On the extremal behavior of a nonstationary normal random field

ARTICLE IN PRESS

Contents lists available at ScienceDirect

Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference 140 (2010) 3567–3576

0378-37

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/jspi

Short communication

On the extremal behavior of a nonstationary normal random field

Luısa Pereira

University of Beira Interior, Department of Mathematics, Portugal

a r t i c l e i n f o

Article history:

Received 2 December 2007

Received in revised form

16 November 2009

Accepted 27 April 2010Available online 6 May 2010

Keywords:

Random field

Dependence conditions

Nonstationarity

Normal random field

58/$ - see front matter & 2010 Published by

016/j.jspi.2010.04.049

ail address: [email protected]

a b s t r a c t

Let X¼ fXngn�1 be a nonstationary random field satisfying a long range weak

dependence for each coordinate at a time and a local dependence condition that avoids

clustering of exceedances of high values. For these random fields, the probability of no

exceedances of high values can be approximated by expð�tÞ, where t is the limiting

mean number of exceedances. We present a class of nonstationary normal random

fields for which this result can be applied.

& 2010 Published by Elsevier B.V.

1. Introduction

Let X¼ fXngnZ1 be a random field on Zdþ , where Zþ is the set of all positive integers and dZ2. We shall consider the

conditions and results for d=2 since it is notationally simplest and the results for higher dimensions follow analogousarguments. Random fields in two and three dimensions are encountered in a wide range of sciences such as hydrology,agriculture and geology.

For i=(i1,i2) and j=(j1,j2), irj means ikr jk, k=1,2, and n¼ ðn1,n2Þ-‘ means nk-1, k¼ 1,2.For a family of real levels fun,i : irngnZ1 and a subset I of the rectangle of points Rn ¼ f1, . . . ,n1g � f1, . . . ,n2g, we will

denote the event fXirun,i : i 2 Ig by fMnðIÞrug or simply by fMnrug when I=Rn.We say the pair I and J is in SiðlÞ, for each i=1,2, if the distance between PiðIÞ and PiðJÞ is greater or equal to l, where

Pi, i¼ 1,2, denote the cartesian projections. The distance d(I,J) between sets I and J of Z2þ , is the minimum of distances

dði,jÞ ¼maxfjis�jsj, s¼ 1,2g, i 2 I and j 2 J.The dependence structure used here for nonstationary random fields is a coordinatewise-mixing type condition as the

DðunÞ�condition introduced in Leadbetter and Rootzen (1998), which restrict dependence by limitingjPðMnðI1Þru,MnðI2ÞruÞ�PðMnðI1ÞruÞPðMnðI1ÞruÞj with the two indexes sets I1 and I2 being ‘‘separated’’ from eachother by a certain distance along each coordinate direction.

Definition 1.1. Let F be a family of indexes sets in Rn. The nonstationary random field X on Z2þ satisfies the condition

Dðun,iÞ over F if there exist sequences of integer valued constants fknigni Z1, flni

gni Z1, i=1,2, such that, as n¼ ðn1,n2Þ�!‘,we have

ðkn1,kn2Þ�!‘,

kn1ln1

n1,kn2

ln2

n2

� ��!0,

Elsevier B.V.

www.elsevier.com/locate/jspi

dx.doi.org/10.1016/j.jspi.2010.04.049

mailto:[email protected]

ARTICLE IN PRESS

L. Pereira / Journal of Statistical Planning and Inference 140 (2010) 3567–35763568

and ðkn1Dð1Þn,ln1

,kn1kn2

Dð2Þn,ln2Þ�!0, where DðiÞn,lni

, i¼ 1,2, are the components of the mixing coefficient, defined as follows:

Dð1Þn,ln1¼ supjPðMnðI1Þru,MnðI2ÞruÞ�PðMnðI1ÞruÞPðMnðI2ÞruÞj,

where the supremum is taken over pairs of I1 and I2 in S1ðln1Þ \ F ,

Dð2Þn,ln2

¼ supjPðMnðI1Þru,MnðI2ÞruÞ�PðMnðI1ÞruÞPðMnðI2ÞruÞj,

where the supremum is taken over pairs of I1 and I2 in S2ðln2Þ \ F .

This condition was used to guarantee the asymptotic independence for maxima over disjoint rectangles of indexes(Pereira and Ferreira, 2006) which is a fundamental result for extending some results of the extreme value theory ofstationary random fields to nonstationary case.

