) holds.
If (A.1) holds, according to (19), we can get E^Yni1+δ≤CVX1+δ<∞. Then,(48)I22≪∑n=1∞n−1gnexpε6d−ε6dln1+εd6∑i=1nE^aniYni−E^Yni2≤c∑n=1∞n−1gnexp−ε6dlnεdc∑i=1nani2E^Yni2≤c∑n=1∞n−1gnexp−ε6dlnεdc∑i=1nani2E^Yni1+δYni1−δ≤c∑n=1∞n−1gnexp−ε6dlnεdcn1−δ∑i=1nani2≤c∑n=1∞n−1gnexp−ε6dlnεdcn−δ≤c∑n=1∞n−1gnn−ε/6dδ≤c∑n=1∞n−1+τn−ε/6dδ≤c∑n=1∞n−1+τ−ε/6dδ=c∑i=1∞n−1+τ−ε/6εδ/121+γδ≤c∑n=1∞n−2−γ<∞.
If (A.2) holds, by (42) and Cr inequality, we have(49)I22≤c∑n=1∞n−1gnexp−ε6dlnεdc∑i=1nani2E^Yni2≪∑n=1∞n−1gn∑i=1nani2E^Yni2ε/6d≤∑n=1∞n−1−ε/6dgnE^X2zμXn+n2E^1−zXnε/6d≤∑n=1∞n−1−ε/6dgnE^X2zμXnε/6d+∑n=1∞n−1−ε/6dgnn2E^1−zXnε/6d≕I4+I5.
Then, in order to proveI22<∞, it suffices to show I4<∞ and I5<∞.
By (36), we can obtain zμX/n≤IX≤n/μ. Since hx↑ and hx/x↓, we can get xhx↑ and x/hx↑. Combine ∑n=1∞gn/nhγμn<∞ and (20), and taking d=ε/6γ, we can obtain(50)I4=∑n=1∞n−1−γgnE^XhXXhXzμXnγ≤∑n=1∞n−1−γgnn/μhn/μE^XhXγ≪∑n=1∞gnnhγμnE^XhXγ<∞.
And, by (36), we also can get 1−zX/n≤IX≥μn. And, combine x/hx↑; then,(51)I5=∑n=1∞n−1−γgnn2E^XhXXhX1−zXnγ≤∑n=1∞n−1−γgnn2E^XhXμnhμnγ≪∑n=1∞n−1−γgnnE^XhXhμnγ=∑n=1∞n−1gnE^XhXhμnγ=∑n=1∞gnnhγμnE^XhXγ<∞.
So, we have provedI22<∞.
Next, we proveI3<∞. From (3), (22), and (27), we have(52)∑i=1naniE^Xi′−E^Xi≤∑n=1naniE^Xi′−Xi≤max1≤i≤nani∑n=1nE^XIX>N≤E^XIX>N<ε6.
So, we can obtainV∑i=1naniE^Xi′−E^Xi>ε/3=0, when n>N. It follows that I3=∑n=1∞np−2V∑i=1naniE^Xi′−E^Xi>ε/3<∞. Then, (23) holds.
Obviously,−Xn,n≥1 also satisfies the conditions of Theorem 1. Considering −Xn,n≥1 instead of Xn,n≥1 in (23), we can obtain (24).
In particular, ifE^Xi=ε^Xi, then (25) follows from (23), and(53)V∑i=1naniXi−ε^Xi>ε≤V∑i=1naniXi−ε^Xi>ε+V∑i=1naniXi−ε^Xi<−ε.
Together with (23)–(25) holds. The proof of Theorem 1 is completed.
We give an example satisfying the conditions of Theorem1 and obtain corresponding results.Example 1.
LetXn;n≥1 be a sequence of WA and identically distributed random variables with dominating coefficient gn=n1/2 under sublinear expectations and E^ is countably subadditive. Assume ani=in−1 and VX>x∼1/x1+δ1 for some δ1>0. Then, for all ε>0, it will show that(54)∑n=1∞n−1V∑i=1nin−1Xi−E^Xi>ε<∞,∑n=1∞n−1V∑i=1nin−1Xi−ε^Xi<−ε<∞.
IfE^Xi=ε^Xi, then(55)∑n=1∞n−1V∑i=1nin−1Xi−E^Xi>ε<∞.
Obviously,gx=x1/2 satisfies (A.1). ∃0<δ<δ1, we have CVX1+δ∼∫1∞VX1+δ>xdx∼∫1∞1/x1+δ1/1+δdx<∞. Furthermore, E^ is countably subadditive, and by [6], then we have E^X1+δ≤CVX1+δ<∞. Therefore, all the conditions of Theorem 1 are satisfied; then, Theorem 1 holds.



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