Model Fields in Crossing Theory: A Weak Convergence Perspective

Model Fields in Crossing Theory: A Weak Convergence PerspectiveAuthor(s): Richard J. WilsonSource: Advances in Applied Probability, Vol. 20, No. 4 (Dec., 1988), pp. 756-774Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1427359 .Accessed: 12/06/2014 13:55

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Adv. Appl. Prob. 20, 756-774 (1988) Printed in N. Ireland

( Applied Probability Trust 1988

MODEL FIELDS IN CROSSING THEORY: A WEAK CONVERGENCE PERSPECTIVE

RICHARD J. WILSON,* University of Queensland

Abstract

In this paper, the behavior of a Gaussian random field near an 'upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2

using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

TIGHTNESS; SMOOTH FUNCTIONS; HOMOGENEOUS; GAUSSIAN RANDOM FIELD;

UPCROSSING; HORIZONTAL WINDOW CONDITIONING; PALM DISTRIBUTION; ERGODIC;

EXCURSION SET; MAXIMUM

1. Introduction

Scientists and engineers are often interested in random phenomena which can be modelled by multiparameter stochastic processes, or random fields. Some examples of such phenomena are ocean and land surfaces, geological structures, seismic patterns, the surface of a ground metal plate and turbulence in fluids. In many instances, the data on a random field provided to statisticians consist of families of contour lines; that is, sets of level crossing points for a number of different levels. From this information, it is necessary to construct models for the behaviour of the random field throughout its region of definition and especially near these level crossings. In particular, such models are required for high levels in order to determine the probabilities of events of interest (such as 'catastrophes').

The purpose of this paper is to use a framework of weak convergence of

probability measures on spaces of smooth functions to develop the results of Wilson and Adler (1982) for the above types of models. In that paper, the horizontal window (h.w.) conditioning method (introduced by Kac and Slepian (1959)) was used to construct a random field which modelled the behavior of a homogeneous Gaussian random field {X(t), t e RN}, Ne = {1, 2, -}, in the vicinity of an

'upcrossing' of a fixed level u E R1. Denote this model field by {X,(t), tE RN}. It was shown that, if X was ergodic, then the distribution of X, could be associated

Received 10 March 1987; revision received 15 October 1987. * Postal address: Department of Mathematics, University of Queensland, St. Lucia, QLD 4067,

Australia. Research supported in part by AFOSR Grant No. F49620 85 C 0144 while the author was visiting the

Center for Stochastic Processes, University of North Carolina, Chapel Hill.

756


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Model fields in crossing theory 757

with appropriate empirical distributions so that X, had a natural frequency interpretation which motivated its use. Some of X,'s properties and asymptotic properties (as u- oo) were also studied.

These results were obtained by showing certain sequences of finite-dimensional distributions converged weakly and then using the limit finite-dimensional distributions to define limit random fields. However, as is well known, convergence of finite-dimensional distributions is not sufficient for weak convergence of the

corresponding distributions on some function space. Relative compactness or

tightness is also needed (see pp. 35-37 of Billingsley (1968)). Hence, if the

properties of X are to be examined by studying the distribution of X, on some function space, then it is necessary to choose an appropriate function space, and construct Xu and study its properties using full weak convergence on that function

space. The continuous mapping theorem and related results (see pp. 29-34 of

Billingsley (1968)) can then be used to obtain distributions of functionals of the

processes under consideration. In this paper, the four cases of weak convergence of finite-dimensional distribu-

tions obtained in Wilson and Adler (1982) will be strengthened to full weak

convergence in (C2, C2) where C2 = C2(~N) is the space of functions from R N to R1 which have continuous partial derivatives up to order two and 12 = 62(R N) is a

a-field of sets in C2 (to be defined in Section 1.2). (Criteria for weak convergence of

probability measures on spaces of smooth functions can be found in Wilson (1983), (1986), which present similar results to those available for spaces of continuous

functions.) In Section 2, the model field X, will be constructed using the h.w.

conditioning method and full weak convergence in (C2, C2). A subspace of C2 Will be used to show that a certain empirical random field also convergences in distribution to X, with probability 1. It will be seen that the distribution of X, is the

appropriate Palm distribution. Section 2 will be concluded by showing that, at a

'large' distance from t = 0 (that is, the location of the 'upcrossing'), X, has

asymptotically the same distribution on (C2, 2) as X, provided certain weak

dependence conditions hold. In Section 3, it will be shown that the distribution of a normalization of X, converges weakly to that of a certain random elliptical paraboloid as u --> . Finally, the asymptotic distributions of the Lebesgue measure of a particular excursion set of X, and the height of X, will be obtained for large u.

Before obtaining these results, a brief review of the literature relating weak

convergence to model processes will be given and then this section will be concluded with the necessary notation and assumptions for Sections 2 and 3.

