# The expected Euler characteristic approximation to excursion probabilities
of smooth Gaussian random fields with general variance functions

Dan Cheng
Arizona State University

###### Abstract

Consider a centered smooth Gaussian random field $\\{X(t),t\in T\\}$ with a
general (nonconstant) variance function. In this work, we demonstrate that as
$u\to\infty$, the excursion probability ${\mathbb{P}}\\{\sup_{t\in T}X(t)\geq
u\\}$ can be accurately approximated by ${\mathbb{E}}\\{\chi(A_{u})\\}$ such
that the error decays at a super-exponential rate. Here, $A_{u}=\\{t\in
T:X(t)\geq u\\}$ represents the excursion set above $u$, and
${\mathbb{E}}\\{\chi(A_{u})\\}$ is the expectation of its Euler characteristic
$\chi(A_{u})$. This result substantiates the expected Euler characteristic
heuristic for a broad class of smooth Gaussian random fields with diverse
covariance structures. In addition, we employ the Laplace method to derive
explicit approximations to the excursion probabilities.

Keywords: Gaussian random fields, excursion probability, excursion set, Euler
characteristic, nonconstant variance, asymptotics, super-exponentially small.

Mathematics Subject Classification: 60G15, 60G60, 60G70.

##  1 Introduction

Let $X=\\{X(t),\,t\in T\\}$ represent a real-valued Gaussian random field
defined on the probability space $(\Omega,\mathcal{F},{\mathbb{P}})$, where
$T$ denotes the parameter space. The study of excursion probabilities, denoted
as ${\mathbb{P}}\\{\sup_{t\in T}X(t)\geq u\\}$, is a classical and fundamental
problem in both probability and statistics. It finds extensive applications
across numerous domains, including $p$-value computations, risk control and
extreme event analysis, etc.

In the field of statistics, excursion probabilities play a critical role in
tasks such as controlling family-wise error rates [13, 14], constructing
confidence bands [10], and detecting signals in noisy data [8, 13]. However,
except for only a few examples, computing the exact values of these
probabilities is almost impossible. To address this challenge, many
researchers have developed various methods for precise approximations of
${\mathbb{P}}\\{\sup_{t\in T}X(t)\geq u\\}$. These methods encompass
techniques like the double sum method [6], the tube method [9] and the Rice
method [3, 4]. For comprehensive theoretical insights and related
applications, we refer readers to the survey by Adler [1] and the monographs
by Piterbarg [6], Adler and Taylor [2], and Azaïs and Wschebor [4], as well as
the references therein.

In recent years, the expected Euler characteristic (EEC) method has emerged as
a powerful tool for approximating excursion probabilities. This method,
originating from the works of Taylor et al. [12] and Adler and Taylor [2],
provides the following approximation:

${\mathbb{P}}\bigg{\\{}\sup_{t\in T}X(t)\geq
u\bigg{\\}}={\mathbb{E}}\\{\chi(A_{u})\\}+{\rm error},\quad{\rm as}\
u\to\infty,$ (1.1)

where $\chi(A_{u})$ represents the Euler characteristic of the excursion set
$A_{u}=\\{t\in T:X(t)\geq u\\}$. This approximation (1.1) is highly elegant
and accurate, primarily due to the fact that the principle term
${\mathbb{E}}\\{\chi(A_{u})\\}$ is computable and the error term decays
exponentially faster than the major component. However, it is essential to
note that this method assumes a Gaussian field with constant variance,
limiting its applicability in various scenarios.

In this paper, we extend the EEC method to accommodate smooth Gaussian random
fields with general (nonconstant) variance functions. Our main objective is to
demonstrate that the EEC approximation (1.1) remains valid under these
conditions, with the error term exhibiting super-exponential decay. For a
precise description of our findings, please refer to Theorem 3.1 below. Our
derived approximation result shows that the maximum variance of $X(t)$,
denoted by $\sigma_{T}^{2}$ (see (2.1) below), plays a pivotal role in both
${\mathbb{E}}\\{\chi(A_{u})\\}$ and the super-exponentially small error. In
our analysis, we observe that the points where $\sigma_{T}^{2}$ is attained
make the most substantial contributions to ${\mathbb{E}}\\{\chi(A_{u})\\}$.
Building on this observation, we establish two simpler approximations: one in
Theorem 3.2, which incorporates boundary conditions on nonzero derivatives of
the variance function over points where $\sigma_{T}^{2}$ is attained, and
another in Theorem 3.3, assuming only a single point attains $\sigma_{T}^{2}$.

