# Almost Sure Limit Theorems for Multivariate General Standard Normal Sequences and Applications

Authors: Zhicheng Chen; Xinsheng Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. The simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples.


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## Body

## 1. Introduction

The classical extreme value theory studies extreme phenomena in nature and human society, such as floods, hurricane, stock market crash, megaseism, and so on [1–3]. Almost sure (a.s.) convergence is a nice behavior of the various ways of convergence [4, 5]. The almost sure central limit theorem (ASCLT) on partial sums is initiated by Brosamler and Schatte for independent identically distributed (i.i.d.) random variables [6, 7]. Let X1,X2,… be i.i.d. random variables with EXn=0,VarXn=1 and Tn=∑k=1nXk. Under some regularity conditions, we have(1)limn⟶∞1logn∑k=1n1kITkk≤x=Φx,a.s.,for any x, where I, logn, and Φx denote the indicator function, the natural logarithm, and the standard normal distribution function. Later, Lacey and Philipp prove that equation (1) holds when f is a bound Lipschitz function [8]. The almost sure central limit theorem on maximum of i.i.d random variables is firstly discovered by Fahrnar and Stadtmüller and Cheng et al., respectively [5, 9]. Berkes et al. consider the general nonlinear functionals case [10]. Csáki and Gondigdanzan investigate ASCLT for the maximum of a stationary Gaussian sequences [11]. Chen and Lin extend to the nonstationary Gaussian sequences [12]. Chen and Peng provide an ASCLT for the maxima of multivariate stationary Gaussian sequences under some mild conditions [13, 14].In recent years, the almost sure convergence of the joint distribution is explored. The almost sure limit theorems related to the maxima of the complete and incomplete samples for stationary sequences are considered [15]. Peng et al. extend the result to the maxima and minima of the complete and incomplete samples for weakly dependent stationary Gaussian sequences [16]. The ASCLT for the maxima and sums of i.i.d. random variables is investigated by Zang et al. [17]. Tan and Wang consider ASCLT for the maxima and sums of standardized stationary Gaussian sequences under some conditions [18]. The almost sure convergence of the maxima and minima for stationary sequences holds under some dependence conditions [19]. The behavior of almost sure convergence of the maxima and the minima for a strongly dependent stationary Gaussian sequence is extended to multivariate vectors [20, 21].In this paper, we study ASCLT of the multivariate general standard normal sequences under some suitable conditions. Throughout this paper,X1,X2,… is a standardized normal sequence of d-dimensional random vector, i.e., each component of the random vector has a zero mean and a unit standard deviation. Set(2)rijp=CovXip,Xjp,rijp,q=CovXip,Xjq,1≤p≠q≤d,Mk,n=Mk,n1,…,Mk,nd,Mn=M0,n,Mk,np=maxk+1≤i≤nXip,Mnp=M0,np,p=1,…,d,and un=un1,…,und and vn=vn1,…,vnd are two real vectors. un>vn implies unp>vnp for all p=1,…,d, and a≪b stands for a=Ob. Finally, write an=2logn1/2 and bn=an−1/2an−1log4πlogn. In this paper, the following results are obtained (proofs are given in Section 2).


## 2. Results

### 2.1. Theorems

Theorem 1.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector, satisfying rijp≤ρj−ip, rijp,q≤ρj−ip,q for i,j=1,2,… and(3)supn≥11≤p≤dρnp<1,supn≥01≤p≠q≤dρnp,q<1,as n⟶∞ and(4)ρnplognloglogn1+ε=O1,ρnp,qlognloglogn1+ε=O1.Let the levelsunip be such that n1−Φλnp is bounded and λnp=min1≤i≤nunip≥clogn1/2 for some c>0. If ∑i=1n1−Φunip⟶τp for some τp≥0 and p=1,…,d, then(5)limn⟶∞1logn∑k=1n1kI∩i=1kXi≤uki=∏p=1dexp−τp,a.s.In particular, letunp=an−1xp+bn,xp∈ℝ for p=1,…,d, and then(6)limn⟶∞1logn∑k=1n1kIMk≤uk=∏p=1dexp−exp−xp,a.s.Theorem 2.
LetXnn=1∞ be a standardized normal d-dimensional random vector sequence satisfying(a)
δ=maxp≠qsuprnpn≥1,suprnp,qn≥0<1(b)
there existsγ≥21+δ/1−δ, such that(7)1n2∑p=1d∑1≤i<j≤nrijpexpγrijplogj−i≪loglogn−1+ε,1n2∑1≤p≠q≤nn∑1≤i≤j≤nrijp,qexpγrijp,qlogj−i≪loglogn−1+ε,where ε>0.Ifn1−Φunp⟶τp,0≤τp<∞,forp=1,2,…,d, then(8)limn⟶∞1logn∑k=1n1kIMk≤Uk=∏p=1dexp−τp,a.s.Letunp=an−1xp+bn, where xp are real numbers for p=1,…,d, and then(9)limn⟶∞1logn∑k=1n1kIMk≤uk=∏p=1dexp−exp−xp,a.s.We shall be concerned with the maximaMn∗=max1≤i≤nYi for a normal sequence Ynn=1∞ given by Yn=Xn+mn where Xnn=1∞ is a standardized normal sequence of d-dimensional vector with each component having a zero mean and unit standard deviation, and mn are added deterministic components. Define mn=mn1,…,mnd, and we shall assume that the constant mnp is such that(10)βnp=max1≤i≤nmip=ologn,p=1,…,d,asn⟶∞.Hence, the mean and the deviation of each component ofYn are mnp and 1, and(11)CovYip,Yjp=rijp,CovYip,Yjq=rijp,q,1≤p≠q≤d.Letmn∗=mn∗1,…,mn∗d be chosen such that mn∗p≤βnp and(12)anmax1≤i≤nmip−mi∗p≤D,p=1,…,d,where D is a constant.Theorem 3.
LetYnn=1∞ be a normal sequence of d-dimensional vector defined above with Xnn=1∞ satisfying the conditions of Theorem 1. Then,(13)limn⟶∞1logn∑k=1n1kIakMk∗−bk∗−mk∗≤x=∏p=1dexp−e−xp,a.s.,where Mn∗=Mn∗1,…,Mn∗d with Mn∗p=max1≤i≤nYip and bn∗=bnId with Id=1,…,1︸d and x=x1,…,xd.