1.1. Model processes and weak convergence. The methods used in Wilson and

Adler (1982) have often been used to construct processes which model the behavior of particular random processes near level crossings and local maxima. Some of the instances of these and their applications are in Slepian (1962)-zero upcrossings, Nosko (1969a-b, 1986)--crossings and maxima for random fields, Lindgren (1970),



758 RICHARD J. WILSON

(1971), (1972a-c)-maxima, Berman (1972)--crossings, Lindgren (1973)--zero- upcrossings, maxima and discrete sampling, Lindgren (1975a-b), (1979), (1980), (1981), (1983), (1984), (1985), de Mare (1980) and Leadbetter et al. (1983)- upcrossings and prediction, and Aronowich and Adler (1986)--X2-processes. How-

ever, some authors have chosen full weak convergence on some function space as an

appropriate tool for constructing and studying model processes. Lamperti (1965) used vertical window (v.w.) conditioning (see Kac and Slepian

(1959)) and C1[0, 1], the space of functions from [0, 1] to R' which have continuous first-order derivatives, to construct a process which modelled the behavior of a random process near a high level crossing. He used this result to obtain similar results for high upcrossings with both v.w. conditioning and h.w. conditioning. De Mare (1977) used C[0, 1], the space of continuous functions from [0, 1] to R, to obtain a process which modelled the behavior of a non-differentiable random

process near a level crossing. Lindgren (1977) used C(R1), the space of continuous functions from R1 to '1, to

show that, with probability 1, the appropriate empirical process converged weakly to the h.w. model process constructed to describe the behavior of a random process near an upcrossing. Lindgren (1983), (1984) and Leadbetter et al. (1983) strengthened this result using C1(D1), the space of functions from R1 to R1 which have continuous first-order derivatives.

Some asymptotic results have been obtained by showing that certain normalized model processes converge almost surely. Lindgren (1983) and Leadbetter et al.

(1983) have done so for the model process of the previous paragraph, while Aronowich and Adler (1986) have done so for h.w. model processes constructed to describe the behavior of X2-processes near upcrossings and downcrossings.

Partial reviews of the above results can be found in Cramer and Leadbetter

(1967), Adler (1981a-b), Leadbetter et al. (1983), Lindgren (1983), (1984) and Wilson (1983).

As mentioned earlier, in Sections 2 and 3, the model field of Wilson and Adler

(1982) will be constructed and studied using weak convergence in (C2, 162). This paper will present a unified approach to the application of weak convergence results to model fields. Finally, the results of this paper, along with detailed proofs, are contained in Wilson (1983).

1.2. Notation and assumptions. The notation of this paper will generally follow that of Wilson and Adler (1982) and Wilson (1986). Similarly to C2, let Ck = Ck( RN) denote the space of functions from RN to R' which have continuous derivatives up to order k e N0 = {0, 1, 2,

- - -}. Denote the derivatives of x e Ck by

(1.1) Dax(t) = . . tN

for each t=(t, ..., tN)T ER N and a e 9Uk where tT denotes the transpose of t,




=?k U U.o U ~1U q U

- k, and

j = {a E RN:cr, ---, aCE No0 and , +- -- + N=j} is the set of all multi-indices of length j No. Suppose from this point on that k = 2. Define a metric on C2 by

d2(x, y)= 2-r/ 1 + sup max ID"x(t) - Dy(t)lI

r=1 l \t r aB 2

for each x, yE C2, where Br denotes the closed ball in RN with centre t = 0 and radius r > 0. Let V2 = ?2(RN) denote the a-field generated by the open sets in

(C2, d2) and, for each x e Co, let

B,(X,6) = sup Ix(s) - x(t)l Is-t10, where Itl =

(t2 + -- + t2N). Let {X(t), t E RN} denote a real, separable, homogeneous, Gaussian random field

with zero mean and covariance function R satisfying R(O) = 1. Assume that RE C4 and that there exists some E* > 0 and finite K* > 0 such that

(1.2) ID R (t) - D "R(0)I _ K* Iln Itl I-(1+ *)

as Itj -0, for all a E W4. It follows that Pr (X e C2) = 1 (see, for example, Theorem 3.4.1 of Adler (1981a)) and hence, that the distribution of X is tight on (C2,

C2). Thus, it follows that, given E > 0, there exists a finite K, > 0 such that

(1.3) Pr max ID"X(O)I > K - E

and

(1.4) lim Pr max WB,(D "X, 6) E = 6--*0 aEa ?2

for each re N.

(See the note after Theorem 2.1 of Wilson (1986).) For convenience, the derivatives of X and R will be denoted in the following way

as well as via (1.1). Let the sample path derivatives of X, and the first- and second-order partial derivatives of R be given by

ax(t) 82X(t) at

X ) =at at

and

aR(t) 82R(t) R (t) , R (t)

=

ti ati ati




respectively for all t E RN and i, j = 1, 2, -

? ?, N. The first-order derivatives will be

written in vector form as

X'= (Xi, i=1, 2,---, N)T,

and

R' = (Ri, i = 1, 2, - - - , N)T.

