# Note on Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of I.I.D.: Random Variables under Sublinear Expectations

Authors: Mingzhou Xu; Kun Cheng
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

LetX,Xn,n≥1 be a sequence of independent, identically distributed random variables under sublinear expectations with CVX2<∞, limc⟶∞EX2−c+=0, and E˘X=E˘−X=0. Write S0=0, Sn=∑k=1nXn, and Mn=max0≤k≤nSk, n≥1. For d>0 and an=ologlogn−d, we obtain the exact rates in the law of iterated logarithm of a kind of weighted infinite series of CVMn−ε+anσ¯nloglognd+ as ε↓0.


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## Body

## 1. Introduction

To study the uncertainty in risk and financial problems, Peng [1, 2] created the concept of the sublinear expectations space. Since Peng created this important concept, people try to study many limit theorems under sublinear expectations space, corresponded to those in classic probability space. Zhang [3–5] investigated the convergence of the sums of independent random variables, Lindeberg’s central limit theorems for martingale like sequences, and Heyde’s theorem under sublinear expectations. Liu and Zhang [6, 7] proved the law of the iterated logarithm for linear processes generated by a sequence of stationary independent random variables, central limit theorem for linear processes generated by independent, and identically distributed (i. i. d.) random variables under sublinear expectations. For more relevant works under sublinear expectations, the reader could refer to Gao and Xu [8], Zhang [9–12], Wu [13], Xu and Cheng [14–16], Zhong and Wu [17], Xu and Zhang [18, 19], Wu and Jiang [20], Chen [21], Fang et al. [22], Hu et al. [23], Hu and Yang [24], Kuczmaszewska [25], Ding [26], and references therein.Since Gut and Spătaru [27] studied precise asymptotics in the law of the iterated logarithm, people established lots of results corresponded to precise asymptotics in the law of the iterated logarithm, for which the readers could refer to Zhang [5, 28], Huang et al. [29], Jiang and Yang [30], Wu and Wen [31], Xiao et al. [32], Xiao and Yin [33], Xu et al. [34], and Xu and Cheng [35, 36]. Recently, Zhang [5] obtained Heyde’s theorem under sublinear expectations and Xiao and Yin [33] proved precise rates in the law of iterated logarithm for the moment convergence of φ-mixing sequence in classic probability space. In this note, motivated by the works of Zhang [5] and Xiao and Yin [33], we try to prove precise rates in the law of iterated logarithm for the moment convergence of maxima of sums of i. i. d. random variables under sublinear expectations, which complement that of Xiao and Yin [33] under sublinear expectations and that of Xu and Cheng [14]. Our result perhaps has potential applications in nature and society (cf. Peng [2] and references therein).The remainder of this paper is organized as follows. In the next section, we present necessary basic notions, concepts and properties, and cite necessary lemmas under sublinear expectations. In Section3, we present our main result, Theorem 1, and we give its proofs in Section 4.


