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

Authors: Mingzhou Xu; Kun Cheng
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

By using Lebesgue bounded convergence theorem, we prove precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectation.


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## Body

## 1. Introduction

Inspired by the phenomena of uncertainty in risk measure, super-hedging problems, Peng et al. [1, 2] initiated the sublinear expectations space nontrivially. Motivated by the works of Peng, people try to investigate whether or not the corresponding limit theorems in classic probability space could hold under sublinear expectations space. For the corresponding works under sublinear expectations, the interested readers could refer to Gao and Xu [3], Zhang et al. [4–10], Wu [11], Xu and Cheng [12], Zhong and Wu [13], Xu et al. [14, 15], Yu and Wu [16], Wu and Jiang [17], Ma and Wu [18], Chen [19], Fang et al. [20], Hu et al. [21], Hu and Yang [22], Kuczmaszewska [23], Wang and Wu [24], and references therein. Since Gut and Spătaru [25] investigated precise asymptotics in the law of the iterated logarithm, people obtained lots of corresponding results related to precise asymptotics in the law of the iterated logarithm, for which the interested readers could refer to Zhang [26], Huang et al. [27], Jiang and Yang [28], Wu and Wen [29], Xiao et al. [30], and Xu et al. [31–33]. Recently, by using Lebesgue bounded convergence theorem together with the results of Zhang et al. [8, 9], Zhang [10] obtained Heyde’s theorem under sublinear expectations. The methods of Zhang [10] are different from that of Xiao et al. [30]. By using the central limit theorem obtained by Wu [11] and the methods of Xiao et al. [30], we investigated precise asymptotics in the law of the iterated logarithm under sublinear expectations in Xu and Cheng [12], which complement that of Xiao et al. [30] under sublinear expectations. It is interesting to wonder whether or not precise asymptotics in the law of the iterated logarithm under sublinear expectations hold under the same conditions as in Zhang [10]. In this note, we try to use the methods in Zhang [10] to investigate precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations under conditions different from that of Xu and Cheng [12], which complement that obtained by Xu and Cheng [12]. Our results may have potential applications in engineering or finance fields (cf. Peng et al. [2, 34, 35]).The rest of this paper is organized as follows. In the next section, we summarize necessary basic notions, concepts, and relevant properties and give necessary lemmas under sublinear expectations. In Section3, we give our main results, Theorems 1 and 2, and we present the proof of Theorem 1 in Section 4. The proof of Theorem 2 is similar to that of Theorem 1, so it is omitted.


