# Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations

Authors: Mingzhou Xu; Kun Cheng
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

By an inequality of partial sum and uniform convergence of the central limit theorem under sublinear expectations, we establish precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations.


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## Body

## 1. Introduction

Motivated by the work of g-expectation of Peng [1], Peng [2, 3] initiated the concept of the sublinear expectation space, which is a powerful tool to model the uncertainty of probability and distribution. We could consider sublinear expectation as an extension of the classical linear expectation. Peng [2, 3] constructed the basic framework, investigated basic properties, and proved the law of large number and central limit theorem under sublinear expectations. Motivated by the seminal work of Peng [2, 3], more and more limit theorems under sublinear expectation space have been established, which generalize the corresponding fundamental, important limit theorems in probability and statistics. Zhang [4–6] proved the exponential inequalities and Rosenthal’s inequalities and obtained an extension of the central limit theorem and Donsker’s invariance principle under sublinear expectations. Wu [7] established precise asymptotics for complete integral convergence under sublinear expectations. Yu and Wu [8] studied Marcinkiewicz-type complete convergence for weighted sums under sublinear expectations. Wu and Jiang [9] obtained a strong law of large numbers and Chover’s law of the iterated logarithm under sublinear expectations. Ma and Wu [10] studied the limiting behavior of weighted sums of extended negatively dependent random variables under sublinear expectations. Xu and Zhang [11, 12] studied three series theorem for independent random variables and the law of logarithm for arrays of random variables under sublinear expectations. Chen [13] proved strong laws of large numbers for sublinear expectations. For more results about limit theorems under sublinear expectations, the interested reader could refer to the studies of Hu et al. [14], Fang et al. [15], Kuczmaszewska [16], Wang and Wu [17], Hu and Yang [18], Zhang [19], and references therein.Precise asymptotics in the law of the iterated logarithm is one of the fundamental problems in probability theory. Many related results have been derived in the probabilistic space. Their results can be found in the work of Gut and Spătaru [20]; Zhang [21]; Xiao et al. [22]; Huang et al. [23]; Jiang and Yang [24]; Wu and Wen [25]; Xu et al. [26]; Xu [27, 28]; and Xu [29]. However, in sublinear expectations, due to the uncertainty of sublinear expectation and related capacity, the precise asymptotics in the law of the iterated logarithm under sublinear expectations have not been reported. Motivated by the work of Wu [7], Xiao et al. [22], Xu et al. [26], and Xu [29], we try to investigate precise asymptotics in the law of the iterated logarithm under sublinear expectations. The aim of this paper is to prove the precise asymptotics in the law of the iterated logarithm for independent, identically distributed random variables under sublinear expectations. The main contribution of this paper is that we prove an useful inequality under sublinear expectations in Lemma 1, and we extend the results of Xiao et al. [22], Xu et al. [26], and Xu [29] to those of the sublinear expectation spaces. Our results may have the potential applications in finance or engineering fields (cf. Wu [7], Peng [3], Zhang [19], and references therein). Our basic idea in this paper comes from that of Wu [7], Xiao et al. [22], Xu et al. [26], Xu [29], Spătaru [30], and Fuk and Nagaev [31]. In conclusion, our results combined with the work of Wu [7] imply heuristically that many results about precise asymptotics in the law of the iterated logarithm in probability spaces may still hold under sublinear expectations.The rest of this paper is organized as follows: in Section2, we summarize necessary basic notions, concepts, and relevant properties and give necessary lemmas under sublinear expectations. In Section 3, we give our main results, Theorems 1 and 2, whose proofs are presented in Sections 4 and 5, respectively.