### 2.2. Proofs of the Main Results

We need the following lemmas for the proofs of the main results.Lemma 1.
Letξnn=1∞ and ηnn=1∞ be standardized normal sequences of d-dimensional random vector with rij0p=Covξip,ξjp, rij0p,q=Covξip,ξjq and rij′p=Covηip,ηjp, rij′p,q=Covηip,ηjq. Let ρijp=maxrij0p,rij′p, ρijp,q=maxrij0p,q,rij′p,q and u1,…,un be real vectors. Then,(14)Pξj≤unj,j=1,…,n−Pηj≤unj,j=1,…,n≤K1∑p=1d∑1≤i<j≤nrij0p−rij′p1−ρij2p−1/2exp−uni2p+unj2p21+ρijp+K2∑1≤p≠q≤d∑1≤i≤j≤nrij0p,q−rij′p,q1−ρij2p,q−1/2exp−uni2p+unj2q21+ρijp,q,for constants K1,K2.Proof.
It follows from Theorem 4.2.1 in [22].Lemma 2.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector satisfying equations (3) and (4), and n1−Φλnp be bounded for p=1,…,d, then(15)Sn∗≔∑p=1d∑1≤i<j≤nrijpexp−uni2p+unj2p21+rijp≪loglogn−1+ε,(16)Sn≔∑1≤p≠q≤d∑1≤i≤j≤nrijp,qexp−uni2p+unj2q21+rijp,q≪loglogn−1+ε.Proof.
By equation (3), we have sup1≤i≤j≤n,1≤p≠q≤drijp,q=δ<1. Define θn=eηλnp2 for η<η0=1−δ/21+δ. Split Sn into two parts, then(17)Sn=∑1≤p≠q≤d∑j−i≤θn1≤i≤j≤nrijp,qexp−uni2p+unj2q21+rijp,q+∑1≤p≠q≤d∑j−i>θn1≤i≤j≤nrijp,qexp−uni2p+unj2q21+rijp,q≔Sn1+Sn2.
Notice thatn1−Φλnp is bounded for p=1,…,d, which implies that(18)exp−λn2p2∼C2πλnpn,λnp∼2logn,for some constant C>0, p=1,…,d, and then(19)Sn1≤∑1≤p≠q≤dδ∑j−i≤θn1≤i≤j≤nexp−uni2p+unj2q21+δ≤∑1≤p≠q≤dδθnnexp−λn2p+λn2q21+δ≪∑1≤p≠q≤dn2η−η0logn1/1+δ−η≪loglogn−1+ε,for η0>η.
Now, considerSn2. Let δt=supj≥tρjp,q,1≤p≠q≤d and t=nα. Since n1−Φλnp is bounded as n⟶∞ such that λn2p>clogn for some c>0 and θn=expηλn2p≥expc2ηlogn=nα,α=c2η>0, and then(20)Sn2≤∑1≤p≠q≤d∑j−i>θn1≤i≤j≤nδtexp−λn2p+λn2q21+δt≤∑1≤p≠q≤dδtn2exp−λn2p+λn2q21+δt≪∑1≤p≠q≤dδtn1−Φλnpn1−Φλnqλnpexpλn2p2δtλnqexpλn2q2δt≪∑1≤p≠q≤dδtλnpexpλn2p2δtλnqexpλn2q2δt≪∑1≤p≠q≤dδtλn2pδtλn2qexpλn2p2δtexpλn2q2δt≪∑1≤p≠q≤dδtlogtexp1αδtlogt.
As(21)δtlogtloglogt−1+ε≤supj≥tρjp,qlogjloglogj−1+ε=O1,we have(22)δtlogt≪loglogn−1+ε⟶0,and expλn2p/2δt is bounded. Then,(23)Sn2≪loglogn−1+ε.Lemma 3.
LetXnn=1∞ be a standardized normal d-dimensional random vector sequence satisfying the conditions (a) and (b) of Theorem 2, and then equations (15) and (16) hold.Proof.
The proof of Lemma3 is similar to that of Lemma 2.Lemma 4.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector satisfying Lemma 2 (Lemma 3), then(24)CovI∩i=1kXi≤uki,I∩i=n+1nXi≤uki≪loglogn−1+ε.Proof.
By Lemma1,(25)CovI∩i=1kXi≤uki,I∩i=k+1nXi≤uni=P∩i=1kXi≤uki,∩i=k+1nXi≤uni−P∩i=1kXi≤ukiP∩i=k+1nXi≤uni≤∑p=1d∑i=1k∑j=k+1nrijpexp−uki2p+unj2p21+rijp+∑1≤p≠q≤d∑i=1k∑j=k+1nrijp,qexp−uki2p+unj2q21+rijp,q≔Dn+En.
BothDn and En are of order loglogn−1+ε, but we will only prove En≪loglogn−1+ε. In fact,(26)En=∑1≤p≠q≤d∑i=1k∑j=k+1nrijp,qexp−uki2p+unj2q21+rijp,q≤∑1≤p≠q≤d∑1≤i≤kk+1≤j≤nj−i<θnrijp,qexp−uki2p+unj2q21+rijp,q+∑1≤p≠q≤d∑1≤i≤kk+1≤j≤nj−i<θnrijp,qexp−uki2p+unj2q21+rijp,q≔En1+En2.
We have(27)En1≤∑1≤p≠q≤dδ∑1≤i≤kk+1≤j≤nj−i<θnexp−λk2p+λn2q21+δ≤∑1≤p≠q≤dδkθnexp−λk2p+λn2q21+δ≤∑1≤p≠q≤dδkθnλkpk1/1+δλnqn1/1+δ≤∑1≤p≠q≤dδk1−1/1+δlogk1/21+δn2η−1/1+δlogn1/21+δ−η≤∑1≤p≠q≤dδn−2η0−ηlogn1/1+δ−η≪loglogn−1+ε,En2≤∑1≤p≠q≤d∑1≤i≤k+1≤j≤nj−i≥θnρijp,qexp−uki2p+unj2q21+ρijp,q≤∑1≤p≠q≤dδt∑i=1kexp−uki2p21+δt∑i=1nexp−uni2p21+δt≤∑1≤p≠q≤dδtkexp−λk22exp12λk2δtnexp−λn22exp12λn2δt≪∑1≤p≠q≤dδtlognexpλn2δt≪loglogn−1+ε.Remark 1.