The second- and fourth-order spectral moments of R are given by

84R(t) Ai = -R i(O) and Aijkl =

(i (tk t (3ti (3tj (3tk 3tl t=0

for i, , k, 1=1, 2, - - , N. The relationships between these spectral moments of R and the elements of the covariance matrix of {X(0), X'(0), (X,;(O),

1=i-j N)} are

given in Section 1.3 of Wilson and Adler (1982). It will be assumed that all the finite-dimensional distributions of X and its first- and second-order partial derivatives are non-singular.

Since the derivatives of X and R with respect to t,, ... , t_, will be treated

differently to those with respect to tN, the following notation will be used for them. Let

X'=(Xi, i= 1,..., N-1)T,

X" = (Xii, i= 1,..., N- 1; X;, 1-5i


o= S2(=R"(O)) gives the covariances between X(O) and X"(O). Note that

S=[S11 SiN] S-l SM

SNN

is the matrix of second-order spectral moments of R.

2. The model field and its properties

The model field constructed in Wilson and Adler (1982) described the behavior of X near the following type of crossing point.

Definition 2.1. The random field X is said to have an upcrossing of the level u in the Nth direction at a 'crossing' point t* if

(2.1) X(t*) = u, X'(t*) = O, XN(t*) > 0 and Dl(t*) < O,

where Dl(t*) < (>)O means that the matrix Dl(t*) is negative (positive) definite.

The application of these crossing points to level crossing theory for random fields can be found in Adler (1981a). A description of them for N = 2 and the rationale for

using them in the construction of model fields can be found in Wilson and Adler

(1982). In a recent paper, Nosko (1986) constructs and studies model fields near similar

crossing points to those given by (2.1), namely points t* E RN such that X(t*) = u and X'(t*)= v where v E RN-1 is fixed. The results obtained are similar to those of Wilson and Adler (1982) and are obtained using finite-dimensional distributions

only. Before obtaining the results of this section, the model field will be defined and its

construction using h.w. conditioning will be briefly discussed.

2.1. Construction of the model field. The model field of Wilson and Adler (1982) is defined as follows. Let

(2.2) Xu(t) = uR(t) - ra(t) - vTb(t) + A(t)

for all tE RN, where

(2.3) a(t) = (RN(t) - R'(t)TS-IlSIN)/lO2

and

(2.4) b(t) = E-1(R"(t) - R(t)S2o) are non-random functions with a2 = SNN - SNS-iiSIN and 2 = S22- 2oS02. is a random variable which has the Rayleigh density

(2.5) (v/u2)exp [-v2/(2a2)] for v >0.




vu is an (N(N - 1)/2)-dimensional random variable whose density is proportional to

(2.6) det (Z + uS11) exp [-1zT-1z]

for Z + uS11 >0

where Z = (Zij)i,j=1,2,,N-1 is a symmetric (N - 1) x (N - 1)-dimensional matrix and

z=(zii, i=1, - - - , N- 1; z,, li


Theorem 2.1.

{XI N(u, Bh) > O} - X in (C2, C2) as h - 0.

Proof. It follows from Theorem 2.1 of Wilson (1986) and Theorems 2.1 and 2.3 of Wilson and Adler (1982) that the theorem is true if conditions (ii) and (iii) of Theorem 2.1 of Wilson (1986) hold for ({X IN(u, Bh) > 0}, hE (0, 1)). These follow from the tightness of X in (C2, C2) and the behavior of N(u, -) through Lemma 2.1. For convenience, these conditions will form the content of the following two lemmas. Since their proofs are very similar, only the proof of the first will be given.

Lemma 2.2. For each E > 0 there exists a finite K > 0 such that

(2.10) Pr max2

D"X(O) > K N(u, Bh)> 0S E

for all hE (0, 1).

Proof. For convenience, denote the left-hand side of (2.10) by PK,h. If NK(u, B) denotes the number of points t E BU for which

(2.11) max sup DDaX(s)I> K CE E?2 S Bl(t)

for all Be ?RN and K > 0, where Br(t) = {s e RN:s - tI r'} then

(2.12) PK, h Pr {N(u, Bh) > 0}/Pr {N(u, Bh) > 0}

for all K

(0, oo) and he (0, 1). Since all the points tE Bu satisfying (2.11) are isolated almost surely for each compact subset B of RN, and NK(u, .) is

homogeneous, it follows from Lemma 2.1 that

lim h-N Pr {NK(u, Bh) > 0} = E{NK(u, B1)} h-0--

for each K E (0, oo). Hence, it follows from (2.9) that

Pr {NK(u, Bh) > 0} E{NK(u, B1)} (2.13) lim =

h--, Pr {N(u, Bh)> 0} E{N(u, B1)}

for each K E (0, oo). If I(K, )(') is the indicator function of (K, oo) then

E{NK(u, B1) }=5 E{N(u,

B1)I(K,o)