## 2. Preliminaries

We use notations similar to those of Peng [3]. Let Ω,ℱ be a given measurable space. Let ℋ be a subset of all random variables on Ω,ℱ such that IA∈ℋ, where A∈ℱ, and if X1,…,Xn∈ℋ, then φX1,…,Xn∈ℋ for each φ∈Cl,Lipℝn, where Cl,Lipℝn denotes the linear space of (local Lipschitz) function φ satisfying(1)φx−φy≤C1+xm+ymx−y,∀x,y∈ℝn,for some C>0, m∈ℕ, depending on φ. We regard ℋ as the space of random variables.Definition 1.
A sublinear expectationE on ℋ is a functional E:ℍ↦ℝ¯:=−∞,∞ satisfying the following properties: for all X,Y∈ℋ, we have the following:(a)
Monotonicity: ifX≥Y, thenEX≥EY(b)
Constant preserving:Ec=c, ∀c∈ℝ(c)
Positive homogeneity:EλX=λEX,∀λ≥0(d)
Subadditivity:EX+Y≥EX+EYwheneverEX+EYis not of the form∞−∞or−∞+∞
A set functionV:ℱ↦0,1 is called a capacity if it satisfies the following:(a)
V∅=0, VΩ1=1(b)
VA≤VB, A⊂B, A,B∈ℱ
A capacityV is said to be subadditive if it satisfies VA+B≤VA+VB, A,B∈ℱ.
In this paper, given a sublinear expectation spaceΩ,ℋ,E, we define a capacity: VA:=infEξ:IA≤ξ,ξ∈ℋ, ∀A∈ℱ (see Zhang [4]). Clearly, V is a subadditive capacity. We also define the Choquet expectations CV by(2)CVX≔∫0∞VX>xdx+∫−∞0VX>x−1dx.
A sublinear expectationE:ℋ↦ℝ is said to be continuous if it satisfies the following:(a)
Lower continuity:EXn↑EX, if 0≤Xn↑X, where Xn,X∈ℋ(b)
Upper continuity:EXn↓EX, if 0≤Xn↓X, where Xn,X∈ℋ
A capacityV:ℱ↦0,1 is said to be continuous capacity if it satisfies the following:(1)
Lower continuity:VAn↑VA, if An↑A, where An,A∈ℱ(2)
Upper continuity:VAn↓VA, if An↓A, where An,A∈ℱ
Assume thatX=X1,…,Xm, Xi∈ℋ, and Y=Y1,…,Yn, Yi∈ℋ, are two random variables on Ω1,ℋ,E. Y is said to be independent of X if for each φ∈Cl,Lipℝm×ℝn, we have EφX,Y=Eφx,Y|x=X whenever φ¯x:=Eφx,Y<∞ for each x and Eφ¯X<∞. Xnn=1∞ is said to be a sequence of independent random variables, if Xn+1 is independent of X1,…,Xn for each n≥1.
Suppose thatX1 and X2 are two n-dimensional random vectors defined, respectively, in sublinear expectation spaces Ω1,ℋ1,E1 and Ω2,ℋ2,E2. They are said to be identically distributed if(3)E1φX1=E2φX2,∀φ∈Cl,Lipℝn,whenever the sublinear expectations are finite. Xnn=1∞ is said to be identically distributed if for each i≥1, Xi and X1 are identically distributed.