Under the conditions of Theorem2, Lemma 4 can be proved by Lemma 1 and Lemma 3.Lemma 5.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector satisfying Lemma 2 (Lemma 3), then (28)EI∩i=1nXi≤uni−I∩i=k+1nXi≤uni≪kn+loglogn−1+ε,where E denotes the mean.Proof.
(29)EI∩i=1nXi≤uni−I∩i=k+1nXi≤uni=P∩i=1nXi≤uni−P∩i=k+1nXi≤uni≤P∩i=k+1nXi≤uni−∏p=1d∏j=k+1nΦunjp+P∩i=1nXi≤uni−∏p=1d∏j=1nΦunjp+∏p=1d∏j=k+1nΦunjp−∏p=1d∏j=1nΦunjp≜A+B+C.
By Lemma1 and Lemma 2, we have(30)A≪loglogn−1+ε,B≪loglogn−1+ε.Meanwhile, (31)0≤C=∏j=k+1n∏p=1dΦunjp−∏j=1n∏p=1dΦunjp≤1−∏j=1k∏p=1dΦunjp≤∑p=1d∑j=1k1−Φunjp≤∑p=1dk1−Φλnp≤∑p=1dknn1−Φλnp≪kn,from which the lemma follows.Remark 2.
Under the conditions of Theorem2, Lemma 5 also can be proved using Lemma 1 and Lemma 3.Lemma 6.
Suppose thatXnn=1∞ is a standardized normal sequence of d-dimensional random vector satisfying equation (3) of Theorem 1, and(32)ρnplogn⟶0,ρnp,qlogn⟶0.
As∑i=1n1−Φunip⟶τp for τp≥0 and all p=1,…,d, then(33)limk⟶∞P∩i=1kXi≤uki=∏p=1dexp−τp.
Especially, letunp=1/anxp+bn with xp∈R for all p=1,…,d; then,(34)limk⟶∞PMk≤uk=∏p=1dexp−e−xp.Proof.
By Lemma1, equation (32) can be obtained under the condition of equation (31), and the process of proof is similar to that of Lemma 2.Lemma 7.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector satisfying (a) of Theorem 3.(c) There existsγ≥21+δ/1−δ, as n⟶∞(35)1n2∑p=1d∑1≤i<j≤nrijpexpγrijplogj−i⟶0,(36)1n2∑1≤p≠q≤d∑1≤i≤j≤nrijp,qexpγrijp,qlogj−i⟶0,where ε>0.Ifn1−Φunp⟶τp,0≤τp<∞,forp=1,2,…,d, then(37)limk⟶∞PMk≤Uk=∏P=1dexp−τp.Letunp=1/anxp+bn with xp∈R for all p=1,…,d, then (38)limk⟶∞PMk≤uk=∏p=1dexp−e−xp.Proof.
By Lemma1, equation (36) can be obtained using equations (34) and (35), the method of proof is similar to that of Lemma 6Lemma 8.
Letϒnn=1∞ be a sequence of bounded random variables. If(39)Var∑k=1n1kΥk≪logn2loglogn−1+ε,then(40)limn⟶∞1logn∑k=1n1kϒk−Eϒk=0,a.s.Proof.
See Lemma 3.1 in [11].Proof of Theorem.
1. Let αk=I∩i=1nXi≤uki, and(41)Var∑k=1n1kαk=∑k=1n1k2Varαk+2∑1≤k<l≤nCovαk,αlkl≤∑k=1n1k2+2∑1≤k<l≤nCovαk,αlkl.
By Lemma4 and Lemma 5,(42)Covαk,αl=CovI∩i=1kXi≤uki,I∩i=1lXi≤uli≤CovI∩i=1kXi≤uki,I∩i=1lXi≤uli−I∩i=k+1lXi≤uli+CovI∩i=1kXi≤uki,I∩i=k+1lXi≤uli≤2EI∩i=1kXi≤uki−I∩i=k+1lXi≤uli+CovI∩i=1kXi≤uki,I∩i=k+1lXi≤uli≪kl+loglogn−1+ε,Var∑k=1n1kαk≤∑k=1n1k2+2∑1≤k<l≤n1klkl+loglogl−1+ε=∑k=1n1k2+2∑1≤k<l≤n1l2+2∑1≤k<l≤n1klloglogl−1+ε≪logn+2∑l=3n1lloglogl1+ε∑k=1l−11k≪logn+2∑l=3nlogllloglogl1+ε≪logn2loglogn−1+ε.
By Lemma6 and Lemma 8, the proof of the theorem is completed.Proof of Theorem.
2. Under the conditions of Theorem 2, using Lemma 4 and Lemma 5, we have(43)Var∑k=1n1kαk≪logn2loglogn−1+ε.
By Lemma7 and Lemma 8, Theorem 2 is obtained.Proof of Theorem.
3. Write unip=unp+mn∗p−mip where unp=xp/an+bn, p=1,…,d.(44)PanMn∗−bn∗−mn∗≤x=PMn∗=1anx+bn∗+mn∗=PMn∗≤un+mn∗=PMn≤uni.
By Theorem1, it is needed to check that n1−Φλnp is bounded with λnp=min1≤i≤nunip=unp+mn∗p−max1≤i≤nmip. Notice that βnp=max1≤i≤nmip=ologn as n⟶∞, mn∗p≤βnp and anmax1≤i≤nmip−mi∗p≤D, p=1,…,d, and(45)unpmax1≤i≤nmip−mn∗p−12max1≤i≤nmip−mn∗p2=anmax1≤i≤nmip−mn∗p1−log4πlognan+xpan2−max1≤i≤nmip−mn∗p2an≤D1+o1,for large n. Hence,(46)n1−Φλnp∼nλn−1pexp−12unp+mn∗p−max1≤i≤nmip2∼n1−Φunpexp−unpmn∗p−max1≤i≤nmip−12mn∗p−max1≤i≤nmip2≤D′.
n1−Φλnp is bounded and satisfies the conditions of Theorem 1. As unp=xp/an+bn, Theorem 3 can be proved.