[max sup ID X(s)

n2o(no

+ 1) --n1?7-% (2.14) 1)Pr max sup ID "X(s) > K - 2 arEJE2 SEB2

+ nPr{N(u,B,)=n} n=no+l




for all no e and K e (0, o). Since E{N(u, B1)} < o (from Section 5.2 of Adler

(1981a) and the assumptions in Section 1.2), the second term of (2.14) can be made small by making no large while, from (1.3) and (1.4), the first term can be made small by making K large for each fixed no. Thus, since PK,h is non-increasing as K--0 for each h, it follows from (2.12) and (2.13) that, given E > 0, there exists Ko = Ko(E) E (0, oo) and ho = ho(E, Ko(E)) E (0, 1) such that PK,h : E for all K e

(Ko, 0.) and h E (0, ho]. Consequently, since

PK,h - Pr max ID X(O)I > KJ/Pr {N(u, Bh) > 0} K, fa EM2 I

and Pr {N(u, Bh) > 0} > 0 for all K e (0, 0 ) and h E (0, 1), it follows from (1.3) that the lemma is true.

Lemma 2.3. For each E > 0 and re N,

lim lim Pr max WBr(D

X,

6)?=

E IN(u, Bh) >o =0. 6--*0

h--0 La E 62 J

This result is obtained in a similar manner to Lemma 2.2 except that, instead of NK(u, .), the following point process is used. Define N6,,(u, E, B) to be the number of points t E Bu such that

max OBs,() (D aX, 6) E

for all B E N, r e N and E, 6 > 0. It can be seen from Theorem 2.1 that Xu

has the same distribution in (C2, Y2) as the conditional field X given that X has an upcrossing of u in the Nth direction if h.w. conditioning is used. Consequently, Xu will be called a model field for X near an upcrossing of the level u in the Nth direction. It follows from the definition of Xu (see (2.2)-(2.7)) that Xu behaves in the appropriate way at t = 0; that is, with

probability 1, Xu has an upcrossing of u in the Nth direction at t = 0. It can also be seen that the derivatives Xu(t)/ItN and

ti ,i=l, -,

N-1; t i< 1

are given by r7 and uS20 - vu respectively at t = 0. For u * 0, the basic shape of Xu near t = 0 is determined by the term uR(t) and, to a lesser extent, the terms -qra(t) and - ub(t), while the term A(t) behaves as a random perturbation. Finally, an obvious consequence of Theorem 2.1 is that Pr (Xu

E C2) = 1.

2.2. The model field and Palm distributions. In Wilson and Adler (1982), it was shown that certain empirical finite-dimensional distributions converged weakly to the finite-dimensional distributions of X, with probability 1. This followed from the

ergodic theorem (see, for example, Theorem 6.5.1 of Adler (1981a)) under the




assumption that X was ergodic. Here, this result will be strengthened to full weak convergence in (C2, 2V) where C2 = {x e C2:x(O)= u} and (2 = {A e 2:A c C2} is the o-field generated by the open sets in (C2, d2). Before doing so, some preliminary results and notation are needed.

Firstly, let N(u, A, B) denote the number of points t E Bu such that {X(t + s), s e DN} eA for all A e 2 and Be RuN, and let Pu

be the probability measure on

(C2, 2) defined by

P,(A) = E{N(u, A, B1)}/E{N(u, B1)}

for all A e . Pu is called the Palm or ergodic distribution of X near an upcrossing of u in the Nth direction. It follows from the definition of Xu (see (2.2)-(2.7)) that Pr (Xu(O) = u) = 1. Furthermore, since the finite-dimensional distributions of Xu are given by those of Pu (see Theorems 2.1 and 2.3 of Wilson and Adler (1982)), and the distribution of

Xu on (C2, 2) is completely determined by its finite-dimensional

distributions (see Section 2 of Wilson (1986)), it follows that Pu is the distribution of X, on (C2, VC2). The following result will be used to define a sequence of empirical distributions and to show that they converge weakly to Pu in (CI, C() with probability 1. Since it is easily obtained by combining the results on pages 144 and 145 of Adler (1981a) and Theorem 3.1 of Leadbetter (1972), no proof will be given.

Lemma 2.4. If X is ergodic, then

lim r-NN(u, A, B,) = E{N(u, A, B1)}

with probability 1 for each A E .

It follows from Lemma 2.4 that, if X is ergodic, then

(2.15) lim T-NN(u, B,) = E{N(u, B,)}

with probability 1 and so, lim_.,

Pr {N(u, B,) > 0} = 1. Hence, a set function P, can be defined on C2 by

Pu,,(A) = N(u, A, B,.)/N(u, B,.)

for all A E

and > 0 where r* = inf {t E [t, o): N(u, B,) > 0}. P,,, is obviously a random probability measure on (C2, (C2) with probability 1, and it can be described as the empirical distribution of X around the locations in B, of upcrossings of u in the Nth direction.

The following result is similar to those obtained by Lindgren (1977), (1983), (1984) and Leadbetter et al. (1983) for one-parameter processes.