For0≤σ¯2≤σ¯2<∞, a random variable ξ under a sublinear expectation space Ω,ℋ,E is called a G-normal N0,σ¯2,σ¯2 distributed random variable, if for any φ∈Cl,Lipℝn, ux,t:=Eφx+tξx∈ℝ,t≥0 is the unique viscosity solution of the following heat equation:(4)∂tu−G∂xx2u=0,u0,x=φx,where Gα=σ¯2α+−σ¯2α−/2.
In the rest of this paper, let {X, Xn, n≥1} be a sequence of i.i.d. random variables under sublinear expectation space Ω,ℋ,E with EX=E−X=0, EX2=σ¯2<∞, and −E−X2=σ¯2, limc⟶∞EX2−c+=0, CVX2<∞. Set Sn=∑i=1nXi. Assume that E is continuous. Let ξ be a G-normal-distributed random variable with Eξ=E−ξ=0, Eξ2=σ¯2, and −E−ξ2=σ¯2. We denote by C a positive constant which may vary from line to line.
To prove our results, we need the following lemmas.Lemma 1.
SupposeEXα<∞,1<α≤2. Then, forx,y>0,(5)VSn≥x≤2nVX>y+2nx/yeEXαnEXα+xyα−1x/y.Proof.
We borrow the proofs from those of Theorem 2 by Fuk and Nagaev [31], and Lemma 2 by Spătaru [30]. Let(6)X˜i=XiforXi≤y,0forXi>y,i=1,…,n,S˜n=∑i=1nX˜i.
Therefore, by the subadditivity property ofV⋅,(7)VSn≥x≤VS˜n≠Sn+VS˜n≥x.
By Markov’s inequality under sublinear expectations, for any positiveh,(8)VS˜n≥x≤e−hxEehS˜n.
From this and (7), it follows that(9)VSn≥x≤∑i=1nVXi≥y+e−hxEehS˜n=nVX≥y+e−hxEehS˜n.
Application of the monotonicity ofu−2ehu−1−hu for u≤y and u−αehu−1−hu for u>0 and the subadditivity property of sublinear expectations yields(10)EehX˜i≤1+EhXiIXi≤y+EehXi−1−hXiXi2Xi2IXi≤y≤1+hEXiIXi≤y+ehy−1−hyy2EXi2IXi≤y≤1+hEXiIXi≤y+ehy−1−hyyαEXiαIXi≤y≤1+hEXiIXi≤y+ehy−1−hyyαEXiα.
Hence, by Lemma 1.1 in the study of Gao and Xu [32],(11)e−hxEehS˜n≤expehy−1−hyy−αnEXα−hx+hnEXIX≤y.
Setting(12)h=1ylogxyα−1nEXα+1,in the right-hand side of (11), we see that(13)e−hxEehS˜n≤expxy−x−nEXI|X|≤yy+nEXαyαlogxyα−1nEXα+1.
SinceEX=E−X=0, by Proposition 3.6 in the study of Peng [3] and Definition 1, we see that(14)EXIX≤y=E−XIX≥y≤EXIX≥y≤1yα−1EXαIX≥y≤1yα−1EXα.
Therefore,(15)1yx−nEXIX≤y+nEXαyα≥xy.
Combining this with (13) and (9), we conclude that(16)VSn≥x≤nVX>y+nx/yeEXαnEXα+xyα−1x/y.
Combining (16) with the inequality derived from it with −X and −Xk in place of X and Xk, respectively, leads to (5).Remark 1.
(see Lemma2 in [7]). For any X∈ℋ, we have(17)CVX2<∞⇔∫1∞xVX>xdx<∞.Lemma 2.
(see Lemma 5 in [7]). Assume thatXn;n≥1is a sequence of independent and identically distributed random variables withEX1=E−X1=0andlimc⟶∞EX2−c+=0. Writeσ¯2=EX12andσ¯2=−E−X12. Suppose thatEis continuous and setΔnx=VSn/n≥x−Vξ≥x,ξ∼N0,σ¯2,σ¯2underE. Then,(18)Δn≔supx≥0Δnx⟶0,asn⟶∞.