In general, the EEC approximation can be expressed as an integral using the
Kac-Rice formula, as outlined in (3.2) in Theorem 3.1. While [12, 2] provided
an elegant expression for ${\mathbb{E}}\\{\chi(A_{u})\\}$ termed the Gaussian
kinematic formula, this expression heavily relies on the assumption of unit
variance, which simplifies the calculation. In our case, where the variance
function of $X(t)$ varies across $T$, deriving an explicit expression for
${\mathbb{E}}\\{\chi(A_{u})\\}$ becomes challenging. Instead, we apply the
Laplace method to extract the term with the leading order of $u$ from the
integral, leaving a remaining error that is
${\mathbb{E}}\\{\chi(A_{u})\\}o(1/u)$. For a more detailed explanation, we
offer specific calculations in Sections 8 and 9. To intuitively grasp the EEC
approximation, one can roughly consider the major term as
$g(u)e^{-u^{2}/(2\sigma_{T}^{2})}$, while the error term diminishes as
$o(e^{-u^{2}/(2\sigma_{T}^{2})-\alpha u^{2}})$, where $g(u)$ is a polynomial
in $u$, and $\alpha>0$ is a constant.

The structure of this paper is as follows: We begin by introducing the
notations and assumptions in Section 2. In Section 3, we present our main
results, including Theorems 3.1, 3.2, and 3.3. To understand our approach, we
outline the main ideas in Section 4 and delve into the analysis of super-
exponentially small errors in Sections 5 and 6. Finally, we provide the proofs
of our main results in Section 7. In Section 8, we apply the Laplace method to
derive explicit approximations (Theorems 8.3 and 8.4) for cases where a unique
maximum point of the variance is present. In Section 9, we demonstrate several
examples that illustrate the evaluation of EEC and the subsequent
approximation of excursion probabilities.

##  2 Notations and assumptions

Let $\\{X(t),\,t\in T\\}$ be a real-valued and centered Gaussian random field,
where $T$ is a compact rectangle in ${\mathbb{R}}^{N}$. We define

$\begin{split}\nu(t)=\sigma_{t}^{2}={\rm Var}(X(t))\quad\text{\rm and
}\quad\sup_{t\in T}\nu(t)=\sigma_{T}^{2}.\end{split}$ (2.1)

Here, $\nu(\cdot)$ represents the variance function of the field and
$\sigma_{T}^{2}$ is the maximum variance over $T$. For a function $f(\cdot)\in
C^{2}({\mathbb{R}}^{N})$ and $t\in{\mathbb{R}}^{N}$, we introduce the
following notations on derivatives:

$\begin{split}f_{i}(t)&=\frac{\partial f(t)}{\partial t_{i}},\quad
f_{ij}(t)=\frac{\partial^{2}f(t)}{\partial t_{i}\partial t_{j}},\quad\forall
i,j=1,\ldots,N;\\\ \nabla
f(t)&=(f_{1}(t),\ldots,f_{N}(t))^{T},\quad\nabla^{2}f(t)=\left(f_{ij}(t)\right)_{i,j=1,\ldots,N}.\end{split}$
(2.2)

Let $B\prec 0$ (negative definite) and $B\preceq 0$ (negative semi-definite)
denote that a symmetric matrix $B$ has all negative or nonpositive
eigenvalues, respectively. Additionally, we use ${\rm Cov}(\xi_{1},\xi_{2})$
and ${\rm Corr}(\xi_{1},\xi_{2})$ to represent the covariance and correlation
between two random variables $\xi_{1}$ and $\xi_{2}$. The density of the
standard Normal distribution is denoted as $\phi(x)$, and its tail probability
is $\Psi(x)=\int_{x}^{\infty}\phi(y)dy$. Let ${\mathbb{S}}^{j}$ be the
$j$-dimensional unit sphere.

Consider the domain $T=\prod^{N}_{i=1}[a_{i},b_{i}]$, where
$-\infty<a_{i}<b_{i}<\infty$. We draw from the notation established by Adler
and Taylor in [2] to demonstrate that $T$ can be decomposed into the union of
its interior and lower-dimensional faces. This decomposition forms the basis
for calculating the Euler characteristic of the excursion set $A_{u}$, as
elaborated in Section 3.

