Extremes 5, 181±212, 2002 # 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields JEAN-MARC AZAIÈS* Laboratoire de Statistique et ProbabiliteÂs, UMR CNRS C5583 Universite Paul-Sabatier, 31062 Toulouse, France E-mail: [email protected] CEÂLINE DELMAS Laboratoire de Statistique et ProbabiliteÂs, UMR CNRS C5583 Universite Paul-Sabatier, 31062 Toulouse, France E-mail: [email protected] [Received February 25, 2002; Revised October 30, 2002; Accepted November 1, 2002] Abstract. Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random ®eld with unit variance on a compact subset S of RN . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian ®eld above an high level inside S and on the border qS. Depending on the geometry of the border we give up to N ‡ 1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown. Key words. asymptotic expansions, Gaussian ®elds, maxima of random ®elds, rice method AMS 2000 Subject Classi®cation.

1.

PrimaryÐ60G15 60G70 SecondaryÐ60E99

Introduction

Let X ˆ fX…t†; t [ S  RN g with S compact, be a Gaussian random ®eld, centered with unit variance. Set M…S† ˆ maxfX…t† : t [ Sg and G…u† ˆ P‰M…S†  uŠ. Many statistical problems can be reduced to the evaluation of G…u† as u ? ‡ ?. It has, for example, direct applications to image processing (see Worsley et al., 1992, and Adler, 2000, for more references). It also occurs in the calculation of the limit distribution of test statistics (see for example, Davies, 1977; Sun, 1991; Piterbarg and Tyurin, 1993; Park and Sun, 1998; Dacunha-Castelle and Gassiat, 1997, 1999). When N ˆ 1 there exists only a few cases of Gaussian processes for which the distribution G…u† is known (see AzaõÈs and Wschebor, 2000, for references and a formula based on the Rice series that converges for some processes). For the other cases, only approximate results are available. In the present paper, we are interested in random ®elds …N > 1†. For this case no exact result is known. Since the initiating work of Belyaev and Piterbarg (1972a, b) and Adler *Corresponding author.

182

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(1981), the ®rst term of the expansion of G…u† as u ? ‡ ? is known for smooth ®elds satisfying some conditions. A rather longer expansion is known only in some particular cases (Piterbarg, 1996a; Sun, 1993; Siegmund and Worsley, 1995). A lot of work has been devoted to the evaluation of the expectation of the differential topology (DT) characteristic and the Euler characteristic of the excursion set Au …X; S† ˆ ft [ S : X…t†  ug (see Worsley, 1994, 1995, 1997, for Gaussian and more general ®elds). A general conjecture is that the expectation of the Euler characteristic gives the behavior of G…u† but this is true only in some particular cases, see Takemura and Kuriki (1999). See the general review by Adler (2000) for a detailed review of this subject. Mainly four methods are used.  The tube formula (Sun, 1993; Takemura and Kuriki, 1999) which is very ef®cient especially for processes with a ®nite Karhunen-LoeÁve expansion: X…t† ˆ

n X iˆ1

xi fi …t†;

where the xi s are standard normal variables and the fi s deterministic functions.  The equivalence method. Piterbarg (1996b) shows that the asymptotic distribution of a Gaussian ®eld with var…X0 …t†† ˆ Id is equivalent to that of an isotropic process. He then uses Hadwiger theorem, that describes additive functionals on convex sets that are invariant by isometry.  The Euler Characteristic method already mentioned.  The Rice method based on the moment of local maxima initiated by Kac (1943) and Rice (1944±1945); see also Cramer and Leadbetter (1967); Wschebor (1985) and AzaõÈs and Wschebor (1997) for random processes and extended to random ®elds by Brillinger (1972); Adler (1981); Piterbarg (1996b); Konakov and Mammen (1997); Konakov and Piterbarg (1995, 1997). In the present paper, we use the Rice method. Our originality is to take into account ``border maxima'' that allows us to get an expansion for G…u† with several terms depending on N and on the geometry of qS in a rather general setting including nonhomogeneous ®elds. The organization of the paper is as follows. In Section 2, we describe the Rice method and the general framework of our results. In Section 3, we give our main results. First we give, in Theorem 1, an asymptotic expansion for the expectation of the number of local maxima of a Gaussian random ®eld above an high level. This result extends Hasofer's one (1976) and Theorem 6.3.1 of Adler (1981) to a much more accurate form. It was presented in Delmas (1998) without its proof. We give this proof in Section 5. This result has an intrinsic interest (see, for example, Adler, 2000, for the link with the expectation of the DT characteristic of the excursion set Au …X; S† and other related results), but it is also the ®rst step in the derivation of our other results. In Theorem 2, we give an asymptotic expansion for G…u† when the Gaussian ®eld X is ``without boundary''. In our point of view it extends

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

183

the result by Sun (1993) though the hypothesis are not exactly comparable. Then we give some asymptotic results for G…u† when S is with regular boundary (Theorem 3.3) and when S is a product of intervals (Theorem 3.4). In Section 4, we compare our asymptotic expansion with classical bound and Monte-Carlo evaluation on a model of sea waves. In Section 5, we give the proofs of our main results. Notations:  lN is the Lebesgue measure on RN and s is the surface measure on some manifold embedded in RN .  m is the set of N6N symmetric matrices, n  m (resp. p) is the set of negative ~ (resp. positive) de®nite matrices. For A [ m, det…A† will denote jdet…A†jIfA [ ng ; MT is the transpose of the matrix M.  j is the standard Gaussian density and F its distribution function. p…X1 ;...;Xn † …x1 ; . . . ; xn † denotes, when it exists, the density of the random variables …X1 ; . . . ; Xn † at point …x1 ; . . . ; xn †. In the same way px1 …x1 =X2 ˆ x2 ; . . . ; Xn ˆ xn † is the density of X1 conditionally to X2 ˆ x2 ; . . . ; Xn ˆ xn .  The notation (const) means some constant that is not needed to be denoted by a precise symbol. Its value may vary from one occurrence to another.  The relation f …u†  …const†6g…u† exp… du2 † for some d > 0 and u suf®ciently large will be denoted: f …u† ˆ oe …g…u††.  var…Z† is the variance±covariance matrix of the random vector Z.  fp …N† is the set of the parts of f1; 2; . . . ; Ng of size p and f…N† the set of all the parts of f1; 2; . . . ; Ng. Let I and J [ fp …N† and M an N6N matrix, we denote DI;J …M† for det …MI;J † and jMj for det(M). I ˆ f1; . . . ; Ng n I and J ˆ f1; . . . ; Ng n J. sI is the permutation of …1; 2; . . . ; N† such that sI j…1;2;...;p† (resp. sI j…p ‡ 1;...;N† ) is an increasing one-to-one mapping from …1; 2; . . . ; p† on I (resp. from …p ‡ 1; . . . ; N† on I). We note e…p† the signature of a permutation p of …1; 2; . . . ; N†. We use the convention e2 …s † ˆ e2 …sf1;2;...;Ng † ˆ 1 and D; …M† ˆ 1.  Xi …t† is qX…t†=qti and Xij …t†; q2 X…t†=qti qtj .  In the sequel, except when mentioned, S is a subset of RN relatively compact with zero Lebesgue measure boundary. S_ denotes the interior of S and S its closure. When it exists, the number of local maxima of a Gaussian ®eld X above the level u in S: MuX …S† is de®ned by MuX …S† ˆ #ft [ S : X…t†  u; X0 …t† ˆ 0; X00 …t† [ ng.

2.

The Rice method

Rice's famous formula for the intensity of level upcrossings by a trajectory of a random process (Rice, 1944±1945) are at the origin of the so-called Rice method. To de®ne the Rice method we could say that it consists in the derivation of some information on the distribution of the maximum of a random ®eld on a set S thanks to some moments of random variables that are solutions of systems of random equations in a set A. Thus, for example, when N ˆ 1 and S ˆ ‰0; LŠ, accurate evaluation of the function G…u† can be

184

AZAIÈS AND DELMAS

obtained from the factorial moments of the random variable UuX …‰0; LŠ† (that is the number of upcrossings of the level u on ‰0; LŠ by the Gaussian random process X): UuX …‰0; LŠ† ˆ #ft [ ‰0; LŠ : X…t† ˆ u; X0 …t† > 0g (see AzaõÈs and Wschebor, 1997). Generally speaking we note that the Rice method is more accurate than other methods for the evaluation of the function G…u†. Now we shall describe roughly the Rice method used in the next sections to derive our results. Assume that X is a Gaussian ®eld that satis®es the H1 a hypothesis: H1 a.