### 2.3. Simulation

The almost sure central limit theorems for the maximum of the multivariate normal sequences are complete. Simulating the almost sure convergence for the maxima, which visually describe the theorems, is not performed in previous papers [12, 13, 15, 16, 23–26]. To construct a suitable standardized nonstationary normal sequence is the key to the simulation.LetZ1,Z2,… be an independent random sequence from the standardized normal distribution N0,1. Set Xi=1/iZ1+i−1/iZi, then EXi=0 and VarXi=1, that is, Xi∼N0,1, i=1,2,…. As the correlation coefficient of Xi and Xj, rij=corrXi,Xj=1/ij=j=i+n1/in+i, X1,X2,… be a standardized nonstationary normal random variable sequence. For rij≤1/n, the sequence satisfies equations (3) and (4). When X1,X2,… are one-dimensional variables and λn=an−1x+bn, equation (6) of Theorem 2 holds.Theorem2 is simulated based on the standardized nonstationary normal random variable sequence X1,X2,… by Matlab. N denotes the number of random variable sequence X1,X2,… and Mk=max1≤i≤kXi. an=2logn1/2, bn=an−1/2an−1log4πlogn, especially a1=1, b1=0 and λn=an−1x+bn. We note that 1/logn∑k=1n1/k is greater than 1 and converges to 1, but the rate of convergence is slow. The value 1/logn∑k=1n1/k and the number of the random variables N are displayed in Table 1. The fitting effect gets better with the number of random variable N increasing. The detailed simulation is shown in Figure 1.Table 1
The rate of convergence.
N103104105106107108109Value1.08361.06271.05011.04081.03581.03181.0279Na denotes the number of random variables.Figure 1
Simulating almost sure central limit theorem for one-dimensional random variable sequence.LetYi=−1/iZ1+i−1/iZi, then EYi=0, and VarYi=1, Yi∼N0,1, i=1,2,…. ξi=Xi,Yi is a two-dimensional random vector.(47)rij2=corrYi,Yj=1iji≠j,rij1,2=corrXi,Yj=1ij,i≠j,i−2i,i=j.ξ1,ξ2,… is a standardized nonstationary normal sequence of two-dimensional random vector satisfying equations (3) and (4), and then Theorem 2 holds as ξ1,ξ2,… is a 2-dimensional random vector. The detailed simulation is shown in Figure 2.Figure 2
Simulating almost sure central limit theorem for two-dimensional random variable sequence.


## 2.1. Theorems

Theorem 1.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector, satisfying rijp≤ρj−ip, rijp,q≤ρj−ip,q for i,j=1,2,… and(3)supn≥11≤p≤dρnp<1,supn≥01≤p≠q≤dρnp,q<1,as n⟶∞ and(4)ρnplognloglogn1+ε=O1,ρnp,qlognloglogn1+ε=O1.Let the levelsunip be such that n1−Φλnp is bounded and λnp=min1≤i≤nunip≥clogn1/2 for some c>0. If ∑i=1n1−Φunip⟶τp for some τp≥0 and p=1,…,d, then(5)limn⟶∞1logn∑k=1n1kI∩i=1kXi≤uki=∏p=1dexp−τp,a.s.In particular, letunp=an−1xp+bn,xp∈ℝ for p=1,…,d, and then(6)limn⟶∞1logn∑k=1n1kIMk≤uk=∏p=1dexp−exp−xp,a.s.Theorem 2.
LetXnn=1∞ be a standardized normal d-dimensional random vector sequence satisfying(a)
δ=maxp≠qsuprnpn≥1,suprnp,qn≥0<1(b)
there existsγ≥21+δ/1−δ, such that(7)1n2∑p=1d∑1≤i<j≤nrijpexpγrijplogj−i≪loglogn−1+ε,1n2∑1≤p≠q≤nn∑1≤i≤j≤nrijp,qexpγrijp,qlogj−i≪loglogn−1+ε,where ε>0.Ifn1−Φunp⟶τp,0≤τp<∞,forp=1,2,…,d, then(8)limn⟶∞1logn∑k=1n1kIMk≤Uk=∏p=1dexp−τp,a.s.Letunp=an−1xp+bn, where xp are real numbers for p=1,…,d, and then(9)limn⟶∞1logn∑k=1n1kIMk≤uk=∏p=1dexp−exp−xp,a.s.We shall be concerned with the maximaMn∗=max1≤i≤nYi for a normal sequence Ynn=1∞ given by Yn=Xn+mn where Xnn=1∞ is a standardized normal sequence of d-dimensional vector with each component having a zero mean and unit standard deviation, and mn are added deterministic components. Define mn=mn1,…,mnd, and we shall assume that the constant mnp is such that(10)βnp=max1≤i≤nmip=ologn,p=1,…,d,asn⟶∞.Hence, the mean and the deviation of each component ofYn are mnp and 1, and(11)CovYip,Yjp=rijp,CovYip,Yjq=rijp,q,1≤p≠q≤d.Letmn∗=mn∗1,…,mn∗d be chosen such that mn∗p≤βnp and(12)anmax1≤i≤nmip−mi∗p≤D,p=1,…,d,where D is a constant.Theorem 3.
LetYnn=1∞ be a normal sequence of d-dimensional vector defined above with Xnn=1∞ satisfying the conditions of Theorem 1. Then,(13)limn⟶∞1logn∑k=1n1kIakMk∗−bk∗−mk∗≤x=∏p=1dexp−e−xp,a.s.,where Mn∗=Mn∗1,…,Mn∗d with Mn∗p=max1≤i≤nYip and bn∗=bnId with Id=1,…,1︸d and x=x1,…,xd.