Proposition 1.1. Suppose that the random field X satisfies the condition D(un,i) over F such that ðI � J4J 2 F Þ ) I 2 F and for

fun,i : irngnZ1 such that

fn1n2 maxfPðXi4un,iÞ : irnggnZ1 is bounded:

If Vr,p ¼ Ir � Jr,p, r¼ 1, . . . ,kn1, p¼ 1, . . . ,kn2

, are disjoint rectangles in F , then, as n-‘,

P\r,p

fMnðVr,pÞrug

!�Yr,p

PðMnðVr,pÞruÞ-0:

Although the maximum may be regarded as the maximum of an approximately independent sequence of submaximathere may be high local dependence (the groups of Xj’s defining a maximum over a rectangle of indexes may be quitedependent) leading to a clustering of high values. In Pereira and Ferreira (2005), in addition to the coordinatewise-mixingcondition, it is restricted the local path behavior with respect to exceedances. It is used the idea of Leadbetter and Rootzen(1998) in combination with Husler (1986) to generalize to the nonstationary case a local dependence condition, D0ðun,iÞ,that avoids clustering of exceedances of un,i.

Definition 1.2. Let Eðun,iÞ denote the family of indexes sets I such that

Xi2I

PðXi4un,iÞr1

kn1kn2

Xirn

PðXi4un,iÞ:

The condition D0ðun,iÞ holds for X if, for each I 2 Eðun,iÞ, we have, as n�!‘,

kn1kn2

Xi,j2I

PðXi4un,i,Xj4un,jÞ-0:

That condition, which bounds the probability of more than one exceedance above the levels un,i in a rectangle with a fewindexes, and the coordinatewise-mixing D(un,i) condition lead to a Poisson approximation for the probability of noexceedances over Rn (Pereira and Ferreira, 2005).

Proposition 1.2. Suppose that the nonstationary random field X satisfies Dðun,iÞ and D0ðun,iÞ over Eðun,iÞ and

fn1n2maxfPðXi4un,iÞ : irnggnZ1 is bounded:

Then,

PðMnrun,iÞ�!n-‘

expð�tÞ, t40,

if and only ifXirn

PðXi4un,iÞ�!n-‘

t:

In Pereira and Ferreira (2006) is discussed the behavior of the maximum when clustering of high values of anonstationary random field is allowed but the local occurrence of two or more crossings of high levels, un,i, is restricted. Foreach one of the eight directions in Z2 the local occurrence of two or more crossings is restricted. These smooth oscillationconditions enabled to compute the asymptotic distribution of the maximum, from the limiting mean number of crossings.It is also proved that only four directions must be inspected since for opposite directions were found the same local pathcrossing behavior and the same limiting mean number of crossings.

In many cases, the coordinatewise-mixing condition is quite difficult to verify. So, it is natural to see if the weakconvergence result mentioned above holds when this condition is dropped and a correlation condition is assumed. The aimof this paper is to establish a correlation condition to ensure that a nonstationary normal random field verifies conditionsD(un,i) and D0ðun,iÞ over Eðun,iÞ, that is, X behaves like an independent normal random field. Normal random fields play animportant role since the specification of their finite-dimensional distributions is simple, they are reasonable models for

ARTICLE IN PRESS

L. Pereira / Journal of Statistical Planning and Inference 140 (2010) 3567–3576 3569

many natural phenomenon, estimation and inference are simple, and the model is specified by expectations andcovariances.

2. Main results

Let X¼ fXngnZ1 be a normal random field on Z2þ , (in general nonstationary) with correlations

ri,j ¼ covðXi,XjÞ, iZ1, jZ1,

and assume, without loss of generality, that, for each i 2 Z2þ ,EðXiÞ ¼ 0 and Var(Xi)=1.

The main tool in the Gaussian extreme value theory, and may be one of the basic important tools of the probabilitytheory, has certainly been the so-called normal comparison theorem. This result, used in the Gaussian case, bounds thedifference between two standardized multidimensional distribution functions by a convenient function of theircovariances. A simple special form of this theorem is given here, for use in next sections.

Lemma 2.1. Let X be a nonstationary normal random field with E(Xn)=0, Var(Xn)=1, nZ1, and d¼ supiajjri,jjo1. Then, for

IDRn,

P\i2I

fXirun,ig

!�Yi2I

PðXirun,iÞ

��rK

Xi,j2I,ir j,iaj

jri,jjexp �

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CA

and K is a constant (depending on d).

We shall be concerned with conditions under which

P\

irn

fXirun,ig

!�!n-‘

expð�tÞ, toþ1: ð2:1Þ

The case with a stationary normal random field and un,i constant in i was treated by Choi (2002). The result depend on aBerman’s type condition on their correlations. Specifically, writing rm=E(XjXj+ m) the extended version of Berman’scondition considered by Choi (2002) was the following:

rn logðn1n2Þ�!n-‘

0: ð2:2Þ

It was shown along the similar lines to Leadbetter et al. (1983, Proof of Theorem 4.3.3) that

jPðMnrunÞ�Fn1n2 ðunÞjrKn1n2

Xj2A

jrjjexp �u2

n

1þjrjj

� ��!n-‘

0, ð2:3Þ

where A¼ fj¼ ðj1,j2Þ : ja0,0r jirni,i¼ 1,2g, F denotes the standard normal distribution function and fungnZ1 is such thatfn1n2ð1�FðunÞÞgnZ1 is bounded. So, (2.2) implies PðMnrunÞ�!

n-‘expð�tÞ for levels un,nZ1, such that n1n2ð1�FðunÞÞ�!

n-‘t.