## 2. Preliminary

We use notations of Peng [2]. Let Ω,ℱ be a given measurable space. Assume that ℋ is a subset of all random variables on Ω,ℱ satisfying IA∈ℋ, where A∈ℱ and if X1,…,Xn∈ℋ, then φX1,…,Xn∈ℋ for each φ∈Cl,Lipℝn, where Cl,Lipℝn stands for 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, then EX≥EY.(b)
Constant preserving:Ec=c, ∀c∈ℝ.(c)
Positive homogeneity:EλX=λEX, ∀λ≥0.(d)
Subadditivity:EX+Y≥EX+EY whenever EX+EY is not of the form ∞−∞ or −∞+∞.
A set functionV:ℱ↦0,1 is named a capacity if it obeys(a)
V∅=0, VΩ=1.(b)
VA≤VB, A⊂B, A,B∈ℱ.
A capacityV is called to be subadditive if VA+B≤VA+VB, A,B∈ℱ.
In this note, given a sublinear expectation spaceΩ,ℋ,E, capacity is defined as follows: VA:=infEξ:IA≤ξ,ξ∈ℋ, ∀A∈ℱ (see Zhang [5]). Clearly, V is a subadditive capacity. We also define the Choquet expectations CV by(2)CVX:=∫0∞VX>xdx+∫−∞0VX>x−1dx.
Suppose thatX=X1,…,Xm, Xi∈ℋ and Y=Y1,…,Yn, Yi∈ℋ are two random variables on Ω,ℋ,E. Y is called to be independent of X, if for each φ∈Cl,Lipℝm×ℝn, EφX,Y=Eφx,Y|x=X whenever φ¯x:=Eφx,Y<∞ for each x and Eφ¯X<∞. Xnn=1∞ is called to be a sequence of independent random variables, if Xn+1 is independent of X1,…,Xn for each n≥1.
Assume thatX1 and X2 are two n-dimensional random vectors defined, respectively, in sublinear expectation spaces Ω1,ℋ1,E1 and Ω2,ℋ2,E2. They are called to be identically distributed if(3)E1φX1=E2φX2,∀φ∈Cl,Lipℝn.
Whenever the sublinear expectations are finite.Xnn=1∞ is called 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 named a 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. We cite two lemmas (cf. Zhang et al. [8–10]) and a corollary (cf. Zhang [10]) as follows.Lemma 1 (see Lemma 2.1 in [10]).
Suppose thatZn,k;k=1,…,kn is an array of independent random variables such that EZn,k≤0 and EZn,k2<∞, k=1,…,kn. Then, for all x,y>0,(5)Vmaxk≤kn∑k=1mZn,k≥x≤Vmaxk≤knZn,k≥y+expxy−xyBnxy+1ln1+xyBn,where Bn=∑k=1knEZn,k2.Lemma 2 (see Lemma 2.4 in [10]).
Assume thatXn;n≥1 is a sequence of independent, identically distributed random variables in a sublinear expectation space Ω,ℋ,E, write Sn=∑k=1nXk. Suppose that (i) limc⟶∞EX12∧c is finite; (iii) x2VX1≥x⟶0 as x⟶∞; and (ii) limc⟶∞E−c∨X1∧c=limc⟶∞E−c∨−X1∧c=0. Set σ¯2=E˘X12≐limc⟶∞EX12∧c and σ¯2=−E˘−X12. Then, for any φ∈Cbℝ,(6)limn⟶∞EφSnn=Eφσ¯ξ,where ξ∼N0,r,1, r=σ¯2/σ¯2. Moreover, for any continuous and bounded functions ψ:D0,1↦ℝ,(7)limn⟶∞EψSn⋅n=Eψσ¯W⋅,where Wt is a G-Brownian motion with W1∼N0,r,1 and D0,1 is the space of right continuous functions which have finite left limits and is endowed with the Skorokhod topology.Corollary 1 (see Corollary 2.1 in [10]).
LetFx=Vξ>x. Under the assumptions in Lemma 2,(8)limn⟶∞VSn>xn=Fx/σ¯ifx/σ¯is a continuous pointofF,and(9)limn⟶∞Vmaxk≤nSk>xn=2Gx/σ¯,x>0,where(10)Gx=∑i=0∞−1iPN≥2i+1x,and N is a standard random variable.In the remainder 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 CVX2<∞, E˘X≐limc⟶∞E−c∨X∧c=0, E˘−X=0, E˘X2=σ¯2<∞, −E˘−X2=σ¯2, and r=σ¯2/σ¯2. Set Sn=∑i=1nXi, n≥1. Let logx=lnx∨e, loglogx=lnlnx∨ee, and x=supl,l≤x,l∈ℤ+; we denote by C a positive constant which may change from line to line.


