Almost Sure Asymptotics for Extremes of Non-stationary Gaussian Random Fields?

Zhongquan TAN Yuebao WANG

1 Introduction

The almost sure central limit theorem(ASCLT for short)was first introduced independently by[3]and[17]for the partial sum,and then the concept was started to have applications in many areas.For example,[4–5]showed applications of ASCLTs for occupation measures of the Brownian motion on a compact Riemannian manifold and for diffusions and its application to path energy and eigenvalues of the Laplacian.His work was also followed up in many other applied areas,including condensed matter physics,statistical mechanics,ergodic theory,dynamical systems,occupational health psychology,control and information sciences and rehabilitation counseling and so on.

In its simplest form the ASCLT states that ifX1,X2,···is an independent and identically distributed(i.i.d.for short)sequence of random variables with mean 0 and variance 1,then

whereIis an indicator function and Φ(x)stands for the standard normal distribution function.

Later on,[11]and independently[7]extended this principle by establishing the ASCLT forthe maximaof i.i.d.random variables.They proved that for anyx∈R,

with real sequencesat>0,bt∈,t≥1 and a non-degenerate distributionG(x).[10]and[6]extended(1.1)for weakly dependent stationary Gaussian sequences.We refer to[13]for the non-stationary Gaussian case,[19]for the more general dependent case and[9]for stationary Gaussian fields.The recent extension is the result of[20].

In this paper,we are interested in the similar problems for extremes of non-stationary Gaussian random fields.It is well-known that Gaussian random fields play a very important role in many applied sciences,such as image analysis,atmospheric sciences and geostatistics,among others.Firstly,we introduce some notations and notions of Gaussian random fields.

Denote the set of all positive integers and the set of all non-negative integers byand,respectively.Letdanddbed-dimensional product spaces ofand,respectively,whered≥2.In this paper,we only consider the case ofd=2 since it is notationally the simplest and the results for higher dimensions follow analogous arguments.For i=(i1,i2)and j=(j1,j2),i≤j and i?j meanik≤jk,k=1,2 and(i1?j1,i2?j2),respectively.|i|and n→∞mean(|i1|,|i2|)andnk→∞,k=1,2,respectively.Let In={j∈Z2:1≤ji≤ni,i=1,2}andχEbe the number of elements in E for any subset E of2.Let fork=(k1,k2)andχ0=1.Note thatχk=χIkwhen k∈2.Also,let logn and loglogn denote(logn1,logn2)and(loglogn1,loglogn2),respectively.Let Φ(·)and?(·)denote the standard Gaussian distribution function and its density function,respectively.

Let X={Xn}n≥1be a non-stationary standardized Gaussian random field on2.Letrij=Cov(Xi,Xj)be the covariance functions of the Gaussian random field X={Xn}n≥1.

[14]studied the extremes for non-stationary Gaussian random fields and obtained the following weak convergence result.

Theorem 1.1LetX={Xn}n≥1be a non-stationary standardized Gaussian random field.Assume that the covariance functions rijsatisfy|rij|<ρ|i?j|for some sequence{ρn}n∈N2?{0}such that

andLet the constants{un,i,i≤n}n≥1be such thatfor some constant c>0andThen

For more detailed limit properties of the extremes and their applications for Gaussian random fields,we refer to[8–9,14–15,18].For further results concerning the extremes in Gaussian random fields we refer the readers to[1–2,8–9,14–16].

In this paper,we concentrate on the almost sure limit theorem on extremes of non-stationary Gaussian random fields.We will extend(1.4)to the almost sure version.As a by-product,we find thatstill holds under weaker conditions.

2 Main Results

Now,we state our main results.

Theorem 2.1LetX={Xn}n≥1be a non-stationary standardized Gaussian random field.Assume that the covariance functions rijsatisfy|rij|<ρ|i?j|for some sequencesuch that for some ε>0,

andhold.Let the constants{un,i,i≤n}n≥1be such that χn(1?Φ(λn))isbounded,whereSuppose thatholds.Then

As a special case,we have the following corollary.