 The paths of X are, with probability one, of class c 2 on S.

Suppose, to simplify, that we know that, with probability one, the maximum of X is not attained on the boundary of S then _  1Š: P‰M…S†  uŠ ˆ P‰MuX …S† Now the following inequality is easy: _ E‰MuX …S†Š

X _ _ E‰MuX …S†…M u …S† 2

1†Š

_  1Š  E‰MuX …S†Š: _  P‰MuX …S†

…1†

_ by a soThus we will work towards three directions. In a ®rst step, we evaluate E‰MuX …S†Š _ called Rice formula (see Proposition 1) and study the asymptotic behavior of E‰MuX …S†Š as u ? ‡ ?. This is the subject of Theorem 1. In a second step, we evaluate X _ _ E‰MuX …S†…M 1†Š and show that it is negligible. In a third step, we eventually have to u …S† study the ®eld on the boundary qS, see Theorems 3 and 4. Proposition 1: Assume that X is a Gaussian centered ®eld on S that satis®es H1 a and the following H1 b and c hypothesis:  H1 b. The distribution of …X…t†; X0 …t†; X00 …t†† is non-degenerate Vt [ S.  H1 c. The moduli of continuity oij of the Xij on S satisfy h i Ve > 0 P max oij …h† > e ˆ o…hN † when h;0: i; j

Then _ ˆ E‰MuX …S†Š

Z Z S

u

‡?

Z n

jdet x00 jpX…t†;X0 …t†;X00 …t† …x; 0; x00 † dx00 dx dt:

Remarks:  The H1 c hypothesis is met, for example, as soon as the paths of Xij are locally HoÈlder with probability one for all i; j ˆ 1; N. _ ˆ E‰MuX …S†Š ˆ E‰MuX …S†Š.   Under the hypothesis of Proposition 1 we have E‰MuX …S†Š

185

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

Proposition 2: Assume that X is a Gaussian centered ®eld on S that satis®es H1 a and b and the following H2 a hypothesis: _ …X…s†; X…t†; X0 …s†; X0 …t†; X00 …s†; X00 …t†† has a non-degenerate distribuH2 a. Vt 6ˆ s [ S; tion. Then X _ _ E‰MuX …S†…M u …S†

Z Z 1†Š 

S2

Z Z ds dt

…u; ‡ ?†2

dx dypX…s†;X…t†;X0 …s†;X0 …t† …x; y; 0; 0†

f 00 …t††=X…s† ˆ x; X…t† ˆ y; X0 …s† ˆ 0; X0 …t† ˆ 0Š: ~ 00 …s†† det…X 6E‰det…X X _ _ Proposition 3 below gives the asymptotic behavior of E‰MuX …S†…M u …S†

…2†

1†Š as u ? ‡ ?.

Proposition 3: Assume that X is a Gaussian centered ®eld with unit variance on S that satis®es the following H2 b and c hypothesis:  H2 b. X has almost surely C3 sample paths on S. 2 0 0 00  H2 c. Vt ˆ 6 s [ S …X…s†; X…t†; X …s†; X …t†; X …s†; X00 …t†† has a non-degenerate distribution. Then as u ? ‡ ? X _ _ E‰MuX …S†…M u …S†

1†Š ˆ oe …j…u††:

…3†

X _ _ Proposition 3 implies that E‰MuX …S†…M 1†Š is negligible as u ? ‡ ?. As a u …S† X _ consequence the expansion of E‰Mu …S†Š gives the expansion of P‰M…S†  uŠ.

Proofs of the proposition above: They are based upon the following proposition that gives us a tool to count the number of local maxima of a Gaussian random ®eld above a _ level u in S. Proposition 4: Assume that X is a Gaussian centered ®eld on S that satis®es H1 a and b. Denote by A the subset ft [ S : X…t† > u; X00 …t† [ ng then with probability one _ ˆ lim MuX …S†

e?0

Z S

de …X0 …t††IA …X…t†; X00 …t††jdet X00 …t†jdt;

where de …x† ˆ 1=l…Se †ISe …x† with Se the ball with radius e. Remark: Under the hypothesis of Proposition 4 we have with probability one that _ ˆ MuX …S† ˆ MuX …S†.  MuX …S†

186

AZAIÈS AND DELMAS

Proposition 4 is an easy consequence of Theorem 5.1.1 of Adler (1981), Proposition 1 is a slight re®nement of Theorem 6.1.1 of Adler (1981) and Proposition 2 is a consequence of Fatou's Lemma and Proposition 4. Proposition 3 is an immediate consequence of a Piterbarg's result (Piterbarg, 1996b; see also AzaõÈs and Delmas, 2001). &

3.

Main results

This section presents the main results of the paper. The ®rst theorem gives an asymptotic _ Hasofer's result (1976) and Theorem 6.3.1 of expansion for the expectation of MuX …S†. _ in the case Adler (1981) only give the ®rst term of the asymptotic expansion of E‰MuX …S†Š of a Gaussian stationary random ®eld. We derive here a more general and accurate form _ under mild conditions that gives a …‰N=2Š ‡ 1†-term approximation formula to E‰MuX …S†Š on the Gaussian random ®eld. Theorem 1: If X is a Gaussian centered ®eld with unit variance on S that satis®es H1 a, b and c then, _ ˆ l…S† E‰MuX …S†Š

‰N=2Š X jˆ0

k2j c2j …u† ‡ oe …j…u††

as

u ! ‡ ?;

with 1 c2j …u† ˆ 1 ‡ j …N ‡ 1†=2 2 p

Z

‡?

u2 2

x…N

1

2j†=2

e

x

dx:

a. When L ˆ varX0 …t† does not depend on t we have   j 1=2 N …2j†! : k2j ˆ … 1† det…L† 2j 2j j! b. In the general case Z X 1 1=2 e…sI †e…sK † det…L…t†† DI; K …L…t††gI;K …t† dt; k2j :ˆ l…S† S I;K [ f …N† 2j

where L…t† :ˆ var…X0 …t††; gI;K …t† :ˆ

X p [ s2j

e…p†

j XY s [ Qj q ˆ 1

vs…2q

1†;pos…2q

1†;s…2q†;pos…2q† ;

…4†

187

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

where Qj is a subset of the symmetric group of permutations s2j that allows to partition 2j elements into j different pairs: …s…2k ‡ 1†; s…2k ‡ 2††; k ˆ 0; j 1: …#Qj ˆ …2j†!=j!2j †. va;b;Z;x ˆ cov‰Xia kb …t†; XiZ kx …t†=X…t†; X0 …t†Š I ˆ …i1 ; . . . ; i2j †;

K ˆ …k1 ; . . . ; k2j †:

We take the convention that g …t† ˆ 1. Fields ``without boundary'': Suppose that we know that the maximum cannot be attained on the boundary of S. In that case Theorem 1 gives the asymptotic behavior of G…u†. This is the case, for example, if the ®eld can be viewed, after reparametrization, as a ®eld on a manifold without boundary. This is the object of Theorem 2. Theorem 2:

Assume that

 fY…t†; t [ ug is a Gaussian ®eld de®ned on a compact smooth manifold u of dimension N without boundary.  Y is centered with unit variance and differentiable sample paths.  There exists a unique ( for simplicity) mapping C : u ? S  RN ; S is supposed relatively compact with zero Lebesgue's measure boundary.  The ®eld X ˆ Y  C 1 can be extended by continuity to S and satis®es H1 b and H2 b and the following H2 a0 hypothesis: …X…t†; X…t0 †; X0 …t†; X0 …t0 †; X00 …t†; X00 …t0 †† C 1 …t† 6ˆ C 1 …t0 †.

is

non-degenerate

Vt; t0 [ S

such

that

Then, _ ‡ oe …j…u†† P‰M…S†  uŠ ˆ E‰MuX …S†Š ˆ l…S†

‰N=2Š X jˆ0

k2j c2j …u† ‡ oe …j…u†† as u ? ‡ ?;

…5† …6†

where the k2j are the same as those given in Theorem 1. Remarks:  The ®rst equality (5) is due to Piterbarg (1996b). The second (6) is new. It gives a …‰N=2Š ‡ 1†-term approximation formula to P‰M…S†  uŠ and could be compared to Sun's results (1993). However, Sun's result requires the strong condition that the Gaussian random ®eld admits a ®nite Karhunen-LoeÁve expansion. In the general case of an in®nite Karhunen-LoeÁve expansion, Sun (1993) could only derive a two-term approximation formula.