## 2. Preliminaries

We use notations similar to that of Peng [2], Chen [21]. Let Ω,ℱ be a given measurable space. Suppose that ℋ is 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. ℋ could be regarded as the space of random variables.Definition 1.
A sublinear expectationEonℋis a functionalE:ℍ↦ℝ¯:=−∞,∞satisfying the following properties; for allX,Y∈ℋ, we have(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+EYwheneverEX+EYis not of the form∞−∞or−∞+∞
A set functionV:ℱ↦0,1 is named a capacity if(a)
V∅=0 and VΩ=1(b)
VA≤VB, A⊂B and 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, we define a capacity: VA:=infEξ:IA≤ξ,ξ∈ℋ, ∀A∈ℱ (see [10]). Obviously, 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 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 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 (written as ξ∼N0,σ¯2,σ¯2) 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.
A continuous random processWtt≥0 in the sublinear expectation space Ω,ℋ,E is named a G-Brownian motion (see definition III.1.2 of [2]) if W0=0 and for any φ∈Cl,Lipℝn, for all 0≤t0≤⋯≤td≤t≤s,(5)EφWt1,…,Wtd,Ws−Wt=EEφx1,…,xd,t−sη|x1=Wt1,…,xd=Wtd,where η∼N0,σ¯2,σ¯2.
In this sequel, we writeS0=0 and Sn=∑k=1nXk. We cite two lemmas (cf. [3–5]) and a corollary (cf. [5]) as follows.Lemma 1 (see Theorem 2.1 and Corollary 2.2 in [11]).
Suppose thatXn;n≥1is an array of independent random variables under sublinear expectation spaceΩ,ℋ,Esuch thatEXk≤0andEXkq<∞,k=1,2,…. Then, forq≥2,(6)Emax0≤k≤nSn−Skq≤Cq∑k=1nEXk2q/2+∑k=1nEXkq,where Cq is a positive constant depending on q.Lemma 2 (see Corollary 2.1 in [5]).
Assume thatXn;n≥1is a sequence of independent, identically distributed random variables in a sublinear expectation spaceΩ,ℋ,E. Suppose that (i) limc⟶∞EX12∧cis finite; (ii) x2VX1||≥x⟶0asx⟶∞, and (iii) limc⟶∞E−c∨X1∧c=limc⟶∞E−c∨−X1∧c=0. Setσ¯2=E˘X12≐limc⟶∞EX12∧candσ¯2=−E˘−X12≐−limc⟶∞E−X12∨−c. Then,(7)limn⟶∞Vmaxk≤nSk>xn=2Gxσ¯,x>0,where(8)Gx=∑i=0∞−1iPN≥2i+1x,and N is a standard random variable under classic probability space.In the remainder of this paper, let {X, Xn, n≥1} be a sequence of independent, identically distributed (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, limc⟶EX2−c+=0, and r=σ¯2/σ¯2. Set Mn=max0≤i≤nSi, n≥1. As pointed in the paragraph above Theorem 1.1 of [5], by Theorem 4.5 (iii) of [10] and the condition limc⟶EX2−c+=0, we see that EX2≤CVX2<∞, EX=E˘X=0, E−X=E˘−X=0, E˘X2=EX2, and E˘−X2˘=E−X2. 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 results.Theorem 1.
Forb,d>0andan=ologlogn−d, we have(9)limε↘0εb/d∑n=3∞loglognb−1n3/2lognCVMn−ε+anσ¯nloglognd+=σ¯∑i=0∞−1i2b+3d/2ddΓxb+2d/2dbb+dπ2i+11+b/d.Remark 1.
Theorem1generalizes Theorem 1.1 in [33] in some extent. When the sublinear expectation space is linear and {Xn, n≥1} is i. i. d., Theorem 1is the same as (1.5) of Theorem 1.1 in [33].Remark 2.
In (3), whenb=1/2,σ¯=d=1, andan=ologlogn−1,(10)limε↘0ε1/2∑n=3∞1n3/2lognloglogn−1/2CVMn−ε+anσ¯nloglogn+,=∑i=0∞23/42Γ1/4−1i32i+13/2π=1.9826849…..
Whenσ¯=b=d=1 and an=ologlogn−1,(11)limε↘0ε∑n=3∞1n3/2lognCVMn−ε+anσ¯nloglogn+,=∑i=0∞−1i2i+12=0.9159656….
Whenb=2, σ¯=d=1, and an=ologlogn−1,(12)limε↘0ε2∑n=3∞loglognn3/2lognCVMn−ε+anσ¯nloglogn+,=∑i=0∞−1i23/2π32i+13=2π5/248=0.5154048….
Whenb=3, σ¯=d=1, and an=ologlogn−1,(13)limε↘0ε3∑n=3∞loglogn2n3/2lognCVMn−ε+anσ¯nloglogn+,=∑i=0∞−1i22i+14=0.4944723.