Remark 2.1. The subsums in (2.3) that involve rð0,j2Þand rðj1 ,0Þ should be treated from assumptions on these correlations

which are excluded from the hypothesis of Choi (2002). It will be also necessary consider the conditions

rðn1 ,0Þ log n1 �!n1-‘

0 and rð0,n2Þlog n2 �!

n2-‘0,

as we could infer from the proofs of the results of nonstationary case, established ahead.

For further related results concerning stationary normal random fields we refer the readers to Piterbarg (1996), Adler(1981) or Berman (1992), and the references therein.

In the following, under a simple condition on the correlation structure of a general nonstationary random field, we giveconditions under which (2.1) holds.

Assume that the correlations ri,j,iZ1,jZ1, of the nonstationary random field X decay to zero at a rate which is not tooslow, that is, satisfy

jri,jjorðji1�j1 j,ji2�j2 jÞ

for some sequence frngn2N20�f0g

such that

rðn1 ,0Þ log n1 �!n1-1

0, rð0,n2Þlog n2 �!

n2-10, ð2:4Þ

and, for n 2 Z2þ ,

rn logðn1n2Þ�!n-‘

0, ð2:5Þ

a generalization of the conditions rn logðn1n2Þ�!n-‘

0, rðn1 ,0Þ logn1 �!n1-1

0, rð0,n2Þlogn2 �!

n2-10, used in the stationary case.

ARTICLE IN PRESS


Our main results concern conditions on the fun,i : irngnZ1 under which X verifies conditions D(un,i) and D0ðun,iÞ overEðun,iÞ. From such results it will then follow simply that if alsoX

i2I

ð1�Fðun,iÞÞ�!n-‘

t,

then (2.1) holds, generalizing the conclusion obtained in the stationary case.The first result treats the case where minnZ1un,i tends to infinity as fast as some multiple of logðn1n2Þ

1=2.

Proposition 2.1. Suppose that the correlations ri,j, iZ1,jZ1, of the normal random field X are such that jri,jjrrðji1�j1 j,ji2�j2jÞ,

where frngn2N20�f0g

satisfies (2.4) and (2.5) and supn2N20�f0gjrnjo1. Let the constants fun,i : irngnZ1 be such that

fP

irnð1�Fðun,iÞÞgnZ1 is bounded and, for some c40, ln ¼minirnun,iZcðlogðn1n2ÞÞ1=2. If, for each I 2 Eðun,iÞ, the sequence

ðkn1kn2Þ1=2

n1n2]fi 2 I : un,iZuðtÞn g

( )nZ1

is bounded, ð2:6Þ

for some fuðtÞn gnZ1 such that

n1n2ð1�FðuðtÞn ÞÞ�!

n-‘t,

then, X verifies the conditions D(un,i) and D0ðun,iÞ over Eðun,iÞ.

The next result generalizes Proposition 2.1 by removing the restriction minirnun,iZclogðn1n2Þ1=2 requiring only that

minirnun,i-1.

Proposition 2.2. Let the correlations ri,j, iZ1,jZ1, of the normal random field X satisfy jri,jjrrðji1�j1j,ji2�j2 jÞ, where frngn2N2

0�f0g

satisfies (2.4) and (2.5) and supn2N20�f0gjrnjo1. Suppose the constants fun,i : irngnZ1 are such that f

Pirnð1�Fðun,iÞÞgnZ1 is

bounded and ln ¼minirnun,i-1. Then X verifies conditions Dðun,iÞ and D0ðun,iÞ over Eðun,iÞ.

Remark 2.2. For practical purposes the condition that jri,jjrrðji1�j1 j,ji2�j2jÞfor i,j 2 N2,iaj, where frngn2N2

0�f0gsatisfies (2.4)

and (2.5) is as useful and as general as is likely to be needed. In fact this condition is rather close to what is necessary forthe maximum of a nonstationary normal random field to behave like that of the associated independent field.

As we are going to see in the next section, it is the correlation condition which makes it possible to prove D(un,i) and

D0ðun,iÞ over Eðun,iÞ.

3. Proofs

The proofs of Propositions 2.1 and 2.2 will be given by means of several lemmas.The next two lemmas enable us to prove that under the conditions of Proposition 2.1, for IDRn,

SnðIÞ ¼X

i,j2I,ir j,iaj

jri,jjexp �

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CA�!

n-‘0:

Lemma 3.1. Let fun,i : irngnZ1 be such that fP

irnð1�Fðun,iÞÞgnZ1 is bounded and ln ¼minirnun,i�!n-‘þ1. Suppose that

the correlations ri,j satisfy jri,jjrd for iaj, where do1 is a constant. Then, for IDRn,

Sð1Þn ðIÞ ¼X

i,j2I,i r j,iajjj1�i1 j�jj2�i2 jrYn

jri,jjexp �

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CA�!

n-‘0,

where Yn ¼ expðZl2nÞ, for any Zo1=4� ð1�dÞ=ð1þdÞ.

