# Precise Asymptotics for Complete Integral Convergence under Sublinear Expectations

Authors: Qunying Wu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The aim of this paper is to study and establish the precise asymptotics for complete integral convergence theorems under a sublinear expectation space. As applications, the precise asymptotics forp 0≤p≤2 order complete integral convergence theorems have been generalized to the sublinear expectation space context. We extend some precise asymptotics for complete moment convergence theorems from the traditional probability space to the sublinear expectation space. Our results generalize corresponding results obtained by Liu and Lin (2006). There is no report on the precise asymptotics under sublinear expectation, and we provide the method to study this subject.


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## Body

## 1. Introduction

The sublinear expectation space has advantages of modelling the uncertainty of probability and distribution. Therefore, limit theorems for sublinear expectations have raised a large number of issues of interest recently. Limit theorems are important research topics in probability and statistics. They were widely used in finance and other fields. Classical limit theorems only hold in the case of model certainty. However, in practice, such model certainty assumption is not realistic in many areas of applications because the uncertainty phenomena cannot be modeled using model certainty. Motivated by modelling uncertainty in practice, Peng [1] introduced a new notion of sublinear expectation. As an alternative to the traditional probability/expectation, capacity/sublinear expectation has been studied in many fields such as statistics, finance, economics, and measures of risk (see Denis and Martini [2]; Gilboa [3]; Marinacci [4]; Peng [1, 5–7], etc.). The general framework of the sublinear expectation in a general function space was introduced by Peng [1, 7, 8], and sublinear expectation is a natural extension of the classical linear expectation.Because the sublinear expectation provides a very flexible framework to model sublinear probability problems, the limit theorems of the sublinear expectation have received more and more attention and research recently. A series of useful results have been established. Peng [1, 7, 8] constructed the basic framework, basic properties, and central limit theorem under sublinear expectations, Zhang [9–11] established the exponential inequalities, Rosenthal’s inequalities, strong law of large numbers, and law of iterated logarithm, Hu [12], Chen [13], and Wu and Jiang [14] studied strong law of large numbers, Wu et al. [15] studied the asymptotic approximation of inverse moment, Xi et al. [16] and Lin and Feng [17] studied complete convergence, and so on. In general, extending the limit properties of conventional probability space to the cases of sublinear expectation is highly desirable and of considerably significance in the theory and application. Because sublinear expectation and capacity is not additive, many powerful tools and common methods for linear expectations and probabilities are no longer valid, so that the study of the limit theorems under sublinear expectation becomes much more complex and difficult.Since the concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [18], there have been extensions in several directions. One of them is to discuss the precise rate, which is more exact than complete convergence. Precise asymptotics for complete convergence and complete moment convergence is one of the most important problems in probability theory. Many related results have been obtained in the probabilistic space. Their recent results can be found in the studies of Heyde [19]; Liu and Lin [20]; Zhao [21]; Li [22]; Zhou [23]; Gut and Steinebach [24]; He and Xie [25]; Wang et al. [26–29]; Spaataru [30]; and Kong and Dai [31]. However, in sublinear expectations, due to the uncertainty of expectation and capacity, the precise asymptotics is essentially different from the ordinary probability space. The study of precise asymptotics of complete convergence and complete integral convergence for sublinear expectations is much more complex and difficult. The precise asymptotics theorems under sublinear expectation have not been reported. The purpose of this paper is to establish the precise asymptotics theorems for p (0≤p≤2) order complete integral convergence for independent and identically distributed random variables under sublinear expectation. As a result, the corresponding results obtained by Liu and Lin [20] have been generalized to the sublinear expectation space context.In the next section, we summarize some basic notations and concepts and related properties under the sublinear expectations.


## 2. Preliminaries

We use the framework and notations of Peng [8]. 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,…,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.