188

AZAIÈS AND DELMAS

 S is not supposed to be open. Theorem 2 can be applied, for example, to processes de®ned on sphere or torus of any dimensions, using classical parametrizations.  S cannot be supposed to be compact because even in the simplest case where u is the circle, there exists no compact parametrization.  The mapping C 1 : S ? u can be de®ned by the Karhunen-LoeÁve expansion P? X…t† ˆ l ˆ 1 Cl 1 …t†xl (see Adler, 1990 and Sun, 1993). Theorem 3: Assume that S is convex and compact with a smooth boundary qS. Assume that X is a Gaussian centered ®eld with unit variance on S that satis®es H2 b and c and such that var…X0 …t†† is constant equal to L. Then, denoting X~ the restriction of X to qS, as u ? ‡ ?: ~

P‰M…S†  uŠ ˆ E‰MuX …S†Š ‡ E‰RXu …qS†Š ‡ oe …j…u†† 1 u ~ ˆ E‰MuX …S†Š ‡ E‰MuX …qS†Š ‡ o…uN 2 e 2 † 2 1=2 u l…S†…det…L†† ˆ uN 1 e 2 N‡1 …2p† 2 Z 1 u 1=2 N 2 2 ‡u e …det…Pt …L††† ds…t†…1 ‡ o…1††; N 2…2p† 2 qS

…7†

2

…8†

2

2

…9† ~

where Pt …L† is the projection of L on the space tangent to qS at point t [ qS and RXu …qS† is de®ned below in (12). When N ˆ 2 and S is the unit ball we have a better remainder in the expansion: Corollary 1: Under the assumptions and the notations of Theorem 3, when N ˆ 2 and S is the unit ball, we have as u ? ‡ ?: u Z  …det…L††1=2 e 2 2p p p ‡ P‰M…S†  uŠ ˆ ue vT …y†Lv…y† dy 4p 0 2 2p Z 2p 1=2  …det…L†† F…u† dy ‡ oe …j…u†† ‡ T 2p 0 v …y†Lv…y† 2

u2 2

where vT …y† ˆ …

…10†

sin y; cos y†.

In both cases, as u ? ‡ ?: ~

P‰M…S†  uŠ ˆ E‰MuX …S†Š ‡ E‰RXu …qS†Š ‡ oe …j…u††;

…11†

~

where RXu …qS† is the number of ``border maxima'' de®ned by ~ ~ RXu …qS† ˆ #ft [ qS : X…t† > u; n…t† ? X0 …t†  0; t is a local maximum for Xg;

…12†

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ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

where n…t† is the unit normal vector pointing outwards. This can be interpreted using Morse's theorem (Adler, 1981) that states that the Euler characteristic w…Au …X; S†† of the excursion set above the level u is given by

a:s:

w…Au …X; S†† ˆ … 1†

N

N X

k

… 1† #ft [ S : X…t† > u; X0 …t† ˆ 0; index…X00 …t†† ˆ kg

kˆ0

‡ … 1†

N

1

N X1

k … 1† #ft [ qS : X…t† > u; n…t† ? X0 …t†  0; X~0 …t†

kˆ0 ~00

ˆ 0; index…X …t†† ˆ kg;

…13†

where the index is the number of negative eigenvalues. The key point is that, as u ? ‡ ?, as shown in the proof of Theorem 1, up to a term which is oe …j…u††, critical points are all local maxima and we get ~

E‰w…Au …X; S††Š ˆ E‰MuX …S†Š ‡ E‰RXu …qS†Š ‡ oe …j…u††: This explains why the expected Euler characteristic gives an accurate approximation of P‰M…S†  uŠ. QN Theorem 4: Assume that S is a product of ®nite closed intervals S ˆ i ˆ 1 ‰ai ; bi Š. Assume that X is a Gaussian centered ®eld with unit variance on S that satis®es H2 b and c 0 ~ and such that var…X Q …t†† ˆ …const† Q ˆ L. Let I  f1; . . . ; Ng then, denoting XI the restriction of X to i [ I ‰ai ; bi Š6 i [ Ifai g (the choice of fai g instead of fbi g is arbitrary), as u ? ‡ ?: P‰M…S†  uŠ ˆ

"

N X X

E

n ˆ 1 I [ fn …N†

ˆ

‰n=2Š N X X nˆ1 jˆ0

~ MuXI

0

cn2j …u†@

Y Y ‰ai ; bi Š6 fai g i[I

X

Y

I [ fn …N†

i[I

where

I k2j

ˆ … 1†

j

1=2 DI;I …L†

i [ I

  N …2j†! : 2j 2j j!

jbi

ai j

!

!#  ‡ F…u† ‡ oe …j…u††

…14†

1

I A k2j

 ‡ F…u† ‡ oe …j…u††;

…15†

190

AZAIÈS AND DELMAS

As a consequence of Theorem 4, when S is the rectangle ‰a1 ; b1 Š6‰a2 ; b2 Š we obtain that: P‰M…S†  uŠ ˆ ue

u2 2

jb1 "

‡e

u2 2

a1 jjb2

a2 j…det…L††1=2

…2p†3=2 p a1 j L11 ‡ jb2 2p

jb1

a2 j

p# L22

 ‡ F…u† ‡ oe …j…u††:

Remarks:  The equalities (7), (8) and (14) of Theorem 3 and 4 are news. Their asymptotic expansions (9), (10) and (15) are the same as those obtained by Piterbarg (1996a, Theorem 5.1). However, the methods are quite different. Piterbarg's proof is based on processes close to isotropic ones and uses the Hadwiger theorem on invariant additive functionals. In comparison our proof, using the Rice method, is more elementary and leads to an interpretation in term of ``border maxima''. Note also that by geometric arguments and Weyl's formula, Sun (1993) seems to have obtained in an unpublished paper (see Adler, 2000, Theorem 6.5.3) the three-term approximation formula for P‰M…S†  uŠ in a rather general setting. The Rice method appears to be more accurate for examples in Corollary 1 and Theorem 1.  As proved for example in Dacunha-Castelle and Gassiat (1999), maxima of Gaussian processes appear as the square root of the distribution of the likelihood ratio test statistics for some special models with nuisance parameter not de®ned under H0 . The expression given in Theorem 4 can be rewritten, for example, P‰M…S†  uŠ ˆ

‰n=2Š N X X G…n nˆ1 jˆ0

0

6@

X

2j ‡ 1=2† P‰w2 …n 2j ‡ 1 pn ‡ 1=2 1 …Pi [ I kbi

2j ‡ 1†  u2 Š

I A  ‡ F…u† ‡ oe …j…u††: ai k†k2j

I [ fn …N†

P‰M…S†  uŠ is a linear combination of P‰w2 …d†  u2 Š for some value of d. Remember that for regular statistical models the distribution of the likelihood ratio test statistics is given by a unique w2 distribution.

4.

Application to wave data

After removing deterministic components like tide and long term effects like surge, sea waves' height can be modeled under speci®c conditions by a stationary Gaussian random ®eld X depending on two spatial coordinates x and y and one time coordinate. Here we will consider only the space dependence model and use a power spectrum measured at

191

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

Figure 1. Spectral density of X.