## 2.2. Proofs of the Main Results

We need the following lemmas for the proofs of the main results.Lemma 1.
Letξnn=1∞ and ηnn=1∞ be standardized normal sequences of d-dimensional random vector with rij0p=Covξip,ξjp, rij0p,q=Covξip,ξjq and rij′p=Covηip,ηjp, rij′p,q=Covηip,ηjq. Let ρijp=maxrij0p,rij′p, ρijp,q=maxrij0p,q,rij′p,q and u1,…,un be real vectors. Then,(14)Pξj≤unj,j=1,…,n−Pηj≤unj,j=1,…,n≤K1∑p=1d∑1≤i<j≤nrij0p−rij′p1−ρij2p−1/2exp−uni2p+unj2p21+ρijp+K2∑1≤p≠q≤d∑1≤i≤j≤nrij0p,q−rij′p,q1−ρij2p,q−1/2exp−uni2p+unj2q21+ρijp,q,for constants K1,K2.Proof.
It follows from Theorem 4.2.1 in [22].Lemma 2.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector satisfying equations (3) and (4), and n1−Φλnp be bounded for p=1,…,d, then(15)Sn∗≔∑p=1d∑1≤i<j≤nrijpexp−uni2p+unj2p21+rijp≪loglogn−1+ε,(16)Sn≔∑1≤p≠q≤d∑1≤i≤j≤nrijp,qexp−uni2p+unj2q21+rijp,q≪loglogn−1+ε.Proof.
By equation (3), we have sup1≤i≤j≤n,1≤p≠q≤drijp,q=δ<1. Define θn=eηλnp2 for η<η0=1−δ/21+δ. Split Sn into two parts, then(17)Sn=∑1≤p≠q≤d∑j−i≤θn1≤i≤j≤nrijp,qexp−uni2p+unj2q21+rijp,q+∑1≤p≠q≤d∑j−i>θn1≤i≤j≤nrijp,qexp−uni2p+unj2q21+rijp,q≔Sn1+Sn2.
Notice thatn1−Φλnp is bounded for p=1,…,d, which implies that(18)exp−λn2p2∼C2πλnpn,λnp∼2logn,for some constant C>0, p=1,…,d, and then(19)Sn1≤∑1≤p≠q≤dδ∑j−i≤θn1≤i≤j≤nexp−uni2p+unj2q21+δ≤∑1≤p≠q≤dδθnnexp−λn2p+λn2q21+δ≪∑1≤p≠q≤dn2η−η0logn1/1+δ−η≪loglogn−1+ε,for η0>η.
Now, considerSn2. Let δt=supj≥tρjp,q,1≤p≠q≤d and t=nα. Since n1−Φλnp is bounded as n⟶∞ such that λn2p>clogn for some c>0 and θn=expηλn2p≥expc2ηlogn=nα,α=c2η>0, and then(20)Sn2≤∑1≤p≠q≤d∑j−i>θn1≤i≤j≤nδtexp−λn2p+λn2q21+δt≤∑1≤p≠q≤dδtn2exp−λn2p+λn2q21+δt≪∑1≤p≠q≤dδtn1−Φλnpn1−Φλnqλnpexpλn2p2δtλnqexpλn2q2δt≪∑1≤p≠q≤dδtλnpexpλn2p2δtλnqexpλn2q2δt≪∑1≤p≠q≤dδtλn2pδtλn2qexpλn2p2δtexpλn2q2δt≪∑1≤p≠q≤dδtlogtexp1αδtlogt.
As(21)δtlogtloglogt−1+ε≤supj≥tρjp,qlogjloglogj−1+ε=O1,we have(22)δtlogt≪loglogn−1+ε⟶0,and expλn2p/2δt is bounded. Then,(23)Sn2≪loglogn−1+ε.Lemma 3.
LetXnn=1∞ be a standardized normal d-dimensional random vector sequence satisfying the conditions (a) and (b) of Theorem 2, and then equations (15) and (16) hold.Proof.