Theorem 2.2. If X is ergodic, then, with probability 1, P,,, converges weakly to P, in (Ci, Vi) as z- oc.




Proof. It follows from Lemma 2.4 and (2.15) that

(2.16) lim Pu,(A) = Pu(A) t---oo

with probability 1 for each A e 2. Since (C2, d2) is separable, (see, for example, Section 2 of Wilson (1986)), it follows from Corollary 1 on page 14 of Billingsley (1968) that (2.16) holds for all A e 2 with probability 1 and so, the theorem is true. (The class V01 of Corollary 1 can be taken to be the subclass of T2 of all finite intersections of open spheres in C2 with rational radii and centers belonging to some countable dense subset of C2.)

It follows from Theorem 2.2 that Xu has a relative frequency interpretation as the empirical distribution of X around the locations (in RN) of X's upcrossings of u in the Nth direction. Thus, the distribution of Xu (that is, Pu) allows frequency statements to be made about a single realization of X over a large domain. This motivates the use of Xu as an appropriate model for the behavior of X in the neighborhood of a random location of an upcrossing of u in the Nth direction. (Note that the equivalent v.w. model would describe the behavior of X near a fixed location of an upcrossing of a random level in the Nth direction.) Finally, it follows from the remarks at the end of the previous section that ri and uS20 - vu represent the derivatives XN(t) and X"(t) respectively at the points t at which X has an upcrossing of u in the Nth direction.

2.3. The distribution of Xu(t) as Itl - oo. In this section, the behavior of Xu at a large distance from the upcrossing of u in the Nth direction at t = 0 is examined. For this purpose, let X,, denote the translated model field given by Xu,,(t) = Xu(s + t) for t

e N. Under appropriate weak dependence or mixing conditions, it would be

expected that, for large Isi, the distribution of X,~ in (C2, T2) would be asymptotically the same as that of X. Since X is Gaussian, such conditions should be expressible in terms of conditions on the covariance function of X (and hence, on the covariance functions of its derivatives).

For convenience, let ,1s, 2,- - - be a monotonic sequence in RN (that is, if

Sn,= (Sln,''', sNn)T, then

s1l, Sj2, '"

is monotonic for each j = 1, 2, - - -, N) and assume that IsI--, oo as n -- oo. The following result will be obtained from Theorem 2.1 of Wilson (1986).

Theorem 2.3. If

(2.17) lim max sup ID R(s, + t)l = 0 n---*oo aE?4 tEB,

for all r e N then X,~,~~4 X in (C2, ~e2) as n -oo

Proof. Firstly, (2.3), (2.4), and (2.17) imply that

lim max sup ID "a(s, + t)l = 0 n--oo ae 2 t~EBr




and

lim max sup ID "b(s, + t)I = 0 n-i-oo

aaER2 tEBr

for all re .

Therefore, it follows from (2.17) and the distributions of r7 and vu

(see (2.5)-(2.6)) that

d2(Xu,s,, An)-* 0 as n

-* oo with probability 1 where A,(t) = A(s, +

t) for t E N. Hence, it is sufficient to show An, X in (C2, 2)

as n --, o in order to prove the theorem. This will be done by showing the conditions of Theorem 2.1 of Wilson (1986) hold for (A,, n E N).

It follows from the definition of C, the covariance function of A (see (2.7)), and

(2.17) that

(2.18) lim max sup E{D "An (s)D "A(t)} - E{D "X(s)D "X(t)}I = 0 n---oo aoa 2 s,tEBr

for all re N. Therefore, the characteristic function of the finite-dimensional distributions of A, converges to that of the finite-dimensional distributions of X; that is, condition (i) of Theorem 2.1 of Wilson (1986) holds. (Proposition 2.1 of Wilson and Adler (1982) states a similar result to this.) Furthermore, (2.18) implies (by the same method) that D "A,(0) 0 D "X(O) as n -- oo for each a E 9X 2. Hence, it follows from (1.3) that, given E > 0, there exists a finite K > 0 such that

Pr max D"A.(0) 1 K} < f a E ?2

2

for all n E N; that is, condition (ii) of Theorem 2.1 of Wilson (1986) holds. It is easy to see that the third condition of Theorem 2.1 of Wilson (1986) will hold

if

(2.19) lim lim Pr {(OB,(D aAn, 6) E} = 0 6-0 n---oo

for all a E j2, r E N and E > 0. Hence, assume throughout the following that a E 22 and re E . Theorem 3.3.3 of Adler (1981a) will be used to show (2.19).