## 4. Proof of Theorem 1

In this section, forM≥3 and 0<ε<1, set bε=expexpMε−1/d. Without loss of restriction, we suppose that σ¯=1.Proposition 1.
Forb,d>0, we have(14)limε↘0εb/d∑n=3∞loglognb−1nlognCVsup0≤t≤1Wt−εloglognd+=σ¯∑i=0∞−1i2b+3d/2ddΓb+2d/2dbb+dπ2i+11+b/d,where Wt is a G-Brownian motion with W1∼N0,r,1.Proof.
By Lemma 6.1 of [9], for any m≥1 and x>0, we have(15)2∑k=02m+1−1kPNv≥2k+1x,≤Vsup0≤t≤1Wt>x,≤2∑k=02m−1kPN≥2k+1x.
Hence, it follows that(16)CVsup0≤t≤1Wt−y+=∫0∞Vsup0≤t≤1Wt>y+xdx≤2∑k=02m−1k∫0∞PN≥2k+1x+ydx=2∑k=02m−1k2k+1∫0∞PN≥2k+1y+xdx=2∑k=02m−1k2k+1EN−2k+1y+and(17)CVsup0≤t≤1Wt−y+≥2∑k=02m+1−1k2k+1EN−2k+1y+,where E⋅ is the linear expectation and N is a standard random variable under classic probability space. Hence, for q≥1 and b>0, it is enough to prove that(18)limε↘0εb/d∑n=3∞loglognb−1nlognEN−qεloglognd+=d2b+d/2dΓb+2d/2dπqb/dbb+d.
On the contrary, we obtain(19)limε↘0εb/d∑n=3∞loglognb−1nlognEN−qεloglognd+,=limε↘0εb/dεb/d∫e∞loglogyb−1ylogydy∫qεloglogyd∞PN≥xdx,=q−b/d1dlimε↘0∫qε∞tb/d−1dt∫t∞PN≥xdx,=q−b/d1dlimε↘0∫qε∞PN||≥xdx∫qεxtb/d−1dt,=q−b/ddENb/d+1bb+d=d2b+d/2dΓb+2d/2dπqb/dbb+d.
Now, takingq=2k+1, the proof is finished.Proposition 2.
Forb,d>0, we have(20)limε↘0εb/d∑n≤bεloglognb−1nlognCVsup0≤t≤1Wt−εloglognd+−CVMnn−εloglognd+=0.Proof.
LetΔn=Supx∈ℝVsup0≤t≤1Wt>x−VMn/n>x. Then, Δn⟶0 as n⟶∞ from Lemma 6.1 of [9] and Lemma 2; we see that(21)εb/d∑n≤bεloglognb−1nlogn|CVsup0≤t≤1Wt−εloglognd+−CVMnn−εloglognd+|=εb/d∑n≤bεloglognb−1nlogn|∫0∞Vsup0≤t≤1Wt>x+εloglognddx−∫0∞VMnn>x+εloglognddx|≤εb/d∑n≤bεloglognb−1+dnlogn∫0∞|Vsup0≤t≤1Wt>x+εloglognd−VMnn>x+εloglognd|dx≤εb/d∑n≤bεloglognb−1+dnlogn∫0l|nVsup0≤t≤1Wt>x+εloglognd−VMnn>x+εloglognd|dx+εb/d∑n≤bεloglognb−1+dnlogn∫ln∞Vsup0≤t≤1Wt>x+εloglognddx+εb/d∑n≤bεloglognb−1+dnlogn∫ln∞VMnn>x+εloglognddx≕∑i=13Ji,where ln=loglogn−dΔn−1/2. First, for J1, by Lemma 2.4 of [33], we see that(22)J1≤Cεb/d∑n≤bεloglognb−1+dnlognlnΔn,=Cεb/d∑n≤bεloglognb−1nlognΔn1/2,≤CMbloglogbεb∑n≤bεloglognb−1nlognΔn1/2⟶0,asε↘0.
As forJ2, by Lemma 6.1 of [9], Markov’s inequality, and Lemma 2.4 of [33], we see that(23)J2≤Cεb/d∑n≤bεloglognb−1+dnlogn∫ln∞1x+ε2loglogn2ddx=Cεb/d∑n≤bεloglognb−1−dnlognln−1≤Cεb/d∑n≤bεloglognb−1nlognΔn1/2⟶0,asε↘0.
Next, forJ3, by Lemma 1, we obtain EMn2≤Cn. Then, by Markov’s inequality, under sublinear expectations and Lemma 2.4 of [33], we see that(24)J3≤εb/d∑n≤bεloglognb−1−dnlogn∫ln∞1x+ε2loglogn2ddx⟶0,asε↘0.
The proof is complete.Proposition 3.
Forb,d>0, we have(25)limM⟶∞limsupε↘0εb/d∑n>bεloglognb−1nlognCVsup0≤t≤1Wt−εloglognd+=0.Proof.