Each face $K$ of dimension $k$ is defined by fixing a subset
$\tau(K)\subset\\{1,\ldots,N\\}$ of size $k$ and a subset
$\varepsilon(K)=\\{\varepsilon_{j},j\notin\tau(K)\\}\subset\\{0,1\\}^{N-k}$ of
size $N-k$ so that

$\begin{split}K=\\{t=(t_{1},\ldots,t_{N})\in T:\,&a_{j}<t_{j}<b_{j}\ {\rm if}\
j\in\tau(K),\\\ &t_{j}=(1-\varepsilon_{j})a_{j}+\varepsilon_{j}b_{j}\ {\rm
if}\ j\notin\tau(K)\\}.\end{split}$

Denote by $\partial_{k}T$ the collection of all $k$-dimensional faces in $T$.
The interior of $T$ is designated as $\overset{\circ}{T}=\partial_{N}T$, while
the boundary of $T$ is formulated as $\partial
T=\cup^{N-1}_{k=0}\cup_{K\in\partial_{k}T}K$. This allows us to partition $T$
in the following manner:

$T=\bigcup_{k=0}^{N}\partial_{k}T=\bigcup_{k=0}^{N}\bigcup_{K\in\partial_{k}T}K.$

For each $t\in T$, let

$\begin{split}\nabla
X_{|K}(t)&=(X_{i_{1}}(t),\ldots,X_{i_{k}}(t))^{T}_{i_{1},\ldots,i_{k}\in\tau(K)},\quad\nabla^{2}X_{|K}(t)=(X_{mn}(t))_{m,n\in\tau(K)},\\\
\Sigma(t)&={\mathbb{E}}\\{X(t)\nabla^{2}X(t)\\}=({\mathbb{E}}\\{X(t)X_{ij}(t)\\})_{1\leq
i,j\leq N},\\\
\Sigma_{K}(t)&={\mathbb{E}}\\{X(t)\nabla^{2}X_{|K}(t)\\}=({\mathbb{E}}\\{X(t)X_{ij}(t)\\})_{i,j\in\tau(K)},\\\
\Lambda(t)&={\rm Cov}(\nabla X(t))=({\mathbb{E}}\\{X_{i}(t)X_{j}(t)\\})_{1\leq
i,j\leq N},\\\ \Lambda_{K}(t)&={\rm Cov}(\nabla
X_{|K}(t))=({\mathbb{E}}\\{X_{i}(t)X_{j}(t)\\})_{i,j\in\tau(K)}.\end{split}$
(2.3)

For each $K\in\partial_{k}T$, we define the _number of extended outward maxima
above $u$ on face $K$_ as

$\begin{split}M_{u}^{E}(K)&:=\\#\\{t\in K:X(t)\geq u,\nabla
X_{|K}(t)=0,\nabla^{2}X_{|K}(t)\prec 0,\varepsilon^{*}_{j}X_{j}(t)\geq
0,\forall j\notin\tau(K)\\},\end{split}$

where $\varepsilon^{*}_{j}=2\varepsilon_{j}-1$, and define the _number of
local maxima above $u$ on face $K$_ as

$\begin{split}M_{u}(K)&:=\\#\\{t\in K:X(t)\geq u,\nabla
X_{|K}(t)=0,\nabla^{2}X_{|K}(t)\prec 0\\}.\end{split}$

Clearly, $M_{u}^{E}(X,K)\leq M_{u}(X,K)$.

For each $t\in T$ with $\nu(t)=\sigma_{T}^{2}$, we define the index set
$\mathcal{I}(t)=\\{\ell:\nu_{\ell}(t)=0\\}$ representing the directions along
which the partial derivatives of $\nu(t)$ vanish. If $t\in K\in\partial_{k}T$
with $\nu(t)=\sigma_{T}^{2}$, then we have $\tau(K)\subset\mathcal{I}(t)$
since $\nu_{\ell}(t)=0$ for all $\ell\in\tau(K)$. It is worth noting that
since $\nu_{i}(t)=2{\mathbb{E}}\\{X_{i}(t)X(t)\\}$, we can also express this
index set as $\mathcal{I}(t)=\\{\ell:{\mathbb{E}}\\{X(t)X_{\ell}(s)\\}=0\\}$.