Table 1. Spectral moments of the Gaussian ®eld X. L00

L11

L12

1.337913

0.01184173

L22

0:003036529

0.005462745

Draupner platform in the North Sea (data kindly provided by Marc Prevosto from IFREMER). The spectrum after symmetrization is shown on Figure 1. From these data we deduce the spectral moments of the Gaussian ®eld X (Table 1). Waves critical heights at level 0.9 are approximated by two methods:  The ®rst one uses Corollary 1 or Theorem 4.  The second one, between parentheses, uses the classical bound by Adler (1981). They are given in Table 2 in meters for different space domains, namely: the square of size a denoted by a6a; the rectangle a6b and the disc with radius a denoted by p6a2. To Table 2. Critical heights for X at level 0.9 obtained by Corollary 1 or Theorem 4 or, between parentheses, by the classical bound by Adler (1981). Empirical levels of these methods obtained with 1000 replications of X. S is a rectangle a6b or a disc of radius a; p6a2 . S: Critical heights at level a ˆ 0:9 Empirical levels

p p 20 p620 p

p p 20 2p620 p2

p6202

p p 30 p630 p

p p 30 2p630 p2

p6302

2.86 (2.45) 0.903 (0.794)

2.89 (2.45) 0.905 (0.782)

2.84 (2.45) 0.908 (0.803)

3.16 (2.94) 0.906 (0.841)

3.18 (2.94) 0.895 (0.839)

3.15 (2.94) 0.91 (0.85)

192

AZAIÈS AND DELMAS

Figure 2. A realization of the Gaussian random ®eld X.

check the exactness of these levels, a Monte-Carlo experiment has been conducted with 1000 replications to evaluate the empirical level of each method. These results are given in Table 2. Note that [0.881; 0.919] is a 95% con®dence interval for a binomial proportion of 0.9 over 1000 replications. Figure 2 gives an example of such a realization. These results show the improvement due to the extra term in the asymptotic expansion.

5. 5.1.

Proofs Proof of Theorem 1

Proof in case a First we check that for all t in S; X0 …t† and X00 …t† are independent. Set r…s; t† ˆ E‰X…s†X…t†Š; rij;kl …s; t† ˆ

q4 r…s; t† : qsi qsj qtk qtl

ri; j …s; t† ˆ

q2 r…s; t† ; qsi qtj

rij;k …s; t† ˆ

q3 r…s; t† ; qsi qsj qtk

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ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

By hypothesis, ri;j …t; t† ˆ E‰Xi …t†Xj …t†Š ˆ Lij . By differentiation with respect to tk : rik;j …t; t† ‡ ri;jk …t; t† ˆ 0:

…16†

Since r…s; t† ˆ r…t; s† and since differentiation is symmetric rik;j …t; t† ˆ rki;j …t; t† ˆ rj;ik …t; t† ˆ rj;ki …t; t†: Exchanging j and k in equation (16) and calculating the difference yields rik;j …t; t† ˆ rij;k …t; t†; and now it is direct that all these terms are zero. As a consequence Vi; j; k ˆ 1; N; ri;jk …t; t† ˆ 0 ˆ E‰Xij …t†Xk …t†Š: Without loss of generality, using the change of variables Z…t† ˆ X…L 1=2 t† that does not modify the properties of S, we may assume that L ˆ Id. We have _ ˆ E‰MuX …S†Š

Z

Z

S

dt

u

?

~ 00 …t††=X…t† ˆ x† dx: j…x†pX0 …t† …0†E…det…X

Due to regression formulae, conditionally to X…t† ˆ x; X00 …t† is Gaussian with expectation xId. Set G…t† ˆ X00 …t† ‡ xId under the conditional distribution. G…t† has a distribution which does not depend on x, is non degenerated due to condition H1 b and varies  Besides, for any choice of norm on m, there exists a continuously on the compact set S. constant c such that kgk 

x implies g c

xId [ n:

We have f 00 …t††=X…t† ˆ x† ˆ … 1† E…det…X

N

ˆ … 1†N

Z g [ m:g

Z

m

with Z jZj 

kgkx=c

jdet…g

xId†jdPG…t† …g†;

xId [ n

det…g

det…g

xId† dPG…t† …g†

xId† dPG…t† …g† ‡ Z;

194

AZAIÈS AND DELMAS

where PG…t† is the probability distribution of G…t†. By compactness the norm of the variance±covariance matrix of G…t† is uniformly bounded on S so it is easy to prove that jZj ˆ oe …1†: Due to regression formulae for i; j; k; l ˆ 1; N E…Gij …t†Gkl …t†† ˆ rij;kl …t; t†

dij dkl ;

where d is the Kronecker symbol. We are in condition to apply Lemma 5.3.2 by Adler (1981) that states that E‰det…G

xId†Š ˆ … 1†

N

‰N=2Š X

… 1† j

jˆ0

  N …2j†! N x 2j j!2j

2j

:

Direct calculations now lead to the result. Proof in case b  the expression of E‰MuX …S†Š _ given by Since the Gaussian ®eld X is with unit variance on S, Proposition 1 becomes _ E‰MuX …S†Š

Z ˆ

S

Z dt

u

?

j…x†p0X …t†…0†

Z n

jdet x00 jpX00 …t† …x00 =X…t† ˆ x; X0 …t† ˆ 0†dx00 dx: …17†

Set L…t† ˆ var X0 …t†

rij;kl …t; t† ˆ E‰Xij …t†Xkl …t†Š

rij;m …t; t† ˆ E‰Xij …t†Xm …t†Š

rn;kl …t; t† ˆ E‰Xn …t†Xkl …t†Š:

Conditionally to X…t† ˆ x and X0 …t† ˆ 0; X00 …t† is Gaussian with mean xL…t†. Set G ˆ X00 …t† ‡ xL…t† under the conditional distribution. G is Gaussian centered with: E‰Gij Gkl Š ˆ rij;kl …t; t†

X m;n

rij;m …t; t†Lm;n1 …t†rn;kl …t; t†

lij …t†lkl …t†:

By the same arguments as those previously used and (17) we obtain that the integral over the set …X00 …t† [ nC † is oe …j…u†† and we get

195

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

_ E‰MuX …S†Š

Z ˆ

Z

S

dt

?

u

Let us evaluate E‰det…G

E‰det…G

N

j…x†p0X …t†…0†… 1† E‰det…G

xL…t††Š dx ‡ oe …j…u††:

xL…t††Š:

xL…t††Š ˆ

N X

X

j ˆ 0 I;K [ fj …N†

e…sI †e…sK †… 1†

N‡j N

x

j

E‰DIK …G†ŠDIK …L…t††:

Since DIJ …G† ˆ

X p [ S2m

e…p†Gi1 jp…1† 6


6Gi2m jp…2m† ;

and since we have, for …Y1 ; . . . ; Y2m † centered Gaussian (see, for example, Lemma 5.3.1 of Adler, 1981): E‰Y1 Y2 ; . . . ; Y2m Š ˆ

X s [ Qm

E‰Ys…1† Ys…2† Š6


6E‰Ys…2m

1† Ys…2m† Š;

we obtain E‰DIK …G†Š ˆ 0 m Y X X e…p† vs…2q E‰DIK …G†Š ˆ p[s s[Q 2m

m

qˆ1

when j ˆ 2m ‡ 1; 1†;p0 s…2q

1†;s…2q†;p0 s…2q†

when j ˆ 2m;

with the notations of the introduction and where I ˆ fi1 ; . . . ; ij g; K ˆ fk1 ; . . . ; kj g; Qm is de®ned in the statement of Theorem 1 and va;b;Z;x ˆ E‰Gia kb GiZ kx Š ˆ cov‰Xia kb ; XiZ kx =X…t†; X0 …t†Š: Then direct calculations lead to Theorem 1(b).

5.2.

&

Proof of Theorem 2

Before proceeding to the proof of Theorem 2 we shall recall the following proposition whose proof can be found in Adler (1981).

196

AZAIÈS AND DELMAS

Proposition 5: Let X a Gaussian centered ®eld on S satisfy H1 a and b and let S be relatively compact with zero measure boundary. Then, with probability one, there exists no point belonging to qS such that X0 …t† ˆ 0. To prove Theorem 2 consider a realization such that the maximum M…u† of Y on u is greater than u. By continuity  ˆ 1: P‰M…S† ˆ M…u† ˆ M…S†Š Let ~t0 a point where Y attains its maximum on u and set t0 ˆ c…~t0 †. Since Y 0 …~t0 † ˆ 0, X0 …t0 † ˆ 0. Proposition 5 proves that a.s. t0 does not belong to qS and that there exists a local maximum greater than u inside S: _ > 0Š: P‰M…S† > uŠ ˆ P‰MuX …S† Theorem 2 will be a direct consequence of relation (1) as soon as we prove that X _ _ E‰MuX …S†…M u …S†

1†Š ˆ oe …j…u†† as u ? ‡ ?:

…18†

The proof of this last point is identical to that of Proposition 3 except that the compactness argument must be conducted on the manifold u using hypothesis H2 a0 . &

5.3.