By Lemma 6.1 of [9], we see that(26)limsupε↘0εb/d∑n>bεloglognb−1nlognCVsup0≤t≤1Wt−εloglognd+,=limsupε↘0εb/d∫bε∞loglogyb−1ylogydy∫εloglogyd∞Vsup0≤t≤1Wt>xdx,=1d∫Md∞tb/d−1dt∫t∞Vsup0≤t≤1Wt>xdx,≤C∫Md∞tb/d−1dt∫t∞PN≥xdx,=C∫Md∞PN≥xdx∫Mdxtb/d−1dt,≤C∫Md∞PN≥xxb/ddx⟶0,asM⟶∞.
Hence, (25) is established.Proposition 4.
Forb,d>0, we have(27)limM⟶∞limsupε↘0εb/d∑n>bεloglognb−1nlognCVMnn−εloglognd+=0.Proof.
Whenb<d, since EMn2≤Cn, then by Markov’s inequality under sublinear expectations, we obtain(28)limsupε↘0εb/d∑n>bεloglognb−1nlognCVMnn−εloglognd+=limsupε↘0εb/d∑n>bεloglognb−1+dnlogn∫0∞VMn>x+εnloglognddx≤limsupε↘0εb/d∑n>bεloglognb−1−dnlogn∫0∞Cx+ε2dx≤Climsupε↘0εb/d−1loglogbεb−d≤limsupε↘0CMb−d⟶0,asM⟶∞.
Next, we considerb≥d. Let Sk′x=∑i=1kXiIXi≤x+εnloglognd. Since, then by Proposition 3.6 in Chapter I of [2] and Markov’s inequality under sublinear expectations, we see that(29)∑i=1kEXiIXi≤x+εnloglognd=∑i=1kE−XiIXi>x+εnloglognd≤∑i=1kEXiIXi>x+εnloglognd≤∑i=1kEXi21/2VXi>x+εnloglognd≤Ckx+εnloglognd.
Hence, forx≥0 and n>bε large enough, we obtain(30)max1≤k≤n∑i=1kEXiIXi≤x+εnloglogndx+εnloglognd≤Cx+ε2loglogn2d≤1M2d<ε.
Since(31)Mn≤max1≤k≤n∑i=1kEXiIXi≤x+εnloglognd+max1≤k≤nSk′x−∑i=1kEXiIXi≤x+εnloglognd+max1≤k≤n∑i=1kXiIXi>x+εnloglognd,then we obtain(32)CVMnn−εloglognd+=∫0∞VMnn>εloglognd+xdx=loglognd∫0∞VMn>x+εnloglognddx≤loglognd∫0∞Vmax1≤k≤nXk>x+εnloglognddx+loglognd∫0∞×Vmax1≤k≤nSk′x−∑i=1kEXiIXi≤x+εnloglognd,≥x+εnloglognddx≕I1+I2.
ForI1, we see that(33)∑n>bεloglognb−1nlognI1≤∑n>bεloglognb+d−1logn∫0∞VX1>x+εnloglognddx=∑n>bεloglognb+d−1lognCVX1nloglognd−ε+≤∑n>bεloglognb−1nlognCVX1IX1>εnloglognd=∑n>bεloglognb−1nlogn∫εnloglognd∞VX1>xdx+∫0εnloglogndVX1I|X1|>εnloglognd>xdx≤∑n>bεloglognb−1nlogn∫εnloglognd∞VX1>xdx+∑n>bεloglognb−1nlognVX1>εnloglogndεnloglognd.
Note that(34)∑n>bεεloglognb+d−1lognVX1>εnloglognd≤C∫bε∞εloglogxb+d−1logxVX1>εxloglogxddx≤C∫Mdbε∞VX1>yyloglogyb−d−1εlogydySinceloglogyb−d−1logy⟶0asy⟶∞≤C∫M2dbε∞VX12>yεdy∑n>bεloglognb−1nlogn∫εnloglognd∞VX1>xdx≤∫bε∞loglogxb−1xlogx∫εxloglogxd∞VX1>ydydx≤C∫y>bεMdVX1>ydy∫bεy/ε2loglogxb−1xlogxdxSinceloglogxb−1logx⟶0as x⟶∞=C∫y>bεMdVX1>yyεdy.
Therefore, whenb≥d, it follows that(35)limsupε↘0εb/d∑n>bεloglognb−1nlognI1,≤Climsupε↘0εb/d−1∫M2dbε∞VX12>ydy=0.
Now, we considerI2 for b≥d. Using Markov’s inequality under sublinear expectations and Lemma 1,(36)Vmax1≤k≤nSk′x−∑i=1kEXiIXi≤x+εnloglognd≥x+εnloglognd≤Vmax0≤k≤nSn′x−Sk′x−∑i=k+1nEXiIXi≤x+εnloglognd≥x+εnloglognd/2≤CEmax0≤k≤nSn′x−Sk′x−∑i=k+1nEXiIXi≤x+εnloglogndq×x+ε−qn−q/2loglogn−qd≤Cx+ε−qn−q/2loglogn−qdnq/2+nEX1qIX1≤x+εnloglognd≤Cx+ε−qloglogn−qd+Cx+ε−qn1−q/2loglogn−qd×X1qIX1≤x+εnloglognd≕I21+I22.
We takeq>1+b/d≥2, implying b+d−qd<0; then, for I21, we obtain(37)limsupε↘0εb/d∑n>bεloglognb−1nlognloglognd∫0∞I21dx,≤limsupε↘0Cεb/d∑n>bεloglognb+d−qd−1nlogn∫0∞1x+εqdx,≤limsupε↘0Cεb/d+1−qloglogbεb+d−qd≤CMb+d−qd⟶0,asM⟶∞.