Our analytical framework relies on the following conditions for smoothness
(H1) and regularity (H2), in addition to curvature conditions (H3) or $({\bf
H}3^{\prime})$.

  * (H1)

$X\in C^{2}({\mathbb{R}}^{N})$ almost surely and the second derivatives
satisfy the _uniform mean-square Hölder condition_ : there exist constants
$C,\delta>0$ such that

$\begin{split}{\mathbb{E}}(X_{ij}(t)-X_{ij}(t^{\prime}))^{2}&\leq
C\|t-t^{\prime}\|^{2\delta},\quad\forall t,t^{\prime}\in T,\
i,j=1,\ldots,N.\end{split}$

  * (H2)

For every pair $(t,t^{\prime})\in T^{2}$ with $t\neq t^{\prime}$, the Gaussian
vector

$\begin{split}\big{(}&X(t),\nabla X(t),X_{ij}(t),X(t^{\prime}),\nabla
X(t^{\prime}),X_{ij}(t^{\prime}),1\leq i\leq j\leq N\big{)}\end{split}$

is non-degenerate.

  * (H3)

For every $t\in K\in\partial_{k}T$, $0\leq k\leq N-2$, such that
$\nu(t)=\sigma_{T}^{2}$ and $\mathcal{I}(t)$ contains at least two indices, we
have

$({\mathbb{E}}\\{X(t)X_{ij}(t)\\})_{i,j\in\mathcal{I}(t)}\prec 0.$ (2.4)

  * $({\bf H}3^{\prime})$

For every $t\in K\in\partial_{k}T$, $0\leq k\leq N-2$, such that
$\nu(t)=\sigma_{T}^{2}$ and $\mathcal{I}(t)$ contains at least two indices, we
have

$\left(\nu_{ij}(t)\right)_{i,j\in\mathcal{I}(t)}\preceq 0.$ (2.5)

Conditions (H3) and $({\bf H}3^{\prime})$ involve the behavior of the variance
function $\nu(t)$ at critical points, and they are closely related, as shown
in Proposition 2.1 below. Here we provide some additional insights into $({\bf
H}3^{\prime})$. Despite its initially technical appearance, $({\bf
H}3^{\prime})$ is in fact a mild condition that specifically applies to lower-
dimensional boundary points $t$ where $\nu(t)=\sigma_{T}^{2}$. In essence, it
indicates that the variance function should possess a negative semi-definite
Hessian matrix at these boundary critical points where $\nu(t)=\sigma_{T}^{2}$
while concurrently exhibiting at least two zero partial derivatives.

For example, in the 1D case, since $\mathcal{I}(t)$ contains at most one
index, there is no need to check $({\bf H}3^{\prime})$. Similarly, in the 2D
case, we only need to check $({\bf H}3^{\prime})$ or (2.5) when
$\sigma_{T}^{2}$ is achieved at corner points $t\in\partial_{0}T$ with
$\mathcal{I}(t)=\\{1,2\\}$. Moreover, if the variance function $\nu(t)$
demonstrates strict monotonicity in all directions across ${\mathbb{R}}^{N}$,
then $\mathcal{I}(t)=\emptyset$ and there is no need to verify $({\bf
H}3^{\prime})$.

######  Proposition 2.1.

The condition $({\bf H}3^{\prime})$ implies $({\bf H}3)$. In addition, $({\bf
H}3)$ implies that

$({\mathbb{E}}\\{X(t)X_{ij}(t)\\})_{i,j\in\mathcal{I}(t)}\prec 0,\quad\forall
t\in T\text{ with }\nu(t)=\sigma_{T}^{2}.$ (2.6)

###### Proof.

Taking the second derivative on both sides of
$\nu(t)={\mathbb{E}}\\{X(t)^{2}\\}$, we obtain
$\nu_{ij}(t)/2={\mathbb{E}}\\{X(t)X_{ij}(t)\\}+{\mathbb{E}}\\{X_{i}(t)X_{j}(t)\\}$,
implying

$({\mathbb{E}}\\{X(t)X_{ij}(t)\\})_{i,j\in\mathcal{I}(t)}=\frac{1}{2}(\nu_{ij}(t))_{i,j\in\mathcal{I}(t)}-({\mathbb{E}}\\{X_{i}(t)X_{j}(t)\\})_{i,j\in\mathcal{I}(t)}.$
(2.7)

Note that, as a covariance matrix,
$({\mathbb{E}}\\{X_{i}(t)X_{j}(t)\\})_{i,j\in\mathcal{I}(t)}$ is positive
definite by $({\bf H}2)$. Therefore, (2.5) implies (2.4), or equivalently
$({\bf H}3^{\prime})$ implies $({\bf H}3)$.