A sublinear expectationE^ on ℋ is a function E^:ℋ⟶−∞,∞ satisfying the following properties: for all X,Y∈ℋ,(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
The tripleΩ,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. LetG⊂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 to be 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,ν of capacities by(5)VA≔infE^ξ;IA≤ξ,ξ∈ℋ,νA≔1−VAc,∀A∈ℱ,where Ac is the complement set of A. By definition of V and ν, it is obvious that V is subadditive, and(6)νA≤VA,∀A∈ℱ;VA=E^IA,νA=ε^IA,ifIA∈ℋ,(7)E^f≤VA≤E^g,ε^f≤νA≤ε^g,iff≤IA≤g,f,g∈ℋ.
This implies Markov inequality:∀X∈ℋ,(8)VX≥x≤E^Xpxp,∀x>0,p>0,from IX≥x≤Xp/xp∈ℋ. By Lemma 4.1 in Zhang [10]; we have Holder inequality: ∀X,Y∈ℋ,p,q>1 satisfying p−1+q−1=1,(9)E^XY≤E^Xp1/pE^Yq1/q.And particularly, we have Jensen inequality: ∀X∈ℋ,(10)E^Xr1/r≤E^Xs1/s,for0<r≤s.
Also, we define the Choquet integrals/expectationsCV,Cν by(11)CVX≔∫0∞VX>xdx+∫−∞0VX>x−1dx,with V being replaced by V and ν, respectively.Definition 2.
(Peng [1] and Zhang [9]).(i)
Identical distribution: letX1 and X2 be two n-dimensional random vectors defined, respectively, in sublinear expectation spaces Ω1,ℋ1,E^1 and Ω2,ℋ2,E^2. They are called identically distributed if(12)E^1φX1=E^2φX2,∀φ∈Cl,Lipℝn,whenever the subexpectations are finite. A sequenceXn;n≥1 of random variables is said to be identically distributed if for each i≥1, Xi and X1 are identically distributed.(ii)
Independence: in a sublinear expectation spaceΩ,ℋ,E^, a random vector Y=Y1,…,Yn, Yi∈ℋ, is said to be independent of another random vector X=X1,…,Xm,Xi∈ℋ, under E^ if for each test function φ∈Cl,Lipℝm×ℝn, we have E^φX,Y=E^E^φx,Y|x=X, whenever φ¯x≔E^φx,Y<∞ for all x and E^φ¯X<∞.(iii)
Independent random variables: a sequence of random variablesXn;n≥1 is said to be independent, if Xi+1 is independent of X1,…,Xi for each i≥1.
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 denote that there exists a constant c>0 such that ax≤cbx for sufficiently large x, and I⋅ denote an indicator function.
To prove our results, we need the following four lemmas.Lemma 1 (Theorem 3.1 in Zhang [10]).
LetXk;k≥1 be a sequence of independent random variables in Ω,ℋ,E^ with E^Xk≤0. Then,(i)
For anyx,y>0,(13)VSn≥x≤Vmax1≤k≤nXk>y+exp−x22xy+Bn1+23ln1+xyBn.(ii)
IfE^Xk=E^−Xk=0, then(14)VSn≥x≤cBnx2,where Bn=∑k=1nE^Xk2.Lemma 2.
For anyX∈ℋ, we have(15)CVX2<∞⟺∫1∞xVX>xdx<∞,(16)CVX2lnX<∞⟺∫3∞xlnxVX>xdx<∞.Proof.
We only prove (16). Let fx≔x2lnx and f−1⋅ denote the inverse function of f⋅. Then, (16) follows from the following three equations:(17)∫f3∞VX2lnX>xdx≤CVX2lnX≤f3+∫f3∞VX2lnX>xdx,∫f3∞VX2lnX>xdx=∫f3∞VX>f−1xdx=∫3∞2ylny+yVX>ydyletf−1x=y,2ylny≤2ylny+y≤3ylny,fory≥3.
Therefore, (16) holds.