Proof. Clearly,

Sð1Þn ðIÞrdX

i,j2I,i r j,iajjj1�i1 j�jj2�i2 jrYn

exp �

1

2ðu2

n,iþu2n,jÞ

1þd

0B@

1CArd

Xi,j2I,i r j,iaj

jj1�i1 j�jj2�i2 jrYn

exp �u2

n,i

1þd

!þexp �

u2n,j

1þd

! !r2dY2

n

Xi2I

exp �u2

n,i

1þd

!

¼ 2dY2n

Xi2I

u�1n,i exp �

u2n,i

2

!

1�Fðun,iÞð1�Fðun,iÞÞun,iexp �

1

2�

1�d1þd

u2n,i

� �,

ARTICLE IN PRESS


giving for a suitable K,

Sð1Þn ðIÞr2dY2nKXi2I

ð1�Fðun,iÞÞun,iexp �1

2�

1�d1þd

u2n,i

� �,

since un,iZln�!n-‘1. So, by attending that f

Pirnð1�Fðun,iÞÞgnZ1 is bounded and Yn ¼ expðZl2

nÞ, for Zo1=4� 1�d=1þd, itfollows that:

Sð1Þn ðIÞr2K1dlnexp �1

2�

1�d1þd

�2Z� �

l2n

� ��!n-‘

0: &

Remark 3.1. In the above lemma no assumption was made about the correlation structure.

Lemma 3.2. Suppose that the correlations ri,j of the normal random field X are such that jri,jjrrðji1�j1 j,ji2�j2 jÞfor iaj, where

frngn2N20�f0g

satisfies (2.4) and (2.5) and supn2N20�f0gjrnjo1. Let fun,i : irngnZ1 be such that f

Pirnð1�Fðun,iÞÞg is bounded

and ln ¼minirnun,iZcðlogðn1n2ÞÞ1=2, for some c40. Then, for IDRn, we have

Sð2Þn ðIÞ ¼X

i,j2I,i r j,iajjj1�i1 j�jj2�i2 j4Yn

jri,jjexp �

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CA�!

n-‘0, ð3:1Þ

where Yn ¼ expðZl2nÞ, Z is a positive constant.

Proof. Split the sum in (3.1) into three parts, the first for io j, the second for i1 ¼ j14i2o j2 and the third for i2 ¼ j24i1o j1.We will denote them by Sð2Þn,iðIÞ, i=1, 2, 3, respectively.

Since

Yn ¼ expðZl2nÞZexpðZc2logðn1n2ÞÞ ¼ ðn1n2Þ

a, a¼ c2Z

and

supjj1�i1 j�jj2�i2 j4Yn

i o j

jri,jjr supjj1�i1 j�jj2�i2 j4 ðn1n2 Þ

ai o j

jri,jjr supjj1�i1 j�jj2�i2 j4 ðn1 n2 Þ

ai o j

jri,jjrðji1�j1 j,ji2�j2 jÞ¼ dð1ÞYn

,

the first term, Sð2Þn,1ðIÞ, is dominated by

Sð2Þn,1ðIÞrdð1ÞYn

Xi,j2I,io j

exp �

1

2u2

n,i

1þdð1ÞYn

�

1

2u2

n,j

1þdð1ÞYn

0B@

1CArdð1ÞYn

Xi2I

exp �

1

2u2

n,i

1þdð1ÞYn

0B@

1CA

0B@

1CA

2

:

Now, define un by 1�FðunÞ ¼ 1=n1n2 and split the sum in braces into two parts, for un,iZun, and un,ioun. We obtain

Sð2Þn,1ðIÞ ¼ dð1ÞYn

Xi2I

un,i Z un

exp �12 u2

n,i

1þdð1ÞYn

!þX

i2Iun,i o un

exp �

1

2u2

n,i

1þdð1ÞYn

0B@

1CA

0B@

1CA

2

rdð1ÞYnu2

nexpðu2ndð1ÞYnÞ

1

un

Xi2I

un,i Z un

exp �u2

n

2

� �þX

i2Iun,i o un

u�1n,i exp �

u2n,i

2

!264

375

2

,

which tends to zero since we have

(i)
expð�u2n=2Þ � Kun=n1n2 andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip
(ii)
un � 2logðn1n2Þ
and then