Next we demonstrate that $({\bf H}3)$ implies (2.6). It suffices to show (2.4)
for $k=N-1$ and $k=N$, and for the case that $\mathcal{I}(t)$ contains at most
one index, which complement those cases in $({\bf H}3)$.

(i) If $k=N$, then $t$ becomes a maximum point of $\nu$ within the interior of
$T$ and $\mathcal{I}(t)=\tau(K)=\\{1,\cdots,N\\}$, implying (2.5), and hence
(2.4) holds by (2.7).

(ii) For $k=N-1$, we consider two scenarios. If $\mathcal{I}(t)=\tau(K)$, then
$t$ becomes a maximum point of $\nu$ restricted on $K$, hence (2.4) is
satisfied as discussed above. If $\mathcal{I}(t)=\\{1,\cdots,N\\}$, then it
follows from Taylor’s formula that

$\nu(t^{\prime})=\nu(t)+(t^{\prime}-t)^{T}\nabla^{2}\nu(t)(t^{\prime}-t)+o(\|t^{\prime}-t\|^{2}),\quad
t^{\prime}\in T.$

Notice that $\\{(t^{\prime}-t)/\|t^{\prime}-t\|:t^{\prime}\in T\\}$ contains
all directions in ${\mathbb{R}}^{N}$ since $t\in K\in\partial_{N-1}T$,
together with the fact $\nu(t)=\sigma_{T}^{2}$, we see that $\nabla^{2}\nu(t)$
cannot have any positive eigenvalue, thus (2.5) and hence (2.4) hold.

(iii) Finally, it’s evident from the 1D Taylor’s formula that (2.5) is valid
when $\mathcal{I}(t)$ contains only one index. ∎

The condition (2.6) established in Proposition 2.1 serves as the fundamental
requirement for our main results, as demonstrated in Theorems 3.1, 3.2 and 3.3
below. As seen from Proposition 2.1, we can simplify (2.6) to condition $({\bf
H}3)$. Thus our main results will be presented under the assumption of
condition $({\bf H}3)$.

Furthermore, it is worth highlighting that, in practical applications,
verifying $({\bf H}3^{\prime})$ can often be a more straightforward process.
This condition directly pertains to the variance function $\nu(t)$, making it
easier to assess. Thus, Proposition 2.1 provides the flexibility to check
$({\bf H}3^{\prime})$ instead of $({\bf H}3)$. This insight simplifies the
verification procedure, enhancing the practical applicability of our results.

##  3 Main results

Here, we will present our main results Theorems 3.1, 3.2 and 3.3, whose proofs
are given in Section 7. Define the _number of extended outward critical points
of index $i$ above level $u$ on face $K$_ be

$\begin{split}\mu_{i}(K)&:=\\#\\{t\in K:X(t)\geq u,\nabla
X_{|K}(t)=0,\text{index}(\nabla^{2}X_{|K}(t))=i,\\\
&\qquad\qquad\qquad\varepsilon^{*}_{j}X_{j}(t)\geq 0\ {\rm for\ all}\
j\notin\tau(K)\\}.\end{split}$

Recall that $\varepsilon^{*}_{j}=2\varepsilon_{j}-1$ and the index of a matrix
is defined as the number of its negative eigenvalues. It is evident to observe
that $\mu_{N}(K)=M_{u}^{E}(K)$. It follows from (H1), (H2) and the Morse
theorem (see Corollary 9.3.5 or pages 211–212 in Adler and Taylor [2]) that
the Euler characteristic of the excursion set $A_{u}$ can be represented as

$\begin{split}\chi(A_{u})&=\sum^{N}_{k=0}\sum_{K\in\partial_{k}T}(-1)^{k}\sum^{k}_{i=0}(-1)^{i}\mu_{i}(K).\end{split}$
(3.1)

Now we state the following general result on the EEC approximation for the
excursion probability.