Here, we give the notations of G-normal distribution which is introduced by Peng [7].Definition 3.
(G-normal random variable). For0≤σ¯2≤σ¯2<∞, a random variable ξ in a sublinear expectation space Ω,ℋ,E^ is called a normal N0,σ¯2,σ¯2 distributed random variable (write ξ∼N0,σ¯2,σ¯2 under E^); if for any φ∈Cl,Lipℝ, the function ux,t=E^φx+tξ (x∈ℝ,t≥0) is the unique viscosity solution of the following heat equation:(18)∂tu−G∂xx2u=0,u0,x=φx,where Gα=σ¯2α+−σ¯2α−/2.Lemma 3 (Theorem 3.3 and Remark 3.4 in Peng [7] (CLT)).
Suppose thatXn;n≥1 is a sequence of independent and identically distributed random variables with E^X1=E^−X1=0. Write σ¯2=E^X12 and σ¯2=ε^X12. Then, for any continuous function φ satisfying φx≤c1+x,(19)limn⟶∞E^φSnn=E^φξ,where ξ∼N0,σ¯2,σ¯2 under E^.In particular, ifσ¯=σ¯, then Lemma 3 becomes a classical central limit theorem.Remark 1.
For anyx>0, by E^xX12=x2σ¯2, ε^xX12=x2σ¯2, and xξ∼N0,x2σ¯2,x2σ¯2 under E^, (19) becomes(20)limn⟶∞E^φxSnn=E^φxξ.Lemma 4 (Lemma 3 in Chen and Hu [32]).
Suppose thatξ∼N0,σ¯2,σ¯2 under E^. Let P be a probability measure and φ be a bounded continuous function on ℝ. If Btt≥0 is a Brownian motion under P, then(21)E^φξ=supθ∈ΘEPφ∫01θsdBs,where(22)Θ=θ;θtisℱt−adapted process such thatσ¯≤θt≤σ¯,ℱt=σBs;0≤s≤t∨N,Nis the collection ofP−null subsets.From Peng [8], if ξ∼N0,σ¯2,σ¯2 under E^, then for each convex function φ,(23)E^φξ=12π∫−∞∞φσ¯xe−x2/2dx,but if φ is a concave function, the above σ¯ must be replaced by σ¯. If σ=σ¯=σ¯, then N0,σ¯2,σ¯2=N0,σ2 which is a classical normal distribution.In particular, notice thatφx=xp,p≥1, is a convex function; taking φx=xp,p≥1, in (23), we get(24)E^ξp=2σ¯p2π∫0∞xpe−x2/2dx<∞.(24) implies that(25)CVξp=∫0∞Vξp>xdx≤1+∫1∞E^ξ2px2dx<∞,for anyp≥12.Definition 4.
A sublinear expectationE^ is called to be continuous if it satisfiesContinuity from below:E^Xn↑E^X if 0≤Xn↑X, where Xn,X∈ℋContinuity from above:E^Xn↓E^X if 0≤Xn↓X, where Xn,X∈ℋLemma 5.
Suppose that the conditions of Lemma3 hold and E^ is continuous, set Δnx≔VSn/n≥x−Vξ≥x, here and later, ξ∼N0,σ¯2,σ¯2 under E^; then,(26)Δn≔supx≥0Δnx⟶0,asn⟶∞.Remark 2.
Lemma5 is a powerful tool for studying the uniform convergence of the central limit theorem under sublinear expectations, which plays a key role in proving the theorems in this paper.Proof of Lemma5.
Ifσ¯=σ¯, then Lemma 3 is a classical central limit theorem. In the classical probability, (26) follows from the central limit theorem and an important fact that P|ξ|≥x is a continuous function of x. Therefore, we only need to prove the situation σ¯<σ¯.
Obviously,Δn0=1−1=0; thus, Δn=supx>0Δnx.