dð1ÞYnu2

n � dð1ÞYn2logðn1n2Þ ¼

2

adð1ÞYn

logðn1n2Þa¼

2

a supjj1�i1j�jj2�i2 j4 ðn1n2Þ

arðji1�j1 j,ji2�j2jÞ

logðna1na2Þ

r2

a supjj1�i1j�jj2�i2 j4 ðn1n2Þ

arðji1�j1j,ji2�j2 jÞ

logðjj1�i1j � jj2�i2jÞ�!n-‘

0,

1

un

Xi2I

exp �u2

n

2

� �r

1

un

Xirn

exp �u2

n

2

� ��

n1n2

un� K

un

n1n2¼ K ,

ARTICLE IN PRESS


and

Xi2I

1

un,iexp �

u2n,i

2

!rXirn

1

un,iexp �

u2n,i

2

!rXirn

ð1�Fðun,iÞÞrK 0:

Similarly the second term is dominated by

dð2ÞYnu2

nexpðu2ndð2ÞYnÞ

1

un

Xi2I

un,i Z un

exp �u2

n

2

� �þX

i2Iun,i o un

u�1n,i exp �

u2n,i

2

!264

375

2

,

where dð2ÞYn¼ supjj2�i2j4na

2rð0,jj2�i2jÞ

, which also converges to zero, since rð0,n2Þlogn2 �!

n2-10.

Likewise the third term tends to zero. &

Remark 3.2. The correlation condition implies that for IDRn, SnðIÞ�!n-‘

0, when un,i�!n-‘1 in a manner which is not too

slow. This will be used in verifying Dðun,iÞ and D0ðun,iÞ over Eðun,iÞ.

With these lemmas we state now the proof of Proposition 2.1.

Proof of Proposition 2.1. Let I,J 2 F . Denoting the event fXirun,ig by Ai,n, we have that

supI,J2Sðln1

Þ\Sðln2Þ

P\

i2I,j2J

Ai,nAj,n

0@

1A�P

\i2I

Ai,n

!P\j2J

Aj,n

0@

1A

��r sup

I,J2Sðln1Þ\Sðln2

Þ

P\

i2I,j2J

Ai,nAj,n

0@

1A�Y

i2I

P Ai,n

� Yj2J

PðAj,nÞ

��

þ supI,J2Sðln1

Þ\Sðln2Þ

Yi2I

PðAi,nÞ�P\i2I

Ai,n

!��þ sup

I,J2Sðln1Þ\Sðln2

Þ

Yj2j

PðAj,nÞ�P\j2J

Aj,n

0@

1A

��r3SnðRnÞ�!

n-‘0:

Let I 2 Eðun,iÞ satisfying (2.6). Then, we have

kn1kn2

Xi,j2I

PðAi,nAj,nÞrkn1kn2

Xi,j2I

jPðAi,nAj,nÞ�PðAi,nÞPðAj,nÞjþkn1kn2

Xi,j2I

PðAi,nÞPðAj,nÞrkn1kn2

SnðIÞ

þkn1kn2

Xi,j2I

ð1�Fðun,iÞÞð1�Fðun,jÞÞrkn1kn2

SnðIÞþkn1kn2

0B@X

i2I

ð1�Fðun,iÞÞ

1CA

2

rkn1kn2

SnðIÞ

þ1

kn1kn2

Xirn

1�Fðun,i

� Þ

!2

�!n-‘

0,

since

kn1kn2

Sð1Þn ðIÞr2dKlnexp �1�d

2ð1þdÞ�2Z

� �l2

n

� �Xirn

ð1�Fðun,iÞÞ�!n-‘

0

and

kn1kn2

Sð1Þn ðIÞrdð1ÞYnu2

nexpðu2ndð1ÞYnÞ

264ðkn1

kn2Þ1=2]fi 2 I : un,iZung �

1

un�

Kun

n1n2

þðkn1kn2Þ1=2

Xi2I

un,i Z un

u�1n,i exp �

1

2u2

n,i

� �1�Fðun,iÞ

� ð1�Fðun,iÞÞ

375

2

�!n-‘

0: &

We return now to the generalization of Proposition 2.1 by removing the restriction minirnun,iZclogðn1n2Þ1=2 requiring

only that minirnun,i-1.We use the technique developed by Husler (1983), tailored for random fields, in obtaining our result. This involves

grouping the un,i,irn, in rectangles as follows. Write c1 ¼ ln ¼minirnun,i and partition Rn into disjoint subsets J1, . . . ,JL asfollows. Define

J1 ¼ fi : c1run,ir2c1g, d1 ¼maxi2J1

un,i, c2 ¼minfun,i : un,i4d1g,

J2 ¼ fi : c2run,ir2c2g, d2 ¼maxi2J2

un,i, c3 ¼minfun,i : un,i4d2g,

and so on, until a set JL which contains the point i with maxirnun,i. Thus Jk is a nonempty subset of Rn such that theminimum and maximum values of un,i, for i 2 Jk, are ck,dk, respectively, where dkr2ck. Clearly also ckþ142ck for each k.