The proof of Lemma3 is similar to that of Lemma 2.Lemma 4.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector satisfying Lemma 2 (Lemma 3), then(24)CovI∩i=1kXi≤uki,I∩i=n+1nXi≤uki≪loglogn−1+ε.Proof.
By Lemma1,(25)CovI∩i=1kXi≤uki,I∩i=k+1nXi≤uni=P∩i=1kXi≤uki,∩i=k+1nXi≤uni−P∩i=1kXi≤ukiP∩i=k+1nXi≤uni≤∑p=1d∑i=1k∑j=k+1nrijpexp−uki2p+unj2p21+rijp+∑1≤p≠q≤d∑i=1k∑j=k+1nrijp,qexp−uki2p+unj2q21+rijp,q≔Dn+En.
BothDn and En are of order loglogn−1+ε, but we will only prove En≪loglogn−1+ε. In fact,(26)En=∑1≤p≠q≤d∑i=1k∑j=k+1nrijp,qexp−uki2p+unj2q21+rijp,q≤∑1≤p≠q≤d∑1≤i≤kk+1≤j≤nj−i<θnrijp,qexp−uki2p+unj2q21+rijp,q+∑1≤p≠q≤d∑1≤i≤kk+1≤j≤nj−i<θnrijp,qexp−uki2p+unj2q21+rijp,q≔En1+En2.
We have(27)En1≤∑1≤p≠q≤dδ∑1≤i≤kk+1≤j≤nj−i<θnexp−λk2p+λn2q21+δ≤∑1≤p≠q≤dδkθnexp−λk2p+λn2q21+δ≤∑1≤p≠q≤dδkθnλkpk1/1+δλnqn1/1+δ≤∑1≤p≠q≤dδk1−1/1+δlogk1/21+δn2η−1/1+δlogn1/21+δ−η≤∑1≤p≠q≤dδn−2η0−ηlogn1/1+δ−η≪loglogn−1+ε,En2≤∑1≤p≠q≤d∑1≤i≤k+1≤j≤nj−i≥θnρijp,qexp−uki2p+unj2q21+ρijp,q≤∑1≤p≠q≤dδt∑i=1kexp−uki2p21+δt∑i=1nexp−uni2p21+δt≤∑1≤p≠q≤dδtkexp−λk22exp12λk2δtnexp−λn22exp12λn2δt≪∑1≤p≠q≤dδtlognexpλn2δt≪loglogn−1+ε.Remark 1.
Under the conditions of Theorem2, Lemma 4 can be proved by Lemma 1 and Lemma 3.Lemma 5.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector satisfying Lemma 2 (Lemma 3), then (28)EI∩i=1nXi≤uni−I∩i=k+1nXi≤uni≪kn+loglogn−1+ε,where E denotes the mean.Proof.
(29)EI∩i=1nXi≤uni−I∩i=k+1nXi≤uni=P∩i=1nXi≤uni−P∩i=k+1nXi≤uni≤P∩i=k+1nXi≤uni−∏p=1d∏j=k+1nΦunjp+P∩i=1nXi≤uni−∏p=1d∏j=1nΦunjp+∏p=1d∏j=k+1nΦunjp−∏p=1d∏j=1nΦunjp≜A+B+C.
By Lemma1 and Lemma 2, we have(30)A≪loglogn−1+ε,B≪loglogn−1+ε.Meanwhile, (31)0≤C=∏j=k+1n∏p=1dΦunjp−∏j=1n∏p=1dΦunjp≤1−∏j=1k∏p=1dΦunjp≤∑p=1d∑j=1k1−Φunjp≤∑p=1dk1−Φλnp≤∑p=1dknn1−Φλnp≪kn,from which the lemma follows.Remark 2.
Under the conditions of Theorem2, Lemma 5 also can be proved using Lemma 1 and Lemma 3.Lemma 6.
Suppose thatXnn=1∞ is a standardized normal sequence of d-dimensional random vector satisfying equation (3) of Theorem 1, and(32)ρnplogn⟶0,ρnp,qlogn⟶0.
As∑i=1n1−Φunip⟶τp for τp≥0 and all p=1,…,d, then(33)limk⟶∞P∩i=1kXi≤uki=∏p=1dexp−τp.
Especially, letunp=1/anxp+bn with xp∈R for all p=1,…,d; then,(34)limk⟶∞PMk≤uk=∏p=1dexp−e−xp.Proof.