For0<μ<1, let φx be a Lipschitz even function and nondecreasing for x≥0 such that 0≤φx≤1, for all x and φx=0 if x|≤μ and φx=1 if x>1. Then,(27)Ix≥1≤φx≤Ix≥μ.
This combines equation (7), for x>0,(28)Δnx≤E^φSnnx−E^φμξx=E^φSnnx−E^φξx+E^φξx−E^φμξx≔Δn1x+Δ2x≤supx>0Δn1x+supx>0Δ2x,where Δn1x=E^φSn/nx−E^φξ/x and Δ2x=E^φξ/x−E^φμξ/x≥0.
On the other hand,(29)Δnx≥E^φμSnnx−E^φξx=E^φμSnnx−E^φμξx+E^φμξx−E^φξx=Δn1xμ−Δ2x≥−supx>0Δn1x−supx>0Δ2x.
Thus,(30)Δn=supx>0Δnx≤supx>0Δn1x+supx>0Δ2x.
Therefore, in order to prove that (26), it suffices to show that(31)limn⟶∞supx>0Δn1x=0,(32)limμ⟶1−supx>0Δ2x=0.
WriteFnx=E^φSn/nx and Fx=E^φξ/xx>01x=0.
Obviously,0≤FnxandFx≤1; Fnx and Fx are nonincreasing functions on 0,+∞. Thus, for any x0>0, the limit limx⟶x0Fx exists. Actually, taking xn↑x0 and xn′↓x0, by continuity of E^, we have(33)limx⟶x0−Fx=limxn↑x0E^φξxn=E^φξx0=Fx0from0≤φξxn↓φξx0,limx⟶x0+Fx=limxn′↓x0E^φξxn′=E^φξx0=Fx0from0≤φξxn′↑φξx0,limx⟶0+Fx=limn⟶∞E^φnξ=1=F0from0≤φnξ↑1.
Hence,Fx is continuous for 0≤x<∞. As well as limx⟶∞Fx=limn⟶∞E^φξ/n=0 from 0≤φξ/n↓0.
Therefore, letϵ be an arbitrary positive number; there exist points 0<x1<x2,…,xm<∞ satisfying the conditions(34)1−Fx1<ε2,Fxk−Fxk+1<ε2,k=1,…,m−1,Fxm<ε2.
Furthermore, by (19) and Remark 1, there exists a number n0 such that for n>n0 and we have(35)Fnxk−Fxk<ε2,k=1,…,m.
Ifxk≤x<xk+1k=1,…,m−1, then for n>n0, we get(36)Fnx−Fx≤Fnxk−Fxk+Fxk−Fxk+1<ε,Fnx−Fx≥Fnxk+1−Fxk+1+Fxk+1−Fxk>−ε.
If0<x<x1, then for n>n0,(37)Fnx−Fx≤1−Fx1<ε,Fnx−Fx≥Fnx1−1=Fnx1−Fx1+Fx1−1>−ε.
Ifx≥xm, then for n>n0,(38)Fnx−Fx≤Fnxm=Fnxm−Fxm+Fxm<ε,Fnx−Fx≥0−Fxm>−ε.
Thus,Fnx−Fx<ε for all x and n>n0. That is, (31) holds.
Next, we prove that (32).
BecauseF is continuous on 0,∞, F is uniformly continuous on 0,2. Therefore, for any ϵ>0, there is δ>0 (can be assumed δ<2), such that ∀x1,x2∈0,2; if x1−x2<δ, then(39)Fx1−Fx2<ε.
Letmax1/2,1−δ/2,σ¯/σ¯<μ<1, for any x∈0,1; we have 0<x/μ≤2x≤2 and x/μ−x<δ. Hence,(40)Δ2x=E^φξx−E^φμξx=Fx−Fxμ<ε.
Thereby,(41)sup0<x≤1Δ2x≤ε.