ARTICLE IN PRESS


Finally, by way of notation write

Pk ¼Xi2Jk

ð1�Fðun,iÞÞ, k¼ 1,2, . . . ,L,

and define

A¼ k : PkZexp �c2

k

4

!,1rkrL

( ): ð3:2Þ

Of course L,ck,dk,Pk and the sets Jk all depend naturally on n, but this dependence is suppressed in the notation whichwill be used without comment.

The next lemma shows that we have to consider only sets I and J which are subsets ofS

k2AJk.

Lemma 3.3. With the above notation write I� ¼ I \ ðS

k2AJkÞ for any IDRn, where A is defined in (3.2). Suppose that

ln ¼minirnun,i-1 andP

irnð1�Fðun,iÞ is bounded. Then

0rP\i2I�fXirun,ig

!�P

\i2I

fXirun,ig

!�!n-‘

0: ð3:3Þ

Proof. The difference in probabilities in (3.3) is dominated by

P[

i2I��I

fXi4un,ig

!rXk=2A

Pk: ð3:4Þ

But if k=2A and k41, since ck42ck�1, we have

Pkoexp �c2

k

4

!ock�1exp �

1

2c2

k�1

� � exp �1

2c2

k�1

� �ck�1

rKlnexp �1

2l2

n

� �ð1�Fðck�1ÞÞrKlnexp �

1

2l2

n

� �Pk�1,

since x expð�x2=2Þ decreases for large x and ð1�Fðck�1ÞÞ is one of the terms contributing to Pk�1.

If 1=2A, P1rexpð�c21=4Þrexpð�l2

n=4Þ, so that

Xk2A

PkrKlnexp �1

2l2

n

� � X1rkr L

Pkþexp �l2

n

4

!�!n-‘

0,

sinceP

1rkrLPk ¼P

irnð1�Fðun,iÞÞ is bounded. Hence (3.3) follows from (3.4). &

Remark 3.3. Since we do not assume any condition on the dependence of Xi, i 2 N20�f0g, Lemma 3.3 holds obviously for

the associated random field of independent variables.

In view of this lemma and Lemma 2.1 it is sufficient, in obtaining, as n-‘,

P\

i2I,j2J

fXirun,i,Xjrun,jg

0@

1A�P

\i2I

fXirun,ig

!P\j2J

fXjrun,jg

0@

1A-0,

to show that

Xi2I� ,j2J�

jri,jjexp�

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CA�!

n-‘0,

where I� and J� are subsets ofS

k2AJk.Since each term of the sum is greater or equal to zero, the supremum on such I� and J� is bounded by

Sn ¼X

i r j r n

i,j2S

k2AJk

jri,jjexp�

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CA, ð3:5Þ

so, it will be shown, by means of the next two lemmas, that Sn�!n-‘

0.

Lemma 3.4. Suppose that the correlations ri,j of the normal random field X satisfy jri,jjrrðji1�j1 j,ji2�j2jÞfor iaj, where

frngn2N20�f0g

is such that rn�!n-‘0,rðn1 ,0Þ �!n1-1

0 and rð0,n2Þ�!

n2-10.

ARTICLE IN PRESS


Suppose that ln ¼minirnun,i�!n-‘1 and f

Pirnð1�Fðun,iÞÞgnZ1 is bounded. Then, for k,m 2 A,krm,

Sð1Þk,m ¼X

i2Jk ,j2Jm ,i r j,iajYn o ji1�j1 j�ji2�j2 jo gn,m

jri,jjexp �

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CArKexp �

l2n

16

!PkPm,

where Yn ¼ expðZl2nÞ, for Z as in Lemma 3.1 and gn,m ¼ expðc

2m

16Þ.

Proof. We partition the sum Sk,m(1) into three subsums, for io j, for i1 ¼ j14i2o j2 and for i2 ¼ j24i1o j1. We will denote

them by Sk,m,i(1) , i=1, 2, 3, respectively. Since

supjj1�i1 j�jj2�i2 j4Yn

i o j

jri,jjr supjj1�i1 j�jj2�i2 j4Yn

i o j

rðji1�j1 j,ji2�j2jÞ¼ dð1ÞYn

,

the first term Sk,m,1(1) is dominated by

dð1ÞYn

Xjj1�i1 j�jj2�i2 j4Yn

i2Jk ,j2Jm ,i o j

exp �

1

2ðu2

n,iþu2n,jÞ

1þdð1ÞYn

0B@

1CA:

Now the exponential term does not exceed

exp �

1

2ðu2

n,iþu2n,jÞ

1þdð1ÞYn

0B@

1CArexp �

1

2ðu2

n,iþu2n,jÞð1�d

ð1ÞYnÞ

� �rexp �

1

2u2

n,i

� �

� exp �c2m

1

2�

1

2dð1ÞYn�

1

2dð1ÞYn

d2k

c2m

! !rexp �

1

2u2

n,i

� �� exp �c2

m

1

2�

5

2dð1ÞYn

� �� ,

since dkr2ckr2cm.