######  Theorem 3.1.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and $({\bf H}3)$. Then there exists a constant
$\alpha>0$ such that as $u\to\infty$,

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}\\\
&=\sum^{N}_{k=0}\sum_{K\in\partial_{k}T}(-1)^{k}\int_{K}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\mathbbm{1}_{\\{X(t)\geq u,\
\varepsilon^{*}_{\ell}X_{\ell}(t)\geq 0\ {\rm for\ all}\
\ell\notin\tau(K)\\}}\big{|}\nabla X_{|K}(t)=0\big{\\}}\\\ &\quad\times
p_{\nabla
X_{|K}(t)}(0)dt+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right)\\\
&={\mathbb{E}}\\{\chi(A_{u})\\}+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$ (3.2)

In general, computing the EEC approximation ${\mathbb{E}}\\{\chi(A_{u})\\}$ is
a challenging task because it involves conditional expectations over the joint
covariance of the Gaussian field and its Hessian, given zero gradient, which
vary across $T$. However, one can apply the Laplace method to extract the term
with the largest order of $u$ from ${\mathbb{E}}\\{\chi(A_{u})\\}$ such that
the remaining error is $o(1/u){\mathbb{E}}\\{\chi(A_{u})\\}$. Examples
demonstrating the Laplace method are presented in Section 9.

It is important to note that in the expression (3.2), when $k=0$, all terms
involving $\nabla X_{|K}(t)$ and $\nabla^{2}X_{|K}(t)$ vanish. Consequently,
if $k=0$, we treat the integral in (3.2) as the usual Gaussian tail
probabilities. This notation is also adopted in the results presented in
Theorems 3.2 and 3.3 below.

The proof of Theorem 3.1 reveals that points where the maximum variance
$\sigma_{T}^{2}$ is attained make the most significant contribution to
${\mathbb{E}}\\{\chi(A_{u})\\}$. Therefore, in many cases, the general EEC
approximation ${\mathbb{E}}\\{\chi(A_{u})\\}$ can be simplified. The following
result is based on the boundary condition (3.3) and is applicable at boundary
points where nonzero partial derivatives of the variance function occur when
$\sigma_{T}^{2}$ is reached.

######  Theorem 3.2.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and the following boundary condition

$\begin{split}\Big{\\{}t\in
J:\,\nu(t)=\sigma_{T}^{2},\prod_{i\notin\tau(J)}\nu_{i}(t)=0\Big{\\}}=\emptyset,\quad\forall\text{
face }J\subset T.\end{split}$ (3.3)

Then there exists a constant $\alpha>0$ such that as $u\to\infty$,

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}&=\sum^{N}_{k=0}\sum_{K\in\partial_{k}T}(-1)^{k}\int_{K}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\mathbbm{1}_{\\{X(t)\geq u\\}}\big{|}\nabla
X_{|K}(t)=0\big{\\}}\\\ &\quad\times p_{\nabla
X_{|K}(t)}(0)dt+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$

In other words, the boundary condition (3.3) indicates that, for any point
$t\in J$ attaining the maximum variance $\sigma_{T}^{2}$, there must be
$\nu_{i}(t)\neq 0$ for all $i\notin\tau(J)$. In particular, as an important
property, we observe that (3.3) implies the condition $({\bf H}3^{\prime})$
and hence $({\bf H}3)$. The following result provides an asymptotic
approximation for the special case where the variance function attains its
maximum $\sigma_{T}^{2}$ only at a unique point.

######  Theorem 3.3.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and $({\bf H}3)$. Suppose $\nu(t)$ attains its
maximum $\sigma_{T}^{2}$ only at a single point $t^{*}\in K$, where
$K\in\partial_{k}T$ with $k\geq 0$. Then there exists a constant $\alpha>0$
such that as $u\to\infty$,

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}\\\
&=\sum_{J}(-1)^{{\rm dim}(J)}\int_{J}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|J}(t)\mathbbm{1}_{\\{X(t)\geq u,\
\varepsilon^{*}_{\ell}X_{\ell}(t)\geq 0\ {\rm for\ all}\
\ell\in\mathcal{I}(t^{*})\setminus\tau(J)\\}}\big{|}\nabla
X_{|J}(t)=0\big{\\}}\\\ &\quad\times p_{\nabla
X_{|J}(t)}(0)dt+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right),\end{split}$

where the sum is taken over all faces $J$ of $T$ such that $t^{*}\in\bar{J}$
and $\tau(J)\subset\mathcal{I}(t^{*})$.