## 3. Main Results

The following are our main results.Theorem 1.
Forb,d>0, we have(19)limε↘0εb/d∑n=3∞loglognb−1nlognVSn≥εnloglognd=CVξb/db.Theorem 2.
Ford>0, we have(20)limε↘01−logε∑n=3∞1nlognloglognVSn≥εσnloglognd=1d.
In the following two sections, forM≥3 and 0<ε<1, set bε=⌊expexpMε−1/d⌋.


## 4. Proof of Theorem1

Proposition 1.
Forb,d>0, we have(21)limε↘0εb/d∑n=3∞loglognb−1nlognVξ≥εloglognd=CVξb/db.Proof.
(22)limε↘0εb/d∑n=3∞loglognb−1nlognVξ≥εloglognd=limε↘0εb/d∫ee∞loglogyb−1ylogyVξ≥εloglogyddy=limε↘0εb/d∫ε∞y/εb−1/d1/dy/ε1/d−11εVξ≥ydy=limε↘0∫ε∞1/dyb/d−1Vξ≥ydy=limε↘0∫εb/d∞1/bVξb/d≥tdt=∫0∞1/bVξb/d≥tdt=CVξb/db.
Thus, this completes the proof of Proposition1.Remark 2.
By the proof of (24) and (25) in the study by Wu [7], CVξb/d is finite for any b,d>0.Proposition 2.
Forb,d>0, we have(23)limε↘0εb/d∑n≤bεloglognb−1nlognVSn≥εnloglognd−Vξ≥εloglognd=0.Proof.
By Lemma2 and Toeplitz’s lemma,(24)limε↘0εb/d∑n≤bεloglognb−1nlognVSn≥εnloglognd−Vξ≥εloglognd≤limε↘0εb/d∑n≤bεloglognb−1nlognΔn=limε↘0CMbloglogbεb∑n≤bεloglognb−1nlognΔn=0.
The proof is complete.Proposition 3.
Forb,d>0, we have(25)limM⟶∞limsupε↘0εb/d∑n>bεloglognb−1nlognVξ≥εloglognd=0.Proof.
We could obtain that(26)limsupε↘0εb/d∑n>bεloglognb−1nlognVξ≥εloglognd≤Climsupε↘0εb/d∫bε∞loglogyb−1ylogyVξ≥εloglogyddy≤C∫Md∞tb/d−1Vξ≥tdt=C∫Mb∞Vξb/d≥tdt.
Note that∫Mb∞Vξb/d≥tdt is integrable:(27)∫Mb∞Vξb/d≥tdt⟶0,as M⟶∞. Proposition 3 is established.Proposition 4.
Forb,d>0, we have(28)limM⟶∞limsupε↘0εb/d∑n>bεloglognb−1nlognVSn≥εnloglognd=0.Proof.
When0<b<2d, by Markov’s inequality under sublinear expectations, we have(29)limsupε↘0εb/d∑n>bεloglognb−1nlognVSn≥εnloglognd≤Climsupε↘0εb/d−2∑n>bεloglognb−1−2dn2lognESn2=Climsupε↘0εb/d−2∑n>bεloglognb−1−2dnlogn≤Climsupε↘0εb/d−2loglogbεb−2d≤CMb−2d⟶0,asM⟶∞.
Forb≥2d, by Lemma 1, we see that(30)∑n>bεloglognb−1nlognVSn≥εnloglognd≤∑n>bεloglognb−1lognVX>εnloglognd/T+C∑n>bεloglognb−1nlogn1loglogn2dTε2T≕L1+L2,where T is a positive constant to be specified later. On the one hand, we obtain that(31)limsupε↘0εb/dL2≤limsupε↘0εb/d−2T∑n>bεloglognb−1−2dTnlogn≤limsupε↘0Cεb/d−2Tloglogbεb−2dT≤CMb−2dT⟶0,asM⟶∞,for any T>b/2d. On the other hand, for L1, without loss of generality, set T=1. By the countable subadditivity property of sublinear expectations and the fact that loglogxb−1/logx⟶0, as x>bε⟶∞ , we obtain that(32)L1=∑n>bεloglognb−1lognVX>εnloglognd≤C∫x>bεloglogxb−1logxVX>εxloglogxddx≤C∫x>bεVX2>ε2xloglogx2ddx≤Cε−2∫x>bεM2dVX2>ydy.
Hence, forb≥2d, we have(33)limsupε↘0εb/dL1≤Cεb/d−2∫x>bεM2dVX2>ydy=0.
Thus, (28) holds for each b,d>0.
Now, by Proposition1–4 and the triangle inequality, ∀β>0, ∃M>0, which is sufficiently large, such that(34)limε↘0εb/d∑n=3∞loglognb−1nlognVSn≥εnloglognd≤limε↘0εb/d∑n=3∞loglognb−1nlognVξ≥εloglognd+limε↘0εb/d∑n≤bεloglognb−1nlognVSn≥εnloglognd−Vξ≥εloglognd+limsupε↘0εb/d∑n>bεloglognb−1nlognVξ≥εloglognd+limsupε↘0εb/d∑n>bεloglognb−1nlognVSn≥εnloglognd=CVξb/db+β,limε↘0εb/d∑n=3∞loglognb−1nlognVSn≥εnloglognd≥limε↘0εb/d∑n=3∞loglognb−1nlognVξ≥εloglognd−limε↘0εb/d∑n≤bεloglognb−1nlognVSn≥εnloglognd−Vξ≥εloglognd−limsupε↘0εb/d∑n>bεloglognb−1nlognVξ≥εloglognd−limsupε↘0εb/d∑n>bεloglognb−1nlognVSn≥εnloglognd≥CVξb/db−β.
We derive Theorem1 from the arbitrariness of β>0.