Forx>1, let Θ=θ;θtisℱt−adapted process such thatσ¯/x≤θt≤σ¯/x and Θμ=θ;θtisℱt−adapted process such thatσ¯/x≤θt≤μσ¯/x; combining any a>0, ξ/a∼N0,σ¯2/a2,σ¯2/a2 under E^. By Lemma 4,(42)Δ2x=E^φξx−E^φμξx≤supθ∈ΘEPφ∫01θsdBs−supθ∈ΘμEPφ∫01θsdBs≤supθ∈ΘEPφ∫01θsdBs−supθ∈ΘEPφ∫01θs∧μσ¯xdBs≤supθ∈ΘEPφ∫01θsdBs−φ∫01θs∧μσ¯xdBs≤cφsupθ∈ΘEP∫01θsdBs−∫01θs∧μσ¯xdBs≤cφsupθ∈ΘEP∫01θs−θs∧μσ¯xdBs2=cφsupθ∈Θ∫01EPθs−θs∧μσ¯x2ds≤cφσ¯1−μx,where cφ is the Lipschitz constant of φ.
Therefore,(43)supx>1Δ2x≤cφσ¯1−μ⟶0,μ⟶1−.
The combination of (41) and (32) is established. This completes the proof of Lemma 5.


## 3. Main Results and Proofs

Our results are stated as follows.Theorem 1.
LetX,Xn;n≥1 be a sequence of independent and identically distributed random variables in Ω,ℋ,E^. We assume that E^ is continuous and(44)E^X=E^−X=0,limc⟶∞E^X12−c+=0,E^X2=σ¯2<∞,ε^X2=σ¯2,CVX2<∞.
Then,(45)limε⟶0ε2∑n=1∞VSn≥εn=CVξ2,here and later, ξ∼N0,σ¯2,σ¯2 under E^.Theorem 2.
Under the conditions of Theorem1, for 0≤p<2,(46)limε⟶0ε2−p∑n=1∞1npCVSnpISn≥εn=22−pCVξ2.Forp=2, we have the following theorem.Theorem 3.
LetX,Xn;n≥1 be a sequence of independent and identically distributed random variables in Ω,ℋ,E^. We assume that E^ is continuous and(47)E^X=E^−X=0,limc⟶∞E^X12−c+=0,E^X2=σ¯2<∞,ε^X2=σ¯2,CVX2lnX<∞.
Then,(48)limε⟶01lnε−1∑n=1∞1n2CVSn2ISn≥εn=2CVξ2.Remark 3.
Theorems1–3 extend the corresponding results obtained by Liu and Lin [20] from the probability space to sublinear expectation space.Proof of Theorem 1.
Note that(49)ε2∑n=1∞VSn≥εn=ε2∑n=1∞Vξ≥εn+ε2∑n=1∞VSn≥εn−Vξ≥εn≔I1ε+I2ε.
Hence, in order to establish (45), it suffices to prove that(50)limε⟶0I1ε=CVξ2,(51)limε⟶0I2ε=0.
Obviously, (50) follows from(52)limε⟶0I1ε=limε⟶0ε2∫1∞Vξ≥εxdx=limε⟶0∫ε2∞Vξ2≥ydy=∫0∞Vξ2≥ydy=CVξ2.
Without loss of generality, here and later, we assume thatE^X2=1. Let M>12,(53)I2ε≤ε2∑n≤Mε−2VSn≥εn−Vξ≥εn+ε2∑n>Mε−2VSn≥εn+ε2∑n>Mε−2Vξ≥εn≔I21ε+I22ε+I23ε.
Let△n=supx≥0VSn≥nx−Vξ≥x, from (26), △n⟶0 as n⟶∞. So, by Toeplitz’s lemma, if xn⟶x,ωi≥0, and ∑i=1nωi⟶∞, then ∑i=1nωixi/∑i=1nωi⟶x,(54)limε⟶0I21ε≤M,limε⟶0∑n≤Mε−2△nMε−2=0.