Since the matrices in the second and third terms of (2.7) are positive definite, it follows from (2.7) and the homogeneity of X that

sup E{[D A,(s) - D "A,(t)]2} g()2 s,tJB, Is-tl-6V N

where

g'(6)2= sup E{[DaX(t) - DaX(O)]2}

for each 6 > 0. Furthermore, if 6 e (0, 1/N) then it follows from (1.2) that

gf(6) s K(-ln 6)-(1+ y)/2

for some y > 0 and finite K >0. Hence, it follows from Theorem 3.3.3 of Adler




(1981a) (in a similar manner to the proof of the second half of Theorem 3.4.1 of Adler (1981a)) that

(2.20) Pr {oB,(D'A,, 6) E) -- (4aV)N

exp {N (In 6)[EK-1(-ln 6)Y/2- 1]2}

for each E > 0, 6 e (0, 1/N), nE N and some finite K > 0. Since the right-hand side of (2.20) is independent of n and converges to 0 as 6--*0, it follows that (2.19) is true for all a E 2, rEN and E > 0. Thus,

lim lim Pr { max (OB,(D "An, 6) EE =0 6--*0 n-.-oo a E D2

for each rE and E > 0; that is, condition (iii) of Theorem 2.1 of Wilson (1986) holds. Hence, A, 4 X in (C2, 62) as n -- oo and so, combining this with the first part of the proof, it follows that the theorem is true.

It can be seen from the proof of Theorem 2.3 that, if (2.17) holds, then, at large distances from the upcrossing of u in the Nth direction at t = 0, the behavior of A dominates X, and A is behaving like X. Note that (2.17) will hold if, for example, R and its partial derivatives up to order 4 are non-increasing in t for large It[ and converge pointwise to 0 as t --- oo. A simple example would be R(t) = exp [-ltTSt] for all t, where S > 0 is the matrix of second-order spectral moments.

One advantage of showing full weak convergence in (C2, 62) and (C2, (2u)

in Theorems 2.1-2.3 is that the distributions of functionals of the fields of these theorems can be investigated via the continuous mapping theorem and related results (see, for example, Lindgren (1977)). Theorems 2.1-2.3 will not be used in this paper to obtain such results. However, this method will be used in Section 3.2 to obtain asymptotic distributions of certain functionals of Xu as u - oo.

Finally, it was shown in Wilson and Adler (1982) that, when N = 1, Xu is simply the Slepian model process obtained by Lindgren (1977), (1983) to model the behavior of a random process near an upcrossing of the level u.

3. Asymptotic properties of Xu as u --+

In Section 3 of Wilson and Adler (1982), it was shown that Xu could be approximated by elliptical paraboloids in the sense that the finite-dimensional distributions of the normalized field X*(t) = u(Xu(t/u)

- u), t R N, converged weakly to those of X*(t) = rltN - ltTSt, t e RN, as u --- 0>. It was also shown that the

asymptotic distribution of the normalized location and height of Xu's 'closest' maximum to t = 0 was given by that of the location and height of X*'s maximum for large u. In Section 3.1, the first of these results will be strengthened to full weak convergence and, in Section 3.2, a similar result to the continuous mapping theorem will be used to obtain the asymptotic distributions of two functionals of X,.

3.1. The asymptotic distribution of X,. Obviously, the asymptotic distribution in




(C', 1) of Xu* depends on the distribution of vu for large u and on how R, a, b and

A (and their first- and second-order derivatives) behave near t = 0. The first of the following lemmas is given in Wilson and Adler (1982) and shown in Lindgren (1972). It is included for the sake of completeness. The second is an obvious consequence of the behavior of R and its derivatives near t = 0, and follows from (1.2), (2.3), (2.4) and (2.7). Neither of these lemmas will be proved here.

Lemma 3.1. If v is an (N(N - 1)/2)-dimensional, Gaussian random variable with zero-mean and covariance matrix E, then vu-

v as n --* c.

Lemma 3.2. There exists a finite K > 0 such that

(3.1) "Da[R(t) - 1 + tTIst] - K ItI4-i

(3.2) ID a[a(t) + tN] I K It'3-i

(3.3) ID "b(t)I 5 K It12-i as tl---0 for each a E i' and i = 0, 1, 2. Furthermore, there exists a random variable K satisfying Pr (0 < K < o) = 1 and such that, with probability 1,

(3.4) ID aA(t)l K It(2-i

as tf --- 0 for each a Ei' and i = 0, 1, 2 (since Pr (AE C2) = 1 and Pr (D "A(0)= 0) = 1 for all a E E.91).

The following result is similar to Theorem 3.1 of Lindgren (1983) (see Theorem 10.4.2 of Leadbetter et al. (1983) also) and it will be obtained using the space (C2, 62) given in Section 2.2.

Theorem 3.1.

X X*X in (C2, 6) as u---*.

Proof. Firstly, it follows from the definitions of Xu, X* and X* that Pr {Xu e C} = Pr {X* E C } = 1 for all u. Hence, the theorem will be true if

(3.5) d2(X*, X*) O as u --- oo (see pp. 24-27 of Billingsley (1968)). For convenience, let

gu(t)= u2[R(tlu) - 1 + ?(t/u)TS(t/u)]

and

h,(t) = u[a(tlu) + tN/u]

for t =IUN. (3.5) will be obtained by considering each of the terms of

X*(t) - X*(t) = gu(t) - rih,(t) - vuub(t/u)

+ uA(t/u)

for t El lN. Since D "X(t)= D aXu(t/u) for a e '1 and D "X (t)= u-1D "Xu(t/u) for a E 0J2, it follows from (3.1), (3.2) and (3.3) respectively, that g,,

hu and the




components of ub(./u)

all converge to the zero function on RWN with respect to d2 as

u---oo. Furthermore, (3.4) implies that, with probability 1, uA(./u) also converges to the zero function on RWN with respect to d2 as u -- oo. Therefore, it follows from Lemma 3.1 that (3.5) is true.