By Lemma1, equation (32) can be obtained under the condition of equation (31), and the process of proof is similar to that of Lemma 2.Lemma 7.
LetXnn=1∞ be a standardized normal sequence of d-dimensional random vector satisfying (a) of Theorem 3.(c) There existsγ≥21+δ/1−δ, as n⟶∞(35)1n2∑p=1d∑1≤i<j≤nrijpexpγrijplogj−i⟶0,(36)1n2∑1≤p≠q≤d∑1≤i≤j≤nrijp,qexpγrijp,qlogj−i⟶0,where ε>0.Ifn1−Φunp⟶τp,0≤τp<∞,forp=1,2,…,d, then(37)limk⟶∞PMk≤Uk=∏P=1dexp−τp.Letunp=1/anxp+bn with xp∈R for all p=1,…,d, then (38)limk⟶∞PMk≤uk=∏p=1dexp−e−xp.Proof.
By Lemma1, equation (36) can be obtained using equations (34) and (35), the method of proof is similar to that of Lemma 6Lemma 8.
Letϒnn=1∞ be a sequence of bounded random variables. If(39)Var∑k=1n1kΥk≪logn2loglogn−1+ε,then(40)limn⟶∞1logn∑k=1n1kϒk−Eϒk=0,a.s.Proof.
See Lemma 3.1 in [11].Proof of Theorem.
1. Let αk=I∩i=1nXi≤uki, and(41)Var∑k=1n1kαk=∑k=1n1k2Varαk+2∑1≤k<l≤nCovαk,αlkl≤∑k=1n1k2+2∑1≤k<l≤nCovαk,αlkl.
By Lemma4 and Lemma 5,(42)Covαk,αl=CovI∩i=1kXi≤uki,I∩i=1lXi≤uli≤CovI∩i=1kXi≤uki,I∩i=1lXi≤uli−I∩i=k+1lXi≤uli+CovI∩i=1kXi≤uki,I∩i=k+1lXi≤uli≤2EI∩i=1kXi≤uki−I∩i=k+1lXi≤uli+CovI∩i=1kXi≤uki,I∩i=k+1lXi≤uli≪kl+loglogn−1+ε,Var∑k=1n1kαk≤∑k=1n1k2+2∑1≤k<l≤n1klkl+loglogl−1+ε=∑k=1n1k2+2∑1≤k<l≤n1l2+2∑1≤k<l≤n1klloglogl−1+ε≪logn+2∑l=3n1lloglogl1+ε∑k=1l−11k≪logn+2∑l=3nlogllloglogl1+ε≪logn2loglogn−1+ε.
By Lemma6 and Lemma 8, the proof of the theorem is completed.Proof of Theorem.
2. Under the conditions of Theorem 2, using Lemma 4 and Lemma 5, we have(43)Var∑k=1n1kαk≪logn2loglogn−1+ε.
By Lemma7 and Lemma 8, Theorem 2 is obtained.Proof of Theorem.
3. Write unip=unp+mn∗p−mip where unp=xp/an+bn, p=1,…,d.(44)PanMn∗−bn∗−mn∗≤x=PMn∗=1anx+bn∗+mn∗=PMn∗≤un+mn∗=PMn≤uni.
By Theorem1, it is needed to check that n1−Φλnp is bounded with λnp=min1≤i≤nunip=unp+mn∗p−max1≤i≤nmip. Notice that βnp=max1≤i≤nmip=ologn as n⟶∞, mn∗p≤βnp and anmax1≤i≤nmip−mi∗p≤D, p=1,…,d, and(45)unpmax1≤i≤nmip−mn∗p−12max1≤i≤nmip−mn∗p2=anmax1≤i≤nmip−mn∗p1−log4πlognan+xpan2−max1≤i≤nmip−mn∗p2an≤D1+o1,for large n. Hence,(46)n1−Φλnp∼nλn−1pexp−12unp+mn∗p−max1≤i≤nmip2∼n1−Φunpexp−unpmn∗p−max1≤i≤nmip−12mn∗p−max1≤i≤nmip2≤D′.
n1−Φλnp is bounded and satisfies the conditions of Theorem 1. As unp=xp/an+bn, Theorem 3 can be proved.