Takingφ as the proof process of Lemma 5, by (7), (27) and identically distributed of X,Xi, for any x>0,(55)VXi≥x≤E^φXix=E^φXx≤VX≥μx.
Hence, forn>Mε−2>12ε−2, taking x=εn and y=εn/12 in Lemma 1 (i),(56)VSn≥εn≤∑i=1nVXi≥εn12+exp−ε2n22ε2n2/12+n1+23ln1+ε2n12≤nVX≥μεn12+cε4n2,from ε2n2/2ε2n2/12+n1+2/3ln1+ε2n/12≥2ln1+ε2n/12.
Since−X,−Xi also satisfies the conditions of Theorem 1, we replace the X,Xi with the −X,−Xi in the upper form:(57)V−Sn≥εn≪nVX≥μεn12+1ε4n2.
Therefore,(58)VSn≥εn≤VSn≥εn+V−Sn≥εn≪nVX≥μεn12+1ε4n2.
More generally, for anyx>0andn>Mε−2, we have(59)VSn≥ε+xn≪nVX≥με+xn12+1ε+x4n2.
This implies from Markov’s inequality and (24) that(60)I22ε+I23ε≪ε2∑n>Mε−2nVX≥μεn12+1ε4n2+E^ξ4ε4n2≪ε2∫ε−2∞xVX≥cεxdx+M−1letcεx=y∼c∫cε−1∞yVX≥ydy+M−1.
Letε⟶0 first; then, let M⟶∞; we get(61)limε⟶0I22ε+I23ε=0,from (44) and (15).
From this, combining with (53) and (54), (51) is established. This completes the proof of Theorem 1.Proof of Theorem 2.
SinceCVSnpISn≥εn=VSn≥εn when p=0, so by Theorem 1, we only discuss the case 0<p<2. Note that(62)ε2−p∑n=1∞1npCVSnpISn≥εn=ε2∑n=1∞VSn≥εn+ε2−p∑n=1∞1np∫εn∞pxp−1VSn≥xdx.
Hence, from Theorem1, in order to establish (46), it suffices to prove that(63)limε⟶0ε2−p∑n=1∞1np∫εn∞pxp−1VSn≥xdx=p2−pCVξ2.
LetM≥12. Note that(64)ε2−p∑n=1∞1np∫εn∞pxp−1VSn≥xdx=ε2−p∑n=1∞1np∫εn∞pxp−1Vξ≥xndx+ε2−p∑n≤Mε−21np∫εn∞pxp−1VSn≥x−Vξ≥xndx+ε2−p∑n>Mε−21np∫εn∞pxp−1VSn≥x−Vξ≥xndx≔J1ε+J2ε+J3ε.
Hence, in order to establish (63), it suffices to prove that(65)limε⟶0J1ε=p2−pCVξ2,(66)limε⟶0J2ε=0,(67)limM⟶∞limsupε⟶0J3ε=0.
We first prove (65); let y≔x/n, then(68)limε⟶0J1ε=limε⟶0ε2−p∑n=1∞1np/2∫εn∞pyp−1Vξ≥ydy=limε⟶0ε2−p∫1∞1tp/2dt∫εt∞pyp−1Vξ≥ydy=limε⟶0ε2−p∫ε∞pyp−1Vξ≥ydy∫1y2/ε21tp/2dt=limε⟶02p2−pε2−p∫ε∞yp−1Vξ≥yy2ε2−p/2+1−1dy=2p2−p∫0∞yVξ≥ydy−limε⟶02p2−pε2−p∫ε∞yp−1Vξ≥ydy=p2−pCVξ2,from (25).