Unlike Theorem 3.1 of Lindgren (1983), a strong result has not been obtained here due to the dependence of vu on u. However, since Xu is constructed using weak convergence arguments, a stronger result than the above is unnecessary.

It follows from Theorem 3.1 that, for large u, Xu has asymptotically the same distribution in (C2, 62) as that of the random elliptical paraboloid given by

Xu(t) = u(1 - ?tTSt) + /tN

for teR .

Theorem 3.1 is similar to results obtained by Lindgren (1972c), Nosko (1969a-b), (1986) and Berman (1985). In particular, under the assumptions of Section 1.2, the random field given by (8.4) of Berman (1985) is

Y(t) = & + VTt - tTSt,

for tER N, where ? has an exponential distribution with mean one, V is an N-dimensional Gaussian random variable with zero-mean vector and covariance matrix S, and ? and V are independent. Berman (1985) obtains Y as the limit of u(X(-/u) - u) I X(O) > u as u -*oo and so, Y models the behavior of X above high levels. The random variables ?/u and V represent the height X(O) - u of X above u at t = 0 and the vector X'(O) of first-order derivatives of X at t = 0 respectively. (Similar processes to Y are constructed in Berman (1974), (1982a-b), (1983), (1987) for other processes, including some which are non-Gaussian.)

3.2. Characteristics of high excursion sets. In this section, the asymptotic distributions of the Lebesgue measure of a particular excursion set of Xu and the height of X, over this set will be obtained. Some notation is needed first.

Definition 3.1. Let A,(x, B) = {t E B :x(t) -

u} denote the excursion set of x above the level u in B where Be RN and x e C . Furthermore, for each s e A(x, B), let Au,s(x, B) denote the set of all points t E A(x, B) such that s and t can be connected by a continuous path in Au(x, B). Au,s(x, B) will be called the excursion set of x above the level u in B connected to s. For convenience, let A,(x) = Au(x, RN)

and Au,,(x)

= Au,~(x, yRN).

The set Au,,(x) is similar to the definition of the rejection of x at level u containing the point (s, x(s)) given by Nosko (1969b), (1986). In addition, if N = 1 and x has a u-upcrossing at t, and its next u-downcrossing at t2 then Au,s(x)= [tl,

t2] for all s e [tl, t2].

It is of interest to know some of the characteristics of the random set Au,o(X,) as u- oo. Since it is possible for this set to be unbounded with positive probability, the characteristics of the set Au,o(X,, B,) will be studied instead, where t is some




(large) positive finite constant. (Conditions under which Au,o(Xu) will be bounded almost surely are given in Molcanov and Stepanov (1979). However, the methods used here do not require additional conditions on X.) In Theorem 3.2, the asymptotic distributions of the Lebesgue measure of Au,o(Xu, B,) and the maximum height of Xu over Au,o(Xu, B,) will be obtained for large u.

Denote the N-dimensional Lebesgue measure of Au,o(Xu, B,), AO,o(X*, Bu,) and

Ao,o(X*) by L,r,, Lu,L

and L* respectively, and let

M,U, = sup {Xu(t) : t Au,o(Xu, B,)},

M*, = sup {X*(t): t AO,o(X*, B,,)}

M* = sup {X*(t): t Ao,o(X*)}

for all u > 0 and where rte (0, 00) is fixed. It is easy to see that

(3.6) Lu,1=u -N Lu, and

(3.7) M,u, = u + u -1Mu, . Theorem 3.2.(i)

L,-, L* as u-- oo where

L* = (det S)-lAN(Bl)(Jr/I)N

and AN denotes N-dimensional Lebesgue measure. (ii)

M,,4 M* as u---0o and M* = 712/(202) has an exponential distribution with

mean 1.