Proof of Theorem 3

With the de®nitions given in the statement of the theorem we have _  1†|…RXu~…qS†  1†Š P‰M…S† > uŠ ˆ P‰…MuX …S† ~

_  1Š ‡ P‰RXu …qS†  1Š ˆ P‰MuX …S†

~

_ Xu …qS†  1Š: P‰MuX …S†R

We are going to evaluate each of these terms separately. : 5.3.1. Evaluation of P‰MuX …S†  1Š. conditions of Proposition 3. Thus X _ _ E‰MuX …S†…M u …S†

Assumptions on X imply that it satis®es the

1†Š ˆ oe …j…u†† as u ? ‡ ?:

197

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

Then using the inequality (1) and Theorem 1 we get that _  1Š ˆ l…S† P‰MuX …S† ˆ uN

‰N=2Š X jˆ0

1

k2j c2j …u† ‡ oe …j…u†† 

exp

  1=2 u2 l…S†…det…L†† ‡ o uN …N ‡ 1†=2 2 …2p†

2

 exp

u2 2

 :

5.3.2. Evaluation of P‰RXu …qS†  1Š. We partition S into cubes (or subsets of cubes at the boundary) of size e with disjoint interiors: H1 ; . . . ; Hm . Since S is compact this number m is ®nite for all e. Let n be the number of these cubes intersecting qS and set V1 ; . . . ; Vn the corresponding qS \ Hi1 ; . . . ; qS \ Hin . For i ˆ 1; . . . ; n choose a point ti [ Vi and de®ne ji as the projection of Vi onto the tangent space Tti at ti . By compactness e can be chosen suf®ciently small in order to ji being one-to-one with inverse ji c? (as a function from Tti to RN ) on the union of Vi with all other adjacent ~ i …~t††; ~t [ Wi . Applying Vj 's (dist…Vi ; Vj † ˆ 0). De®ne Wi ˆ ji …Vi †; Yi …~t† ˆ X…c Proposition 5 to the process Yi we know that, with probability one: ~

RXu …qS† ˆ

n X iˆ1

~

RXu …Vi †:

Thus the inequality (1) applied to the variables above becomes: n X iˆ1

~

E‰RXu …Vi †Š ~

n 1X ~ ~ E‰RXu …Vi †…RXu …Vi † 2 iˆ1

 P‰RXu …qS†  1Š  Step 1:

n X iˆ1

1†Š

1X ~ ~ E‰RXu …Vi †RXu …Vj †Š 2 i=j

~

E‰RXu …Vi †Š:

We prove that the double-sum appearing in the lower-bound is negligible.

 Suppose that dist …Vi ; Vj † > 0 then ~

~

~

~

Y

E‰RXu …Vi †RXu …Vj †Š  E‰MuX …Vi †MuX …Vj †Š ˆ E‰MuYi …Wi †Mu j …Wj †Š: j , the vector i ; ~t [ W On account of the non degeneracy condition H2 c, for all s~ 6ˆ ~t; s~ [ W s†; Yj …~t†; Yi0 …~ s†; Yj0 …~t†; Yi00 …~ s†; Yj00 …~t†† …Yi …~

198

AZAIÈS AND DELMAS

is non-degenerated and using the same arguments as those involved in the proof of Piterbarg's proposition (Proposition 3), we get Y E‰MuYi …Wi †Mu j …Wj †Š

Z Z 

Wi 6Wj

Z Z ~ d~ s dt

…u; ‡ ?†2

dx dypYi …~s†;Yj …~t†;Yi0 …~s†;Yj0 …~t† …x; y; 0; 0†

~ j00 …~t††=Yi …~ ~ i00 …~ s†† det…Y s† ˆ x; Yj …~t† ˆ y; Yi0 …~ s† ˆ 0; Yj0 …~t† ˆ 0Š ˆ oe …j…u††: 6E‰det…Y ~

~

Y

Y

 Consider now E‰RXu …Vi †…RXu …Vi † 1†Š  E‰Mu i …Wi †…Mu i …Wi † result of Proposition 3.  Consider now the case dist…Vi ; Vj † ˆ 0, then ~

1†Š ˆ oe …j…u†† by the

~

E‰RXu …Vi †RXu …Vj †Š  E‰MuYi …ji …Vi [ Vj ††…MuYi …ji …Vi [ Vj ††

1†Š ˆ oe …j…u††;

by the same argument than for the preceding case. Step 2:

We turn now to the evaluation of

Pn

iˆ1

~

E‰RXu …Vi †Š.

~

E‰RXu …Vi †Š ˆ E‰RYu i …Wi †Š: Using the same arguments as those involved in the proof of Proposition 1, we get E‰RYu i …Wi †Š

Z ˆ

Wi

Z ~ dt

‡?

u

Z dy

‡?

0

Z dz

n

jdet y00 jpYi00 …~t†;Yi …~t†;Yi0 …~t†;Zi …~t† …y00 ; y; 0; z† dy00 ;

where Z…~t† ˆ n…ci …~t†† ? X0 …ci …~t††. Forgetting the subscript i for short, we get E‰RYu …W†Š ˆ

Z W

Z d~t

‡?

u

Z j…y† dy

0

‡?

Z dzpY 0 …~t †;Z…~t† …0; z†

n 00

jdet y00 j

6pY 00 …~t † …y00 =Y…~t † ˆ y; Y 0 …~t † ˆ 0; Z…~t † ˆ z† dy :

Y…~t † and …Y 0 …~t †; Z…~t †† are independent since X…t† and X0 …t† are. Thus, E‰Y 00 …~t †=Y…~t † ˆ y; Y 0 …~t † ˆ 0; Z…~t † ˆ zŠ ˆ E‰Y 00 …~t †=Y…~t † ˆ yŠ ‡ E‰Y 00 …~t †=Y 0 …~t † ˆ 0; Z…~t † ˆ zŠ: The derivative c0 …~t † of c at ~t can be viewed as an N6…N 1† matrix and the second derivative c00 …~t † as a bilinear function RN 1 6RN 1 ? RN . Put t ˆ c…~t †, then combination of derivation implies that: Y 0 …~t † ˆ …c0 …~t ††T X0 …t† Y 00 …~t † ˆ …c0 …~t ††T X00 …t†…c0 …~t †† ‡ A;

199

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

where A is the bilinear function …h; k† ? …c00 …~t †…h; k††T X0 …t†. Since X0 …t† and Y…~t † ˆ X…t† are independent T E‰Y 00 …~t †=Y…~t † ˆ yŠ ˆ E‰…c0 …~t †† X00 …t†c0 …~t †=X…t† ˆ yŠ ˆ

T …c0 …~t †† Lc0 …~t †y:

In conclusion, we have proved that conditionally to …Y…~t † ˆ y; Y 0 …~t † ˆ 0; Z…~t † ˆ z†, the expectation of Y 00 …~t † is …c0 …~t ††T Lc0 …~t †y ‡ za; T with a ˆ E‰Y 00 …~t †=Y 0 …~t † ˆ 0; Z…~t † ˆ 1Š. Put G…~t † ˆ Y 00 …~t † ‡ …c0 …~t †† Lc0 …~t †y Under the condition …Y…~t † ˆ y; Y 0 …~t † ˆ 0; Z…~t † ˆ z†; E‰RYu …W†Š can be written

Z W

Z d~t

‡? u

Z j…y†

‡?

0

Z pY 0 …~t †;Z…~t † …0; z†

m

~ det…g

za.

y…c0 …~t ††T Lc0 …~t † ‡ za†pG…~t † …g† dg dz dy:

Set y ˆ u~ y u1

N

y~N

~ det…g 1

~ y…c0 …~t †† Lc0 …~t † ‡ za† ˆ det T

g

T det……c …~t †† Lc0 …~t †† 0

u

a u! ‡ ? T y~…c0 …~t †† Lc0 …~t † ‡ z ? u

Thus by a dominated convergence argument, as u ? ‡ ? and since: Z 0

‡?