## 3. Main Results

The following are our main results.Theorem 1.
Forb,d>0, we have(11)limε↘0εb/d∑n=3∞loglognb−1nlognVSn≥εnloglognd=σ¯b/dCVξb/db,where ξ∼N0,r,1. Moreover, for b,d>0,(12)limε↘0εb/d∑n=3∞loglognb−1nlognVmaxk≤nSk≥εnloglognd=∑i=0∞σ¯b/d−1i21+b/2dΓb+d/2dbπ2i+1b/d.Remark 1.
In (12), when b=3/2, d=1, σ¯=1,(13)limε↘0ε3/2∑n=3∞loglogn1/2nlognVmaxk≤nSk≥εnloglogn=∑i=0∞−1i23/4Γ1/43π2i+13/2=0.9913425….
Whenb=2, d=1, σ¯=1,(14)limε↘0ε2∑n=3∞loglognnlognVmaxk≤nSk≥εnloglogn=∑i=0∞−1i2i+12=0.9159656….
Whenb=3, d=1, σ¯=1,(15)limε↘0ε3∑n=3∞loglogn2nlognVmaxk≤nSk≥εnloglogn=∑i=0∞−1i25/2Γ23π2i+13=2π5/224=1.03081….
Setlog1x≐logx, log2x≐loglogx, …, logkx≐loglogk−1x, k≥2. Adapting the proofs of Theorem 1, we also have the following theorem, whose proofs are omitted.Theorem 2.
Forb,d>0, for any integer k≥3, we have(16)limε↘0εb/d∑n=3∞logkxb−1nlog1nlog2n⋯logk−1nVSn≥εnlogknd=σ¯b/dCVξb/db.Forb,d>0,(17)limε↘0εb/d∑n=3∞logkxb−1nlog1nlog2n⋯logk−1nVmaxk≤nSk≥εnlogknd=∑i=0∞σ¯b/d−1i21+b/2dΓb+d/2dbπ2i+1b/d.In the following two sections, forM≥3 and 0<ε<1, set bε=expexpMε−1/d.