Corollary 2.1LetX={Xn}n≥1be a non-stationary standardized Gaussian random field.Assume that the covariance functions rijsatisfy|rij|<ρ|i?j|for some sequencesuch that(2.1)–(2.2)andhold.Let the constants{un}n≥1be such thatThen

where

Further,letandand then

Next,we give a weak convergence result whichis an extension of Theorem A.

Theorem 2.2LetX={Xn}n≥1be a non-stationary standardized Gaussian random field.Assume that the covariance functions rijsatisfy|rij|<ρ|i?j|for some sequencesuch that

In addition,assume that(1.3)and<1hold.Let the constants{un,i,i≤n}n≥1besuch that χn(1?Φ(λn))is bounded,whereSuppose thatτ∈[0,∞)holds.Then(1.4)holds.

Remark 2.1The assertions of Theorems 2.1–2.2 still hold for stationary Gaussian fields with the similar conditions on the correlation functions.Note that even for the stationary case,Theorems 2.1–2.2 are still new results.

Example 2.1(1)The assertions of Theorems 2.1–2.2 still hold for independent Gaussian random fields,andm-dependent Gaussian random fields.

(2)LetZ1be a Gaussian field with mean 0,variance 1 andand thenis a non-stationary Gaussian random field which satisfies the conditions of Theorems 2.1–2.2,whereρncan be chosen as follows:

Using Theorem 2.2,we extend Theorem 6.2.1 of[12]to Gaussian random fields.The obtained result also tells us how to construct a non-stationary Gaussian random field by a stationary Gaussian field.

Corollary 2.2LetY={Xn+mn}n≥1,where{Xn}n≥1is the Gaussian random field satisfying the conditions of Theorem2.2and{mn}n≥1satisfies

and letbe such that

and

asn→∞,whereThen

whereand anand bnare defined as in Corollary2.1.

Using Theorem 2.1,Corollary 2.2 can be extended to the almost sure version.

Corollary 2.3LetY={Xn+mn}n≥1,where{Xn}n≥1is the Gaussian random field satisfying the conditions of Theorem2.1and{mn}n≥1satisfies the conditions of Corollary2.2.Letsatisfy(2.8)–(2.9)and

Then

whereand anand bnare defined as in Corollary2.1.

3 Auxiliary Results

In this section,we state and prove several lemmas which will be used in the proofs of our main results.As usual,an?bnmeansan=O(bn).LetKdenote positive constants whose values may vary from place to place.

The first lemma is the so-called normal comparison lemma which can be found in[12].A simple special form of this theorem is given here.

Lemma 3.1(cf.[12])Letandbe standardized Gaussianrandom fields with covariance functionsandrespectively.LetwhereThen,for constantswe have

where K is some constant,depending only on γ.

The second lemma is an extension of Lemma 3.1 of[10]from random sequences to random fields,which will play a crucial role in the proof of Theorem 2.1.

Lemma 3.2Let{ξk}k≥1,k∈d,d≥2be a sequence of uniformly bounded randomvariables,i.e.,there exists somesuch thata.s.for allk∈d.If

for some ε>0,then

ProofWe only prove the case ofd=2.Setting

andwe have

Thus,by applying the Borel-Cantelli lemma,μnk→0 a.s.Since forν<1,(k+1)ν ?kν→0 as k→∞ifν<1,we have fori=1,2,

aski→∞.Obviously for any given n∈2,there exists an integer k∈2,such that nk

and thus

The proofis complete.

In the following lemmas,we will intensively use the following notations and facts.By the assumption onλn,we haveχn(1?Φ(λn))

for large n.Letwhereηis a positive constant satisfying

Lemma 3.3LetX={Xn}n≥1be a non-stationary standardized Gaussian random fieldwith covariance functions rijsatisfyingLet the constants{un,i,i≤n}n≥1besuch that χn(1?Φ(λn))is bounded,whereThen,we have

with the constant σ1>0.