Now, we prove (66). Let bn=n△n1/2p−1. Then,(69)J2ε≤ε2−p∑n≤Mε−21np∫εn∞pxp−1VSn≥x−Vξ≥xndx≤ε2−p∑n≤Mε−2∫0∞py+εp−1VSn≥y+εn−Vξ≥y+εndy≤ε2−p∑n≤Mε−2∫0bnpy+εp−1△ndy+ε2−p∑n≤Mε−2∫bn∞py+εp−1VSn≥y+εn+Vξ≥y+εndy≔ε2−p∑n≤Mε−21np/2J21ε+J22ε,where(70)J21ε=np/2∫0bnpy+εp−1△ndy,J22ε=np/2∫bn∞py+εp−1VSn≥y+εn+Vξ≥y+εndy.
Sincen≤Mε−2 implies εn≤M, one can easily obtain that(71)J21ε≤Δnnp/21n△n1/2p+εp≤Δn1/2p+MΔn1/pp≪Δn1/2⟶0asn⟶∞.
By Lemma1 (ii), Markov’s inequality, and (24), we get(72)J22ε≤cnp/2∫bn∞y+εp−1nE^X2y+ε2n2+E^ξ2y+ε2ndy≪np/2−1∫bn∞1y+ε3−pdy=cnp/2−11n△n1/2p+εp−2≪Δn1/p−1/2⟶0asn⟶∞.
From (69)–(72) and ∑n≤Mε−21/np/2=Oεp−2⟶∞,ε⟶0, using Toeplitz’s lemma, we get(73)J2ε≤ε2−p∑n≤Mε−2J21ε+J22εnp/2≪∑n≤Mε−2△n1/2+△n1/p−1/2/np/2∑n≤Mε−21/np/2⟶0,ε⟶0.
That is, (66) is established.
Finally, we prove (67):(74)J3ε≤ε2−p∑n>Mε−21np∫εn∞pxp−1VSn≥x+Vξ≥xndxlety=xn−ε≤ε2−p∑n>Mε−2∫0∞py+εp−1VSn|≥y+εn+Vξ≥y+εndy.
By (59), Markov’s inequality, and (24),(75)J3ε≪ε2−p∑n>Mε−2∫0∞y+εp−1nVX≥cy+εn+1y+ε4n2+E^|ξ|4y+ε4n2dy≪ε2−p∫Mε−2∞xVX≥cy+εxdx∫0∞y+εp−1dy+ε2−p∑n>Mε−21n2∫0∞y+εp−5dy≪ε2−p∫0∞y+εp−1dy∫cMε−2y+ε∞ty+ε2VX≥tdt+M−1letcy+εx=t≪ε2−p∫0∞y+εp−3dy∫cε−1∞tVX≥tdt+M−1=∫cε−1∞tVX≥tdt+M−1.
Letε⟶0 first, then let M⟶∞, we get (67) from (44) and (15). This completes the proof of Theorem 2.Proof of Theorem 3.
Note that(76)∑n=1∞1n2CVSn2ISn≥εn=ε2∑n=1∞VSn≥εn+∑n=1∞1n2∫εn∞2xVSn≥xdx.
Hence, by Theorem1, in order to establish (48), it suffices to prove that(77)limε⟶01lnε−1∑n=1∞1n2∫εn∞2xVSn≥xdx=2CVξ2.
However,(78)1lnε−1∑n=1∞1n2∫εn∞2xVSn≥xdx=1lnε−1∑n=1∞1n2∫εn∞2xVξ≥xndx+1lnε−1∑n≤Mε−21n2∫εn∞2xVSn≥x−Vξ≥xndx+1lnε−1∑n>Mε−21n2∫εn∞2xVSn≥x−Vξ≥xndx≔K1ε+K2ε+K3ε.
Hence, in order to establish (77), it suffices to prove that(79)limε⟶0K1ε=2CVξ2,(80)limε⟶0K2ε=0,(81)limM⟶∞limsupε⟶0K3ε=0.
We first prove (79). Because limy⟶0+ylny=0, there is c1>0, such that ylny≤c1 for any y∈0,1. Therefore, combining with Markov’s inequality and (24),(82)∫ε∞ylnyVξ≥ydy≤∫01ylnydy+∫1∞ylnyE^ξ3y3dy≤c1+E^ξ3∫1∞lnyy2dy=c<∞.