1

N

…2p† 2 pY 0 …~t †;Z…~t † …0; z† dz ˆ q P‰Z…~t † > 0=Y 0 …~t † ˆ 0Š T det……c0 …~t †† Lc0 …~t †† 1

N

…2p† 2 ˆ q ; 2 det……c0 …~t ††T Lc0 …~t †† E‰RYu …W†Š

Z ˆ

W

ˆ uN

d~t

Z‡ ? 1

2

exp

y~N



1 N q …2p† 2 y…1 ‡ o…1†† det……c0 …~t ††T Lc0 …~t †† d~ 2  Z q u2 1 det……c0 …~t ††T Lc0 …~t †† d~t…1 ‡ o…1††: 2 2…2p†N2 W

1 N

u j…u~ y†

200

AZAIÈS AND DELMAS

In conclusion we have proved that as u ? ‡ ?: ~

P‰RXu …qS†  1Š ˆ uN

Step 3:

2

 exp

 q n Z u2 1 X det……c0i …~t ††T Lc0i …~t †† d~t…1 ‡ o…1††: 2 2…2p†N2 i ˆ 1 Wi …19†

We look for a better expression for the term:

Tn ˆ

n Z q X T det……c0i …~t †† Lc0i …~t †† d~t iˆ1

Wi

appearing in formula (19). Let ~t [ Wi ; t ˆ ci …~t † and recall that ti [ Vi is the point de®ning the tangent space Tti and the projection ji . Then c0 …~t † is the projection Tti ? Tt orthogonal to Tti viewed as a function W ? RN . Since qS is regular, the normal vector n…t† to qS at point t [ qS is a continuous function of t. Thus taking arbitrary small sets Vi we can impose: q det……c0 …~t ††T Lc0 …~t ††

q T det……c0 …~ti †† Lc0 …~ti ††  e:

Then we get that Tn , for suf®ciently small set Vi , is arbitrary close to Tn0 ˆ

n q X T det……c0 …~ti †† Lc0 …~ti ††lN iˆ1

1 …Wi †:

T

Note that …c0 …~ti †† Lc0 …~ti † is the projection Pti …L† of L on the tangent space Tti . We get Tn0 ˆ

n q X det…Pti …L††lN iˆ1

1 …Wi †;

which is a Riemann-type sum associated to the integral Z p det…Pt …L†† ds…t†: qS

Since Tn does not depend on n and on the choice of V1 ; . . . ; Vn , we get: Z p Tn ˆ det…Pt …L†† ds…t†: qS

201

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

5.3.3.

: Proof that P‰RXu …qS†MuX …S†  1Š ˆ oe … u …u††. ~

Step 1: As in step 2 of the evaluation of P‰RXu …qS†  1Š we de®ne H1 ; . . . ; Hm and Vi ˆ 1; . . . ; n; Vi ; ti ; ji ; ci ; Wi ; Yi …t†. For ~t [ Wi we de®ne Zi …~t † by X0 …ci …~t †† ? n…ci …~t ††. With probability one we have ~

RXu …qS† ˆ

n X iˆ1

~ _ ˆ RXu …Vi † and MuX …S†

m X iˆ1

MuX …Hi †:

Thus ~

~

_  1Š  E‰RXu …qS†MuX …S†Š _ ˆ P‰RXu …qS†MuX …S†

X i ˆ 1;...;n; j ˆ 1;...;m

~

E‰RXu …Vi †MuX …Hj †Š:

By the same arguments as those used in the proof of Proposition 2: ~

E‰RXu …Vi †MuX …Hj †Š Z Z Z Z  dt d~t

Z

‰u; ‡ ?†2

Hj 6Wi

dx dy

0

?

pX…t†;Yi …~t †;X0 …t†;Yi0 …~t †;Zi …~t † …x; y; 0; 0; z†

~ i00 …~t ††=X…t† ˆ x; Yi …~t † ˆ y; X0 …t† ˆ 0; Yi0 …~t † ˆ 0; Zi …~t † ˆ zŠ dz ~ 00 …t†† det…Y E‰det…X Z ? Z ? Z Z dt d~t dx pX…t†;X0 …t†;Yi0 …~t †;Zi …~t † …x; 0; 0; z†  Hj 6Wi

u

0

~ i00 …~t ††=X…t† ˆ x; X0 …t† ˆ 0; Yi0 …~t † ˆ 0; Zi …~t † ˆ zŠ dz ~ 00 …t†† det…Y E‰det…X Z Z def ˆ A…t; ~t† dt d~t Hj 6Wi

…20†

Step 2: Suppose that dist…Hj ; Vi † > 0. As in the preceding section and using the argument by Piterbarg we get ~

E‰MuX …Hj †RXu …Vi †Š ˆ oe …j…u†† as u ? ‡ ?: Suppose now that dist…Hj ; Vi † ˆ 0 but that Vi is distinct from Hj \ qS. De®ne Hj as the union of Hj with adjacent Hl s …l ˆ 1; . . . ; m† and Vj as Hj \ qS. Then ~ ~ E‰MuX …Hj †RXu …Vi †Š  E‰MuX …Hj †RXu …Vj †Š:

Thus this case can be included into the next one.

202

AZAIÈS AND DELMAS

Step 3: We suppose now that Vi ˆ Hj \ qS and, for short, we forget the subscripts i and j. De®ne: et;~t ˆ e ˆ

c…~t † kc…~t †

t ; Rt;~t ˆ R ˆ var‰X00 …t0 †eŠ: tk

We divide the integral (20) into two domains: Z Z H6W

A…t; ~t † dt d~t 

Z Z D1

A…t; ~t † dt d~t ‡

Z Z D2

A…t; ~t † dt d~t;

where D1 and D2 are the two compact domains de®ned by: D1 ˆ ft [ H; ~t [ W=eT LR T

D2 ˆ ft [ H; ~t [ W=e LR

1

n…t0 †  eg

1

n…t0 †  eg;

with e > 0 that will be ®xed below. We will use the following relations the proof of which is given in step 4. These relations are valid for t [ H; ~t [ W and H being small enough. 1. There exists constants N1 and N2 such that: E‰jdet X00 …t†kdet Y 00 …~t †j=X…t† ˆ x; X0 …t† ˆ 0; Y 0 …~t † ˆ 0; Z…~t † ˆ zŠ "   N2 # z ‡1 :  …const† xN1 ‡ kc…~t † tk E‰jdet X00 …t†kdet Y 00 …~t †j=X…t† ˆ x; X0 …t† ˆ 0; Y 0 …~t † ˆ 0Š  …const†‰1 ‡ xN1 Š: 2. There exists a constant K; 0 < K < ‡ ? such that: var…Z…~t †=X0 …t†; Y 0 …~t ††  Kkc…~t †

tk2 :

3. There exists a strictly positive constant such that: det var…X…t†; X0 …t†; Y 0 …~t †; Z…~t ††  …const†kc…~t † det var…X…t†; X0 …t†; Y 0 …~t ††  …const†kc…~t †

tk2N : tk

2…N

1†

:

4. There exists a constant K 0 ; 0 < K 0 < ‡ ? such that: var…X…t†=X0 …t†; Y 0 …~t †; Z…~t ††  K 0 < 1: 5. On D1 : E‰X…t†=X0 …t† ˆ 0; Y 0 …~t † ˆ 0; Z…~t † ˆ 1Š  0. 6. On D2 : There exists a constant K 00 ; 0 < K 00 < 1 such that var…X…t†=X0 …t†; Y 0 …~t ††  K 00 < 1.