ProofUsing the facts in(3.1),it is easy to see that

Sinceand 0<δ<1,we haveHence,there exists a constantσ1>0,such that(3.2)holds.

Lemma 3.4Under the conditions of Theorem2.2,we have

asn→∞.Under the conditions of Theorem2.1,we have

ProofDenote the sum in(3.3)and(3.4)bySnand split it into three parts,the first for i0,such thatand hence

whereα=c2η,and

For the first termSn,1,applying the facts in(3.1),we get

Now,using the conditions(1.3)and(2.2),we obtain the desired bounds on the right-hand sides of(3.3)and(3.4),respectively.For the second term,note that

Similarly,applying the facts in(3.1)again,we have

Now,using the condition(2.6)we getSn,2=o(1)as n→∞.Using the condition(2.1),we get

Likewise we can bound the third term.

Lemma 3.5Under the conditions of Theorem2.1,fork,n∈N2such thatk≠nand uk,i≤un,j,we have

ProofSplit the sum into two parts:

For the first term,it follows from the facts in(3.1)that

Sinceand 0<δ<1,we haveHence,there exists a constantσ2>0,such that

As in the proof of Lemma 3.4,splitinto three parts,the first for i

For the first termin view of the facts(3.1),we have

where we have used the condition(2.2)in the last step.

Similarly,taking into account the facts in(3.1),we get

where we have used the condition(2.1)in the last step.Likewise,we can bound the third term.

Lemma 3.6Under the conditions of Theorem2.1,fork,n∈2such thatk≠nand uk,i≤un,j,we have

ProofFor part(i),using Lemmas 3.1 and 3.5,we have

For part(ii),we have

Using Lemmas 3.1 and 3.3–3.4,we know thatRn,1andRn,2are both bounded by

Using the fact thatχn(1?Φ(λn))is bounded,we have

This completes the proof of the lemma.

4 Proof of Main Results

In this section,we give the proofs of our main results.

Proof of Theorem 2.1First,note that conditions(2.1)and(2.2)imply(2.6)and(1.3),respectively,and hence(1.4)holds under the conditions of Theorem 2.1.Then we have

Therefore,it suffices to prove that

Let

Note that|ξk|≤1 for all k∈2.By Lemma 3.2,we only need to show that

Now,we have

Since|ξk|≤1,it follows that

Note that for k ≠l such thatuk,i≤ul,j,

where we have used Lemma 3.6 in the last step.Now,we have

In order to estimate A21,we defineforLet amdenotefor a∈2and m∈Λ.Then,we have

Sincebecomesfor(k,l)∈m,it follows that

for someν>0.For A22,we have

Therefore,

This and(4.2)together establish(4.1).

Proof of Theorem 2.2Let Y={Yn}n≥1be an independent standardized Gaussian random field.It is easy to see that

By Lemmas 3.1 and 3.3–3.4,we have

By Lemma 6.1.1 in[12]and the condition thatthe second sum is alsoo(1).The proofis complete.

Proof of Corollary 2.2LetwhereThen the probability on the left-hand side of(2.10)can be written as

Sincefor sufficiently large n,andit follows that

Thus if it is shown that

the result will follow from Theorem 2.2.To see that(4.3)holds,we note thatas n→∞,

and

uniformly in i≤n.Clearly,we also have

Therefore,according to(2.7),(2.9)and(4.4),we have

sinceby a direct calculation.Hence(4.3)holds and the proof of the corollary is complete.

Proof of Corollary 2.3As in the proof of Corollary 2.2,letwhereThen,by Corollary 2.2 we have

as n→∞.Hence,it suffices to prove that

as n→∞,which will be done by showing that

due to Theorem 2.1,whereBy the definitions ofβnandwe have

for large n.Hence

for large n.Hence(4.5)holds and the proof of the corollary is complete.

AcknowledgementThe authors would like to thank the referees for their careful reading and helpful comments that have helped to improve the quality of the paper.

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