## 5. Proof of Theorem2

Proposition 5.
Ford>0, we have(35)limε↘01−logε∑n=3∞1nlognloglognVξ≥εloglognd=1d.Proof.
We claim that(36)limε↘01−logε∑n=3∞1nlognloglognVξ≥εloglognd=limε↘01−logε∫ee∞1ylogyloglogyVξ≥εloglogyddy=1dlimε↘01−logε∫ε∞1tVξ≥tdt=1d.
Indeed, by Lemma 4 in the study by Wu [7], ∀α>0, ∃δ>0, such that ∀t<δ<1, Vξ>t>1−αd. Therefore,(37)1dlimε↘01−logε∫ε∞1tVξ≥tdt−1≤1dlimε↘01−logε∫εδ1tVξ≥tdt−1+limε↘01−logε∫δ∞1tVξ≥tdt<1dlimε↘01−logε∫εδ1t1−αddt−1+limε↘01−logε∫δ∞1t3Eξ2dt≤α.
This establishes (35).Proposition 6.
Ford>0, we have(38)limε↘01−logε∑n≤bε1nlognloglognVSn≥εnloglognd−Vξ≥εloglognd=0.Proof.
By Lemma2 and Toeplitz’s lemma,(39)limε↘01−logε∑n≤bε1nlognloglognVSn≥εnloglognd−Vξ≥εloglognd=limε↘01−logε∑n≤bε1nlognloglognΔn=limε↘0logM−logε/d−logε1logloglogbε∑n≤bε1nlognloglognΔn=0.
The proof is complete.Proposition 7.
Ford>0, we have(40)limε↘01−logε∑n>bε1nlognloglognVξ≥εloglognd=0.Proof.
By Markov inequality under sublinear expectations, we see that(41)limε↘01−logε∑n>bε1nlognloglognVξ≥εloglognd≤Climε↘01−logε∫bε∞1ylogyloglogyVξ≥εloglogyddy≤Climε↘01−logε∫Md∞1tVξ>tdt≤Climε↘01−logε∫Md∞1t3Eξ2dt=0.
Thus this proves Proposition7.Proposition 8.
Ford>0, we have(42)limε↘01−logε∑n>bε1nlognloglognVSn≥εnloglognd=0.Proof.
By Markov inequality under sublinear expectations, we deduce that(43)limε↘01−logε∑n>bε1nlognloglognVSn≥εnloglognd≤limε↘01−ε2logε∑n>bε1n2lognloglogn1+2dESn2≤Climε↘01−ε2logε∑n>bε1nlognloglogn1+2d≤Climε↘0loglogbε−2d−ε2logε≤Climε↘0M−2d−logε=0.
The proof is complete.
Finally, similar to the proof of Theorem1, by the triangle inequality and Propositions 5–8, we finish the proof of Theorem 2.



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