Next, we estimateI22:(38)∑n>bεloglognb−1nlognloglognd∫0∞I22dx≤∑n>bεloglognb+d−qd−1nq/2logn∫0∞Cx+εqEX1qIX1x+ε≤nloglognddx≤∑n>bεloglognb+d−qd−1nq/2logn∫0∞Cx+εqCVX1qIX1x+ε≤nloglognddx=∫0∞Cx+εq∑n>bεloglognb+d−qd−1nq/2lognCVX1qIX1x+ε≤nloglognddx≤∫0∞Cx+εq∑n>bεloglognb+d−qd−1nq/2logn∫0x+εnloglogndqVX1q>ydydx=∫0∞Cx+εq∑n>bεloglognb+d−qd−1nq/2logn∑k=1n∫x+εk−1loglogk−1dqx+εkloglogkdqVX1q>ydydx=∫0∞Cx+εq∑k>bε∫x+εk−1loglogk−1dqx+εkloglogkdqVX1q>ydy∑n≥kloglognb+d−qd−1nq/2logndx+∫0∞Cx+εq∑k=1bε∫x+εk−1loglogk−1dqx+εkloglogkdqVX1q>ydy∑n>bεloglognb+d−qd−1nq/2logndx=I221+I222.
ForI221, we see that(39)I221≤C∫0∞1x+εq∑k>bε∫x+εk−1loglogk−1dx+εkloglogkdloglogkb+d−qd−1kq/2−1logkVX1>yyq−1dydx,≤C∫0∞1x+ε2∑k>bε∫x+εk−1loglogk−1dx+εkloglogkdloglogkb−d−1logkVX1>yydydx,≤C1εCVX12IX1>εbεloglogbεd.
Sinceloglogkb−1−dlogk−1⟶0 as k>bε⟶∞, then, for b≥d, we obtain(40)limsupε↘0εb/dI221≤limsupε↘0εb/d−1CVX12IX1>εbεloglogbεd=0.
Next, forI222, we see that(41)I222≤C∫0∞1x+εq∑k=1bεloglogbεb+d−qd−1bεq/2−1logbε∫x+εk−1loglogk−1dx+εkloglogkdVX1>yyq−1dydx≤C∫0∞1x+ε2∑k=1bεloglogbεb−d−1logbε∫x+εk−1loglogk−1dx+εkloglogkdVX1>yydydx≤C1εloglogbεb−d−1logbεCVX12.
Since forb≥d, we conclude that(42)limsupε↘0Cεb/d−1loglogbεb−d−1logbε≤limsupε↘0CMb−1−dε1/dlogbε=0.
Then, we obtain(43)limsupε↘0εb/dI222=0.
The proof of Proposition4 is finished.Proposition 5.
Forb,d>0andan=ologlogn−d, we have(44)limsupε↘0εb/d∑n=3∞loglognb−1nlognCVMnn−ε+anloglognd+−CVMnn−εloglognd+=0.Proof.
As in the proof of Proposition 3.5 in [33], when n>bε, we obtain(45)an=ologlogbε−d=oε.
From Proposition4, it follows that, for any α>0,(46)limM⟶∞limsupε↘0εb/d∑n>bεloglognb−1nlognCVMnn−αεloglognd+=0.
Then, it is enough to prove that(47)limsupε↘0εb/d∑n≤bεloglognb−1nlogn|CVMnn−ε+anloglognd+,−CVMnn−εloglognd+=0.
In addition, similar to the proof of Proposition2, we also obtain(48)limsupε↘0εb/d∑n≤bεloglognb−1nlogn|CVMnn−ε+anloglognd+,−CVsup0≤t≤1Wt−ε+anloglognd+|=0.
Hence, it is sufficient to prove that(49)limsupε↘0εb/d∑n≤bεloglognb−1nlogn|CVsup0≤t≤1Wt−ε−anloglognd+,−CVsup0≤t≤1Wt−ε+anloglognd+|=0.
Byan=ologlogn−d and Lemma 2.4 of [33], we see that(50)limsupε↘0εb/d∑n≤bεloglognb−1nlogn|CVsup0≤t≤1Wt−ε−anloglognd+−CVsup0≤t≤1Wt−ε+anloglognd+|≤limsupε↘0εb/d∑n≤bεloglognb−1nlogn∫ε−|an|loglogndε+|an|loglogndVsup0≤t≤1Wt>xdx≤Climsupε↘0εb/d∑n≤bεloglognb−1nlognanloglognd=Climsupε↘0Mbloglogbεb∑n≤bεloglognb−1nlognanloglognd=0.
This completes the proof of Proposition5.
Now, using Propositions1–5, we finish the proof of Theorem 1.



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*Source: 7566141-2022-09-16.xml*