## 4. Proof of Theorem1

Here, we borrow ideas from the proofs in Zhang [10]. Without loss of restrictions, we suppose σ¯=1. We first establish (11). Notice for n large enough,(18)∫nn+1loglogxb−1xlogxVSxx>εloglogxddx≤loglognb−1nlognVSnn>εloglognd≤∫nn+1loglogx−1b−1x−1logx−1VSxx>εloglogx−1ddx,and set Fx=Vξ>x, for a∈0,1,(19)limε↘0εb/d∫3∞loglogx−ab−1x−alogx−aFεloglogx−addx=limε↘0∫0∞1dyb/d−1Fydy=CVξb/db.Therefore, it is enough to establish that(20)limε↘0εb/d∫3∞loglogx−ab−1x−alogx−a×VSxx>εloglogx−ad−Fεloglogx−addx=0.Obviously, we see that for anyδ>0,(21)limε↘0εb/d∫3expexpδ1/bε−1/dloglogx−ab−1x−alogx−a×VSxx>εloglogx−ad−Fεloglogx−addx≤εb/d1bδε−b/d=δb.ForM>1≥δ>ε>0, and 0≤a≤1,(22)εb/d∫expexpδ1/bε−1/dbεloglogx−ab−1x−alogx−a×VSxx>εloglogx−ad−Fεloglogx−addx≤εb/d∫expexpδ1/dε−1/dbεloglogx−ab−1x−alogx−a×maxn≥expexpδ1/bε−1/dVSnn>εloglogx−ad−Fεloglogx−addx≤εb/d∫0Md1bmaxn≥expexpδ1/dε−1/dVSnn>y−Fydy/εb/d≤∫0Mb1bmaxn≥expexpδ1/bε−1/dVSnn>zd/b−Fzd/bdz.As in the proofs of Zhang [10], CVX12<∞ implies the assumptions of Lemma 2. By Lemma 2 and Corollary 1,(23)maxn≥expexpδ1/bε−1/dVSnn>zd/b−Fzd/b⟶0asε⟶0.Ifzd/b is the continuous point of F. Note that Fx is a monotone function, whose discontinuous points are countable, zd/b is one-to-one mapping from 0,∞ to 0,∞, and so the above convergence holds except on a set with null Lebesgue measure. From the Lebesgue bounded convergence theorem it follows that(24)limε↘0∫0Mb1bmaxn≥expexpδ1/bε−1/dVSnn>zd/b−Fzd/bdz=0.Hence, forM≥1,(25)limε↘0εb/d∫expexpδ1/bε−1/dbεloglogx−ab−1x−alogx−a×VSxx>εloglogx−ad−Fεloglogx−addx=0.On the other hand, forM≥2, by the study (24) and (25) in [11],(26)εb/d∫bε∞loglogx−ab−1x−alogx−aFεloglogx−addx≤εb/d∫Md/2∞1bFydy/εb/d=εb/d∫Mb/2b/d∞1bVξb/d>zdz⟶0asM⟶∞.It remains to establish that(27)limM⟶∞limε↘0εb/d∫bε∞loglogx−ab−1x−alogx−aVmaxk≤x|Sk|>b/d2εx−aloglogx−addx=0.Letbx=εd/bx−aloglogx−ad/32, Yi,x=−bx∨Xi∧bx. Then,(28)EY1,x=E˘X1−EY1,x≤E˘−bx+≤bx−1E˘X12≤32εb/dε2x−aloglogx−adE˘X12≤32εb/dloglogx−adM2d/2x≤εb/dloglogx−ad32x,forM2d/2>322,x≥b.Hence, we see that forM2d/2>322 and x≥bε,(29)Vmaxk≤xSk>b/d2εx−aloglogx−ad≤∑k≤xV|Xk|>εd/bx−aloglogx−ad32+Vmaxk≤x∑i=1kYi,x−EYi,x>b/d4εx−aloglogx−ad.NoticeYi,x−EYi,x≤2bx and EYi,x−EYi,x2≤4X12=4. By Lemma 1,(30)Vmaxk≤x∑i=1kYi,x−EYi,x>b/d/4εx−aloglogx−ad≤expb/d/4εx−aloglogx−ad2bx−b/d/4εx−aloglogx−ad2bxln1+2bxb/d/4εx−aloglogx−ad4x≤Ce4256ε2loglogx−a2d4b/d.Therefore, by the proof of Lemma1 in Zhong and Wu [13],(31)εb/d∫bε∞Vmaxk≤xSk>b/d2εx−aloglogx−adloglogx−ab−1x−alogx−adx≤Cεb/d∫bε∞loglogx−ab−1logx−aVX1>εd/bx−aloglogx−ad64dx+Cεb/d∫bε∞loglogx−ab−1x−alogx−a256ε2loglogx−a2d4b/ddx≤C∫Mdε−1∞yVX1>ydy+Cεb/dε−8b/d1Mε−1/d7b≤C∫M2d∞VX12>ydy+C/M7b⟶0.AsM⟶∞, for −Sks, we have same convergence. Thus, (11) is established.Next, we prove (12). With (9) in place of (8), we obtain(32)limε↘0εb/d∫3∞loglogx−ab−1x−alogx−a×Vmaxk≤xSkx>εloglogx−ad−2Gεloglogx−addx=0.Also, for0≤a≤1,(33)limε↘0εb/d∫3∞loglogx−ab−1x−alogx−a2Gεloglogx−addx=limε↘0εb/d∫0∞2Gy1bdyεb/d=∫0∞2dGyyb/d−1dy.We see that for any integerm≥0,(34)∫0∞2dGyyb/d−1dy≤∫0∞2d∑i=02m−1iPN≥2i+1yyb/d−1dy=∑i=02m−1i∫0∞2dPN≥2i+1yyb/d−1dy=∑i=02m−1i21+b/2dbΓb+d/2dπ2i+1b/d,∫0∞2dGyyb/d−1dy≥∫0∞2d∑i=02m+1−1iPN≥2i+1yyb/d−1dy=∑i=02m+1−1i∫0∞2dPN≥2i+1yyb/d−1dy=∑i=02m+1−1i21+b/2dbΓb+d/2dπ2i+1b/d.Hence, lettingm⟶∞ in the above two inequalities results in(35)∫0∞2dyyb/d−1dyG=∑i=0∞−1i21+b/2dbΓb+d/2dπ2i+1b/d.Therefore, (12) is proved.



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