203

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

On D1 write

pX…t†;X0 …t†;Y 0 …~t †;Z…~t † …x; 0; 0; z† ˆ pX…t† …x=X0 …t† ˆ 0; Y 0 …~t † ˆ 0; Z…~t † ˆ z†pX0 …t†;Y 0 …~t †;Z…~t † …0; 0; z†   x2 1=2 0 0 ~ ~ exp  …const†…det var…X…t†; X …t†; Y …t †; Z…t ††† 2K 0   2 z exp 2var…Z…~t †=X0 …t†; Y 0 …~t ††

by relations 4 and 5. Thus by relations 1, 2 and 3: A…t; ~t †    N2 ! Z ‡? Z ‡? z dz x N1 ‡ ‡1 kc…~t † …const† kc…~t † tk 0 u ! z2 dx: exp 2 2Kkc…~t † tk Using the change of variable ~ z ˆ z=kc…~t † tk yields: Z ‡? Z ‡? ~ dz …xN1 ‡ zN2 ‡ 1†kc…~t † A…t; t †  …const† 0 u   z2 exp dx: 2K

tk

tk

1

N

N

 exp

 exp

x2 2K 0

x2 2K 0



The conclusion that Z Z A…t; ~t † dt d~t ˆ oe …j…u†† D1

follows directly from the integrability of kc…~t † tk1 N on H6W. On D2 write: Z ? ~ 00 …t†† det…Y ~ 00 …~t ††=X…t† ˆ x; X0 …t† dxpX…t†;X0 …t†;Y 0 …~t † …x; 0; 0†E‰det…X A…t; ~t †  u

ˆ 0; Y 0 …~t † ˆ 0Š: Relations 1, 3 and 6 imply that: Z ‡? ~ A…t; t †  …const† …1 ‡ xN1 †kc…~t † u

tk

N‡1

 exp

The integrability of kc…~t † tk1 N on H6W imply that: Z Z A…t; ~t † dt d~t ˆ oe …j…u††; D2

which concludes the proof of Theorem 3.

 x2 dx: 2K 00



204

AZAIÈS AND DELMAS

Step 4: Proofs of the relations given in step 3 above. Since the Hi are of arbitrary small size, by a compactness argument it is suf®cient to prove all the relations for …tn ; c…~tn †† tending to …t0 ; t0 †, with t0 [ Vi and for the point of projection ti tending to t0 also. In the following, the Landau symbol o refers to such an asymptotic framework. One key point is that since j is the projection on the tangent space, Y 0 …~t † is arbitrary close to the projection Pt0 …X0 …c…~t ††† of X0 …c…~t †† on the space tangent to t0 . Y Y 0 …~t † ˆ …X0 …c…~t ††† ‡ o…1†: t0

The second key point is that the transformation 0 @

Y

1T T …X0 …c…~t ††† ; Z…~t †A ? X0 …c…~t ††

c…~t †

is unitary thus with determinant 1.  Relation 1 Put C ˆ fX…tn † ˆ x; X0 …tn † ˆ 0; Y 0 …~tn † ˆ 0; Z…~tn † ˆ zg: We have z ~ kc…tn † z 00 ~ ~ E‰Y …tn †=CŠ ˆ Atn ;~tn x ‡ Btn ;~tn ~ kc…tn †

E‰X00 …tn †=CŠ ˆ Atn ;~tn x ‡ Btn ;~tn

tn k tn k

:

…21†

:

…22†

Our aim is to prove that functions Atn ;~tn ; Btn ;~tn ; A~tn ;~tn and B~tn ;~tn are bounded as …tn ; c…~tn †† tends to …t0 ; t0 †. We give the proof for Atn ;~tn and Btn ;~tn . The proof for the two other cases is similar. Atn ;~tn ˆ E‰X00 …tn †=C1 Š with  C1 ˆ

X0 …c…~tn †† X…tn † ˆ 1; X …tn † ˆ 0; kc…~tn † 0

X0 …tn † ˆ0 tn k



ˆ fX…t0 † ‡ o…1† ˆ 1; X0 …t0 † ‡ o…1† ˆ 0; X00 …t0 †etn ;~tn ‡ o…1† ˆ 0g: Since the limit distribution of …X…t0 †; X0 …t0 †; X00 …t0 †etn ;~tn † is uniformly non-degenerate, E‰X00 …tn †=C1 Š ˆ E‰X00 …tn †=X…t0 † ˆ 1; X0 …t0 † ˆ 0; X00 …t0 †etn ;~tn ˆ 0Š ‡ o…1† which gives the result.

205

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

In the same way Btn ;~tn ˆ E‰X00 …tn †=X…t0 † ‡ o…1† ˆ 0; X0 …t0 † ‡ o…1† ˆ 0; X00 …t0 †etn ;~tn ‡ o…1† ˆ n…t0 †Š ˆ E‰X00 …tn †=X…t0 † ˆ 0; X0 …t0 † ˆ 0; X00 …t0 † ˆ n…t0 †Š ‡ o…1†: Now det…X00 …t†† for example can be written as the sum of N! products of N terms. Thus jdet…X00 …t††j is bounded by the sum of the absolute value of these N! terms. The same kind of relation is true for det…Y 00 …t††. Thus jdet…X00 …t†† det…Y 00 …t††j is bounded by the sum of the absolute value of N!…N 1†! products of 2N 1 Gaussian variables with bounded variance and expectation bounded by (const) …x ‡ z=kc…~t † tk†. Thus writing each variable as the sum of its expectation and a centered variable, making the product gives the result with N1 ˆ N2 ˆ 2N 1. For this one may use, for example, Lemma 5.3.1 of Adler (1981) that states that if Y1 ; . . . ; Yn are n centered Gaussian variables E…Y1 ; . . . ; Y2n ‡ 1 † ˆ 0 X E‰Ys…1† Ys…2† Š6


6E‰Y2n E…Y1 ; . . . ; Y2n † ˆ

1 Y2n Š;

s [ Qn

where Qn has been de®ned in the statement of Theorem 1. In the same way we prove that: E‰kdet X00 …tn †kdet Y 00 …~tn †k=X…tn † ˆ x; X0 …tn † ˆ 0; Y 0 …~tn † ˆ 0Š  …const†…1 ‡ xN1 †:  Relation 2. det var‰X0 …tn †; Y 0 …~tn †; Z…~tn †Š det var‰X0 …tn †; Y 0 …~tn †Š det var‰X0 …tn †; V~tn …X0 …c…~tn †† X0 …tn ††Š ˆ det var‰X0 …tn †; c0 …~tn †…X0 …c…~tn †† X0 …tn ††Š kc…~tn † tn k2N …det var‰X0 …t0 †; X00 …t0 †etn ;~tn Š ‡ o…1† ˆ Q 2…N 1† …det var‰X0 …t0 †; t0 X00 …t0 †etn ;~tn Š ‡ o…1† kc…~tn † tn k

var‰Z…~tn †=X0 …tn †; Y 0 …~tn †Š ˆ

ˆ kc…~t † ˆ

tk2

det var‰X00 …t0 †etn ;~tn Š ‡ o…1† Q det var‰ t0 X00 …t0 †etn ;~tn Š ‡ o…1†

kc…~tn †

tn k2 …1 ‡ o…1††

nT …t0 †‰var‰X00 …t0 †etn ;~tn ŠŠ

1

n…t0 † ‡ o…1†

;

where V~t is the N6N matrix such that V~t …X0 …c…~t ††† ˆ …Y 0 …~t †T ; Z…~t ††T . Since n…t0 † and etn ;~tn are unit vectors and var…X00 …t0 †etn ;~tn † is non-degenerate uniformly in tn and ~tn , relation 2 is proved.

206

AZAIÈS AND DELMAS

 Relation 3 By the same arguments det var‰X…tn †; X0 …tn †; Y 0 …~tn †; Z…~tn †Š ˆ kc…~tn †

tn k2N …det var…X…t0 †; X0 …t0 †; X00 …t0 †etn ;~tn † ‡ o…1††:

Since det var…X…t0 †; X0 …t0 †; X00 …t0 †etn ;~tn † is non-degenerate uniformly in tn and ~tn , relation 3 is established. In the same way we prove that: det var‰X…tn †; X0 …tn †; Y 0 …~tn †Š  …const†kc…~tn †

tn k2…N

1†

:

 Relation 4 var…X…tn †=X0 …tn †; Y 0 …~tn †; Z…~tn † ˆ var‰X…t0 †=X00 …t0 †etn ;~tn Š ‡ o…1† ˆ1

eTtn ;~tn L…var‰X00 …t0 †etn ;~tn Š†

1

Letn ;~tn ‡ o…1†:

var‰X00 …t0 †etn ;~tn Š is non-degenerate uniformly in tn and ~tn and kLetn ;~tn ; k is uniformly strictly positive thus relation 4 is valid.  Relation 5 It is easy to see that the condition C2 ˆ fX0 …tn † ˆ 0; Y 0 …~tn † ˆ 0; Z 0 …~tn † ˆ 1g can be written 

 X0 …tn † n…c…~tn †† ‡ o…1† ˆ kc…~tn † tn k tn k   n…c…~tn †† ‡ o…1† ˆ X0 …t0 † ‡ o…1† ˆ 0; X00 …t0 †etn ;~tn ‡ o…1† ˆ : kc…~tn † tn k

C2 ˆ

X0 …tn † ˆ 0;

X0 …c…~tn †† kc…~tn †

Thus E‰X…t0 †=C2 Š ˆ kc…~tn †

tn k

1

‰ eTtn ;~tn LR

and relation 5 is valid for any value of e.