Thus, (79) follows(83)limε⟶0K1ε=limε⟶01lnε−1∑n=1∞1n∫εn∞2yVξ≥ydylety=xn=limε⟶01lnε−1∫1∞1tdt∫εt∞2yVξ≥ydyletu=εt=limε⟶01lnε−1∫ε∞2duu∫u∞2yVξ≥ydy=limε⟶01lnε−1∫ε∞4yVξ≥ydy∫εy1udu=limε⟶01lnε−1∫ε∞4ylnyVξ≥ydy+∫0∞4yVξ≥ydy=2CVξ2.
Now, we prove (80). Let dn=n△n1/4−1. Then,(84)K2ε≤1lnε−1∑n≤Mε−21n2∫εn∞2xVSn≥x−Vξ≥xndxlety+εn=x=1lnε−1∑n≤Mε−2∫0∞2y+εVSn≥y+εn−Vξ≥y+εndy≤1lnε−1∑n≤Mε−2∫0dn2y+ε△ndy+1lnε−1∑n≤Mε−2∫dn∞2y+εVSn≥y+εn+Vξ≥y+εndy≔1lnε−1∑n≤Mε−21nK21ε+K22ε,where(85)K21ε=n∫0dn2y+ε△ndy,K22ε=n∫dn∞2y+εVSn≥y+εn+Vξ≥y+εndy.
Sincen≤Mε−2 implies ϵn≤M, one can easily obtain that(86)K21ε≤Δnn1n△n1/4+ε2≤Δn1/4+MΔn1/22≪Δn1/2⟶0asn⟶∞.
By (59), Markov’s inequality, (24), and cndn+ε≥cn, we get(87)K22ε≤cn∫dn∞2y+εnVX≥cny+ε+1n2y+ε4+E^|ξ|4n2y+ε4dy=c∫cndn+ε∞tVX≥tdt+c∫dn∞1ny+ε3dyletcny+ε=t≪∫cn∞tVX≥tdt+1n1n△n1/4+ε−2⟶0asn⟶∞.
From (84)–(87) and ∑n≤Mε−21/n=Olnε−1⟶∞,ε⟶0, using Toeplitz’s lemma, (80) follows(88)K2ε≤1lnε−1∑n≤Mε−2△n1/2+K22εn≪∑n≤Mε−2△n1/2+K22ε/n∑n≤Mε−21/n⟶0,asε⟶0.
Finally, we prove (81) as follows:(89)K3ε≤1lnε−1∑n>Mε−21n2∫εn∞2xVSn≥x+Vξ≥xndxlety=xn−ε=1lnε−1∑n>Mε−2∫0∞2y+εVSn≥y+εn+Vξ≥y+εndy.
By (59), Markov’s inequality, and (24),(90)K3ε≪1lnε−1∑n>Mε−2∫0∞y+εnVX≥cy+εn+1y+ε4n2+E^|ξ|4y+ε4n2dy∼1lnε−1∫Mε−2∞∫0∞y+εxVX≥cy+εxdydx+clnε−1∑n>Mε−21n2∫0∞y+ε−3dy∼1lnε−1∫0∞y+ε∫cMε−2y+ε∞ty+ε2VX≥tdtdy+2cM−1lnε−1≪1lnε−1∫cε−1∞tVX≥tdt∫0ε2t/cM−ε1y+εdy+M−1=1lnε−1∫cε−1∞tVX≥tlnε2cM+lnt+lnε−1dt+M−1≤1lnε−1∫cε−1∞tlnVX≥tdt+∫cε−1∞tVX≥tdt+M−1.
Letε⟶0 first and then let M⟶∞; we get (81) from (47) and (16). This completes the proof of Theorem 3.



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*Source: 3145935-2020-05-05.xml*