## 2.3. Simulation

The almost sure central limit theorems for the maximum of the multivariate normal sequences are complete. Simulating the almost sure convergence for the maxima, which visually describe the theorems, is not performed in previous papers [12, 13, 15, 16, 23–26]. To construct a suitable standardized nonstationary normal sequence is the key to the simulation.LetZ1,Z2,… be an independent random sequence from the standardized normal distribution N0,1. Set Xi=1/iZ1+i−1/iZi, then EXi=0 and VarXi=1, that is, Xi∼N0,1, i=1,2,…. As the correlation coefficient of Xi and Xj, rij=corrXi,Xj=1/ij=j=i+n1/in+i, X1,X2,… be a standardized nonstationary normal random variable sequence. For rij≤1/n, the sequence satisfies equations (3) and (4). When X1,X2,… are one-dimensional variables and λn=an−1x+bn, equation (6) of Theorem 2 holds.Theorem2 is simulated based on the standardized nonstationary normal random variable sequence X1,X2,… by Matlab. N denotes the number of random variable sequence X1,X2,… and Mk=max1≤i≤kXi. an=2logn1/2, bn=an−1/2an−1log4πlogn, especially a1=1, b1=0 and λn=an−1x+bn. We note that 1/logn∑k=1n1/k is greater than 1 and converges to 1, but the rate of convergence is slow. The value 1/logn∑k=1n1/k and the number of the random variables N are displayed in Table 1. The fitting effect gets better with the number of random variable N increasing. The detailed simulation is shown in Figure 1.Table 1
The rate of convergence.
N103104105106107108109Value1.08361.06271.05011.04081.03581.03181.0279Na denotes the number of random variables.Figure 1
Simulating almost sure central limit theorem for one-dimensional random variable sequence.LetYi=−1/iZ1+i−1/iZi, then EYi=0, and VarYi=1, Yi∼N0,1, i=1,2,…. ξi=Xi,Yi is a two-dimensional random vector.(47)rij2=corrYi,Yj=1iji≠j,rij1,2=corrXi,Yj=1ij,i≠j,i−2i,i=j.ξ1,ξ2,… is a standardized nonstationary normal sequence of two-dimensional random vector satisfying equations (3) and (4), and then Theorem 2 holds as ξ1,ξ2,… is a 2-dimensional random vector. The detailed simulation is shown in Figure 2.Figure 2
Simulating almost sure central limit theorem for two-dimensional random variable sequence.


## 3. Conclusions

The extreme value theory deals with extreme phenomena which are less likely to occur, but more harmful [1–3]. Almost sure limit theorems, nice behavior of convergence, for the maxima of normal vector sequences are put forward and proved. Figures 1 and 2 display good simulation for two examples and show the rates of the convergence visually, which promote the intuitive understanding about the almost sure central limit theorems for the maximum of nonstationary normal sequences of random vectors.



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*Source: 3945946-2020-05-18.xml*