Hence, we have

Sð1Þk,m,1rdð1ÞYn

Xjj1�i1 j�jj2�i2 jo gn,m

i2Jk ,j2Jm ,i o j

exp �1

2u2

n,i

� �� exp �c2

m

1

2�

5

2dð1ÞYn

� ��

rdð1ÞYn

Xjj1�i1 j�jj2�i2 jo gn,m

i2Jk ,j2Jm ,i o j

dk

un,iexp �

1

2u2

n,i

� �� exp �c2

m

1

2�

5

2dð1ÞYn

� ��

rdð1ÞYng2

n,mdk

Xi2Jk

exp �1

2u2

n,i

� �u�1

n,i exp �c2m

1

2�

5

2dð1ÞYn

� ��

rdð1ÞYng2

n,mdkPkexp �c2m

1

2�

5

2dð1ÞYn

� �� rKcmexp �c2

m

1

2�

1

4�

1

8�

5

2dð1ÞYn

� �� PkPmrKexp �

l2n

16

!PkPm,

since PmZexpð�c2m=4Þ, gn,m ¼ expðc2

m=16Þ, dkr2ckr2cm, cmZln and dð1ÞYn�!n-‘

0.

To deal with the second and third sums we define

dð2ÞYn¼ supjj2�i2 j4Yn

rð0,ji2�j2 jÞ, dð3ÞYn

¼ supjj1�i1j4Yn

rðji1�j1j,0Þ,

and note that, for i=1, 2,

Sð1Þk,m,irdðiÞYncmexp �c2

m

1

2�

1

4�

1

8�

5

2dð2ÞYn

� �� PkPmrKiexp �

l2n

16

!PkPm

for some K2,K3, since cmZln, dð2ÞYn�!

n2-10, dð3ÞYn

�!n1-1

0. &

Lemma 3.5. Suppose that the correlations ri,j of the normal random field X are such that jri,jjrrðji1�j1j,ji2�j2 jÞfor iaj, where

frngn2N20�f0g

satisfies (2.4) and (2.5). Let fun,i : irngnZ1 be such that fP

irnð1�Fðun,iÞÞgnZ1 is bounded and

ln ¼minirnun,i-1. Then, for krm, we have

Sð2Þk,m ¼X

i2Jk ,j2Jm ,i r j,iajgn,m o ji1�j1 j�ji2�j2 jr n1 n2

jri,jjexp �

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CArbnPkPm, ð3:6Þ

where gn,m ¼ expðc2m=16Þ and bn-0, bn depending only on n.

ARTICLE IN PRESS


Proof. Split the sum in (3.6) into partial sums, S(2)k,m,i, i=1, 2, 3, the first for io j, the second for i1 ¼ j14i2o j2 and the third

for i2 ¼ j24i1o j1. Since

supjj1�i1 j�jj2�i2 j4 gn,m

i o j

jri,jjr supjj1�i1 j�jj2�i2 j4 gn,m

i o j

rðji1�j1j,ji2�j2jÞ¼ dð1Þgn,m

,

the first sum, Sk,m,1(2) , is dominated by

dð1Þgn,m

Xjj1�i1 j�jj2�i2 jr n1n2

i2Jk ,j2Jm ,i o j

exp �

1

2u2

n,iþu2n,j

�1þdð1Þgn,m

0B@

1CArdð1Þgn,m

Xjj1�i1 j�jj2�i2 jr n1 n2

i2Jk ,j2Jm ,i o j

exp �1

2ðu2

n,iþu2n,jÞð1�d

ð1Þgn,mÞ

� �

rdð1Þgn,m

Yl ¼ k,m

dl

Xi2Jl

expu2

n,i

2dð1Þgn,m

! exp �u2

n,i

2

!

un,i

0BBBB@

1CCCCArdð1Þgn,m

dkdmPkPmexp1

2ðd2

kþd2mÞd

ð1Þgn,m

� �

rKdð1Þgn,md2

mPkPmexpðd2mdð1Þgn,mÞ:

Now

dð1Þgn,md2

mr4dð1Þgn,mc2

m ¼ 64dð1Þgn,mloggn,m,

where gn,m ¼ expðc2m=16ÞZexpðl2

n=16Þ�!n-1

0. Similarly the second and third sums are dominated by

Sð2Þk,m,2rdð2Þgn,md2

m expðd2mdð2Þgn,mÞPkPm and Sð2Þk,m,3rdð3Þgn,m

d2mexpðd2

mdð3Þgn,mÞPkPm,

where dð2Þgn,m¼ supji2�j2j4gn,m

rð0,ji2�j2jÞand dð3Þgn,m

¼ supji1�j1j4gn,mrðji1�j1 j,0Þ

, from which the result follows. &

By collecting these lemmas we can now show Proposition 2.2.