1

n…t0 † ‡ o…1†Š

207

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

 Relation 6 " var‰X…tn †=X0 …tn †; Y 0 …~tn †Š ˆ var X…t0 †= Y

ˆ1

t0

Y t0

Y t0

# X00 …t0 †etn ;~tn ‡ o…1† !T "

Letn ;~tn

Letn ;~tn

var

!

Y t0

!#

1

00

X …t0 †etn ;~tn

‡ o…1†:

…23†

Since Pt0 X00 …t0 †etn ;~tn is non-degenerated uniformly in tn and ~tn ; var‰X…tn †=X0 …tn †; Y 0 …~tn †Š may tend to zero only if Y t0

Letn ;~tn ? 0

which implies that there exists constants atn ;~tn such that Letn ;~tn ˆ atn ;~tn n…t0 † ‡ o…1† ˆ atn ;~tn n…c…~tn †† ‡ o…1†: Since S is convex eTt ;~t n…c…~tn ††  0. Therefore there exists a strictly positive constant a0 n n such that atn ;~tn  a0 . Thus eTtn ;~tn LR

1

n…t0 † ˆ atn ;~tn nT …t0 †R T

 a0 n …t0 †R

1

1

n…t0 † ‡ o…1†

n…t0 † ‡ o…1†:

…24†

For each value e, this is not possible on d2 for suf®ciently small subset s. Thus we conclude that relation (6) is valid. &

5.4.

Proof of Corollary 1

The proof is just a re®nement of the proof of Theorem 3. From this proof we know that  P‰M…S† > uŠ ˆ u exp

 1=2 u2 jLj ~ p ‡ E‰RXu …qS†Š ‡ oe …j…u††: 2 2 2p

208

AZAIÈS AND DELMAS

We use the parametrization Y…y† ˆ X…cos y; sin y†. We have 0

Y …y† ˆ … Y 00 …y† ˆ



0

sin y; cos y†X …cos y; sin y† ˆ

  sin y 0 ; X …cos y; sin y† cos y

…cos y; sin y†X0 …cos y; sin y† sin y; cos y†X00 …cos y; sin y†…

‡…

T

sin y; cos y† :

Let Z…y† be the normal derivative:    cos y 0 Z…y† ˆ ; X …cos y; sin y† : sin y We have: E‰R~u …‰0; 2pŠ†Š ˆ

Z

2p

dy

0

Z

6

?

u

Z dy j…y†

?

0

Z dz pY 0 …y†;Z…y† …0; z†

0 ?

ˆ y; Y 0 …y† ˆ 0; Z…y† ˆ z† dy00 :

jy00 jpY 00 …y† …y00 =Y…y†

Since E‰Y 00 …y†=Y…y† ˆ y; Y 0 …y† ˆ 0; Z…y† ˆ zŠ ˆ

y‰L11 sin2 y

2L12 cos y sin y ‡ L22 cos2 yŠ

z

which is bounded above by zero, it is easy to see using previous method that: E‰R~u …‰0; 2p‰†Š ˆ

Z

2p 0

Z dy

?

u T

Z dy j…y†

0

?

Z dz pY 0 …y†;Z…y† …0; z†

R

…g yv …y†Lv…y† z†pG …g† dg   Z Z  u2 1 2p p 1 2p jLj1=2  ˆ exp dy; vT …y†Lu…y† dy ‡ F…u† 2p 0 vT …y†Lv…y† 2 4p 0 where G is a Gaussian random variable centered with variance var…Y 00 …y†= Y…y†; Y 0 …y†; Z…y†† and v…y† is de®ned in the statement of Corollary 1. Thus Corollary 1 is established. &

ON THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS

5.5.

209

Sketch of the proof of Theorem 4

A face of S can be indexed by:  a set I of coordinates that are free in the face. This set de®nes a class of parallel faces. We denote VI the vector space generated by these faces,  a number i; i [ f1; . . . ; 2N jIj g ˆ KI that indexes the various faces in the preceding class. So a face will be denoted by SI;i ; I [ f…N†; i ˆ 1; . . . ; 2N jIj. When considering a unique face SI;i , we will choose an orthonormal base e1 ; . . . ; eN with e1 ; . . . ; en …n ˆ jIj† parallel to SI;i and en ‡ 1 ; . . . ; eN pointing outward S. Though this base depends on the considered face, to save notations it will always be denoted by the same symbol. We denote by X10 …t†; . . . ; XN0 …t† the coordinates of the gradient X0 …t† in this base, XI0 …t† ˆ PVI …X0 …t††; X0 I …t† ˆ PVI\ …X0 …t†† and XI00 …t† the restriction of X00 …t† to VI . Let X~I;i be the restriction of X to such a face, we de®ne the number of border maxima of the face SI;i . X~

Ru I;i …SI;i † ˆ #ft [ SI;i : X…t†  u; XI0 …t† ˆ 0; Xj0 …t† > 0; Vj > jIj; XI00 …t† [ ng: Note, for example, that if I ˆ ; S;i ˆ fti g for some point ti . X~

Ru 0;i …S0;i † ˆ IfX…ti †>u;Xj0 …ti †>0;Vjg X~

and if I ˆ f1; . . . ; Ng; Ru I …SI † ˆ MuX …S†. We have X~

P‰M…S†  uŠ ˆ P‰|I [ f…N† |i [ KI …Ru I;i …SI;i †  1†Š: Using the same methods as in the proof of Theorem 3, it can be proved (see Delmas 2001, or AzaõÈs and Delmas, 2001, for details) that h  ~ i X X X X~ P |I [ f…N† |i [ KI Ru I;i …SI;i †  1 ˆ P‰Ru I;i …SI;i †  1Š ‡ oe …j…u††; I [ f…N† i [ KI

X X

ˆ

I [ f…N† i [ KI

 When I ˆ : h ~ i X E Ru I;i …SI;i † ˆ P‰X…ti †  u; Xj0 …ti † > 0Š:

X~

E‰Ru I;i …SI;i †  1Š ‡ oe …j…u††:

210

AZAIÈS AND DELMAS

Since the distribution of …X…ti †; X0 …ti †† is N…0; 1†6N…0; L†, it does not depend on i. Let O1 ; . . . ; O2N the 2N orthants. X

X~ E‰Ru I;i …SI;i †Š

i [ KI

 ˆ F…u†

2N X iˆ1

 P‰X0 …ti † [ Oi Š ˆ F…u†:

~

 When I ˆ f1; . . . ; Ng, it is clear that E‰RXu …SI †Š ˆ E‰MuX …S†Š.  When I=; I=f1; . . . ; Ng, we have Z ? Z Z ? h ~ i Z XI;i 0 0 E Ru …SI;i † ˆ dt dxn ‡ 1 . . . dxN dx SI;i

Z n

0

jdet x00 jpX…t†;XI0 …t†;X0

I

u

0 0 00 00 …t†;XI00 …t† …0; 0 . . . 0; xn ‡ 1 ; . . . xN ; x † dx ;

…25†

where n is now the set of negative de®nite matrices of size n ˆ jIj. Since X0 …t† is independent of X…t†; XI00 …t†, the expression above can be written as h ~ i Z X E Ru I;i …SI;i † ˆ Z

?

u

Z dx

n

Z Z SI;i

dt

? 0

dx0n ‡ 1 . . . dx0N

pX0 …t† …0; . . . ; 0; x0n ‡ 1 ; . . . ; x0N † pXI0 …t† …0; . . . ; 0†

jdet x00 jpX…t†;XI0 …t†;XI00 …t† …0; 0 . . . 0; x00 † dx00 :

…26†

By Theorem 1, as u ? ‡ ? Z

?

u

Z dx

n

jdet x00 jpX…t†;XI0 …t†;XI00 …t† …0; 0 . . . 0; x00 † dx00 ˆ

‰n=2Š X jˆ0

I k2j Cn2j …u† ‡ oe …j…u††:

Summing up N n 2X

jˆ1

X~

E‰Ru I;i …SI;i †Š ˆ l…SI;i †

which gives the result.

‰n=2Š X jˆ0

I k2j Cn2j …u† ‡ oe …j…u††:

&

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