Proof. It follows from Theorem 5.5 of Billingsley (1968) and Theorem 3.1 that, in order to prove the theorem, it is sufficient to show that

(3.8) lim N{Ao,o(x,, B,,)} = AAN{Ao,o(x)} u----

and

(3.9) lim sup {x,(t): t Ao,o(x,, B,,)} = sup {x(t): t E Ao,o(x)

for all sequences {x, E C2"u NJ} such that

(3.10) lim d2(xu, x) = 0 u--)oo

and all sample paths x of X*; that is, for all

(3.11) x(t) = vtN - t St

for t E n", where v e (0, oo). Consequently, assume x, {x,)u=a

and v are as above. It will first be shown that Ao,o(x,, B,,), Ao,o(x,) and Ao(x,, B,) are all identical

for some r and large u. Choose r e N such that A_1,o(x) c B,. It follows from (3.10)




and (3.11) that, given E > 0, there exists a positive integer uo = uo(E) - T r/ such that

(3.12) D ax(t) - E - D ax,(t) - D ax(t) + E

for all t E Br, a E q 2 and u - uo. Hence, if E < 1 then Ao(xu, Br) c A-_,o(x) B0 (the interior of Br), and so,

(3.13) A0, o(xu) cAo(x, Br)

for all u - uo. Therefore, Ao,o(x,, Bu,) and Ao,o(x,) are identical for all u - uo (since uo r/nr).

Furthermore, it follows from (3.12) and (3.13) that, if E is small enough so that

(3.14) {t E Br: Ix(t)l E} {tE) Br"

:max DM 'x(t)l > E,

then x, only has stationary points near the location of the maximum of x (that is,

t= S-'(0)) and, in particular, xu

has no stationary points in the neighborhood of

the set {tEBr,:xu(t) = u} for all u -uo. Hence, every point in Ao(xu, Br) can be connected to t = 0 by a continuous curve in Ao(x, Br) and so, from (3.13), Ao,o(x,) and Ao(xu, Br) are identical for all u -> o.

The limits (3.8) and (3.9) will next be established. Firstly, if EE (0, 1) satisfies (3.14) and u - uo then, from (3.11),

IAN{AO(xu, Br)} - AN4{Ao,o(x)}I f AN{t E B,: Ix(t)l E} I

(VN2

N12

_ IV2 N12

= (det S)2,N(B)

+ N/2 2 N/2

which can be made small by making E small. Thus, (3.8) is true and so, part (i) of the theorem holds. The expression for L* can be easily obtained from the form of X*.

Similarly, if E E (0, 1) satisfies (3.14) and u >- u then

sup {x,(t) : t Ao,o(xu, B,)} = sup {Xu(t) : t E Ao(x, B,)}

and so, it follows from (3.12) that

Isup {x(t): t E Ao(xu, Br)} - sup {x(t): t E Ao,o(x)}l - sup IXu(t) - x(t)l e tEB,

for all u - uo. Therefore, (3.9) is true and so, part (ii) of the theorem holds. Again, the expression for M* is easily obtained from the form of X*.

It follows from (3.6), (3.7) and Theorem 3.2 that L,~, and M,~, have asymptotic distributions given by those of

u-NL* = (det S)- iN(Bl)(Jl/u)N




and

u + u-IM* = u + rI2/(2ua2)

respectively if u is large. These are obviously the appropriate functionals of X. These results are similar to those obtained by Nosko (1969a-b), (1986) and Belayev (1972). They are also closely related to the work done on sojourns and maxima by a number of authors (see, for example, Berman (1974), (1982), (1985), (1987) and Adler (1978)).

It is easy to see that the asymptotic distributions of other functionals of X, could also be investigated using the above methods. Since (C2, C2) has been used, these functionals can obviously involve the first- and second-order partial derivatives of X,. Also, since the functions in C2 have domain FRN instead of some bounded subset of RN1, the above results have been stated more succinctly than would otherwise have been possible.

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Article Contentsp. 756p. 757p. 758p. 759p. 760p. 761p. 762p. 763p. 764p. 765p. 766p. 767p. 768p. 769p. 770p. 771p. 772p. 773p. 774

Issue Table of ContentsAdvances in Applied Probability, Vol. 20, No. 4 (Dec., 1988), pp. 695-920Volume Information [pp. 918-920]Front MatterProbabilistic Analysis of a Learning Matrix [pp. 695-705]Distributions That Are Both Subexponential and in the Domain of Attraction of an Extreme-Value Distribution [pp. 706-718]Sample Path Behaviour of Surfaces at Extrema [pp. 719-738]An Age-Dependent Counting Process Generated from a Renewal Process [pp. 739-755]Model Fields in Crossing Theory: A Weak Convergence Perspective [pp. 756-774]Symbolic Computation and the Diffusion of Shapes of Triads [pp. 775-797]Bivariate Exponential and Geometric Autoregressive and Autoregressive Moving Average Models [pp. 798-821]Some ARMA Models for Dependent Sequences of Poisson Counts [pp. 822-835]On a Reduction Principle in Dynamic Programming [pp. 836-851]Analysis of a Non-Preemptive Priority Multiserver Queue [pp. 852-879]Exponential Expansion for the Tail of the Waiting-Time Probability in the Single-Server Queue with Batch Arrivals [pp. 880-895]Perturbation Analysis of a Phase-Type Queue with Weakly Correlated Arrivals [pp. 896-912]Letter to the EditorRuin Probabilities Expressed in Terms of Storage Processes [pp. 913-916]

Acknowledgement [p. 917]Back Matter

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Model Fields in Crossing Theory: A Weak Convergence Perspective