Proof of Proposition 2.2. Letting A defined as in (3.2) we see from Lemmas 3.1, 3.4, 3.5 that

Sn ¼X

i r j r n

i,j2S

k2AJk

jri,jjexp�

1

2ðu2

n,iþu2n,jÞ

1þjri,jj

0B@

1CAren

Xk,m2A

PkPmþoð1Þ,

where en-0 as n-‘. ButP

k,m2APkPmrfP

irnð1�Fðun,iÞÞg2 which is bounded, so Sn�!n-‘

0. It thus follows conditionsDðun,iÞ and D0ðun,iÞ over Eðun,iÞ. &

Those results about the asymptotic behavior of the maximum of a nonstationary normal random field can be applied tosimple forms of nonstationarity like periodic random fields. However, in this case we could consider slightly weakercovariance conditions (Pereira, 2008).

4. Example

The main results may be used to give the asymptotic distribution of the maximum in many cases of interest.Let us consider an isotropic standardized normal random field, fYngnZ1, that is EðYsÞ ¼ 0,VarðYsÞ ¼ 1 and CovðYs,YtÞ

depends alone on distance Jt�sJ, where JUJ denotes the Euclidean norm. Suppose that the correlation function of fYngnZ1

is given by

CovðYs,YtÞ ¼ expð�Jt�sJ2Þ:

From fYngnZ1 we shall define two nonstationary and isotropic random fields whose correlations decay to zero at a ratewhich is not too slow, that is, jri,jjorðji1�j1j,ji2�j2jÞ

, for some sequence frngn2N20�f0g

verifying (2.4) and (2.5).Let fXngnZ1 be such that

Xs ¼msþYs,

where ms is a real valued function. Its covariance function and expectation are

CovðXs,XtÞ ¼ CovðYs,YtÞ, EðXsÞ ¼ms,

and

jCovðYs,YtÞj ¼ expð�½ðt1�s1Þ2þðt2�s2Þ

2�Þorðjt1�s1j,jt2�s2jÞ

,

ARTICLE IN PRESS


with

rðn1 ,n2Þ¼

expð�n1n2Þ, n1,n2 2 N,

expð�n1Þ, n1 2 N4n2 ¼ 0,

expð�n2Þ, n1 ¼ 04n2 2N,

8><>:

which verifies the conditions (2.4) and (2.5).Now, consider the normal random field

Ws ¼ ssYsþZs,

where ss is a real valued function such that, for each s 2 Z2þ , jssjrK , K is a constant, Zs are Gaussian random variables

independent of Ys, with covariance function rðJt�sJÞ ¼ expð�Jt�sJ2Þ. The variance and covariance function of fWsgsZ1 are

VarðWsÞ ¼ s2s , covðWt,WsÞ ¼ ðssstþ1ÞcovðYt,YsÞÞoðK2þ1Þrðjt1�s1j,jt2�s2 jÞ

,

and frngn2N20�f0g

verifies (2.4) and (2.5).

Acknowledgements

We would like to thank the referee’s careful reading of the manuscript which has resulted in improvements to the finalform of this paper. Work partially supported by FCT/POCI2010/FEDER (EVT Project).

References

Adler, R.A., 1981. The Geometry of Random Fields. John Wiley, New York.Berman, S., 1992. Sojourns and Extremes of Stochastic Processes. Taylor & Francis Ltd, USA.Choi, H., 2002. Central limit theory and extremes of random fields. Ph.D. Dissertation in University of North Carolina at Chapel Hill.Husler, J., 1986. Extremes of nonstationary random sequences. Journal of Applied Probability 23, 937–950.Husler, J., 1983. Asymptotic approximation of crossings probabilities of random sequences. Wahrscheinlichkeitstheorie und verwandte Gebiete 63,

257–270.Leadbetter, M.R., Lindgren, G., Rootzen, H., 1983. Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Leadbetter, S., Rootzen, H., 1998. On Extreme Values in Stationary Random Fields. In: Stochastic Processes and Related Topics. Trends in Mathematics.

Birkhauser, Boston, pp. 275–285.Pereira, L., Ferreira, H., 2005. Extremes of quasi-independent random fields and clustering of high values. In: Proceedings of Eighth WSEAS International

Conference on Applied Mathematics.Pereira, L., Ferreira, H., 2006. Limiting crossing probabilities of random fields. Journal of Applied Probability 3, 884–891.Pereira, L., 2008. On the maximum of a periodic normal random field. In: Proceedings of the XV Congress of Portuguese Society of Statistics, pp. 415–424.Piterbarg, V.I., 1996. Asymptotics Methods in Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs, vol. 48. American

Mathematical Society.

Documents

On the extremal behavior of a nonstationary normal random field