Employing the Laplace method, we will provide refined explicit approximation
results in Section 8 under the assumptions in Theorem 3.3. Furthermore, we
demonstrate several examples that illustrate the evaluation of approximating
excursion probabilities in Section 9.

##  4 Outline of the proofs

Here we show the main idea for proving the main results above. Let $f$ be a
smooth real-valued function, then $\sup_{t\in T}f(t)\geq u$ if and only if
there exists at least one extended outward local maximum above $u$ on some
face of $T$. Thus, under conditions $({\bf H}1)$ and $({\bf H}2)$, the
following relation holds for each $u\in{\mathbb{R}}$:

$\begin{split}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}=\bigcup_{k=0}^{N}\bigcup_{K\in\partial_{k}T}\\{M_{u}^{E}(K)\geq
1\\}\quad{\rm a.s.}\end{split}$ (4.1)

This implies that the probability of the supremum of the Gaussian random field
exceeding $u$ is equal to the probability that there exists at least one
extended outward local maximum above $u$ on some face $K$ of $T$. Therefore,
we obtain the following upper bound for the excursion probability:

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}\leq\sum_{k=0}^{N}\sum_{K\in\partial_{k}T}{\mathbb{P}}\\{M_{u}^{E}(K)\geq
1\\}\leq\sum_{k=0}^{N}\sum_{K\in\partial_{k}T}{\mathbb{E}}\\{M_{u}^{E}(K)\\}.\end{split}$
(4.2)

On the other hand, notice that

$\begin{split}{\mathbb{E}}\\{M_{u}^{E}(K)\\}-{\mathbb{P}}\\{M_{u}^{E}(K)\geq
1\\}&=\sum_{i=1}^{\infty}(i-1){\mathbb{P}}\\{M_{u}^{E}(K)=i\\}\\\
&\leq\sum_{i=1}^{\infty}i(i-1){\mathbb{P}}\\{M_{u}^{E}(K)=i\\}={\mathbb{E}}\\{M_{u}^{E}(K)[M_{u}^{E}(K)-1]\\}\end{split}$

and

$\begin{split}{\mathbb{P}}\\{M_{u}^{E}(K)\geq 1,M_{u}^{E}(K^{\prime})\geq
1\\}\leq{\mathbb{E}}\\{M_{u}^{E}(K)M_{u}^{E}(K^{\prime})\\}.\end{split}$

Applying the Bonferroni inequality to (4.1) and combining these two
inequalities, we obtain the following lower bound for the excursion
probability:

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}\\\
&\geq\sum_{k=0}^{N}\sum_{K\in\partial_{k}T}{\mathbb{P}}\\{M_{u}^{E}(K)\geq
1\\}-\sum_{K\neq K^{\prime}}{\mathbb{P}}\\{M_{u}^{E}(K)\geq
1,M_{u}^{E}(K^{\prime})\geq 1\\}\\\
&\geq\sum_{k=0}^{N}\sum_{K\in\partial_{k}T}\left({\mathbb{E}}\\{M_{u}^{E}(K)\\}-{\mathbb{E}}\\{M_{u}^{E}(K)[M_{u}^{E}(K)-1]\\}\right)-\sum_{K\neq
K^{\prime}}{\mathbb{E}}\\{M_{u}^{E}(K)M_{u}^{E}(K^{\prime})\\},\end{split}$
(4.3)

where the last sum is taken over all possible pairs of different faces
$(K,K^{\prime})$.

Remark 4.1  Note that, following the same arguments above, we have that the
expectations on the number of extended outward maxima $M_{u}^{E}(\cdot)$ in
both (4.2) and (4.3) can be replaced by the expectations on the number of
local maxima $M_{u}(\cdot)$. ∎

We call a function $h(u)$ _super-exponentially small_ [when compared with the
excursion probability ${\mathbb{P}}\\{\sup_{t\in T}X(t)\geq u\\}$ or
${\mathbb{E}}\\{\chi(A_{u})\\}$], if there exists a constant $\alpha>0$ such
that $h(u)=o(e^{-u^{2}/{(2\sigma_{T}^{2})-\alpha u^{2}}})$ as $u\to\infty$.
The main idea for proving the EEC approximation Theorem 3.1 consists of the
following two steps: (i) show that, except for the upper bound in (4.2), all
terms in the lower bound in (4.3) are super-exponentially small; and (ii)
demonstrate that the difference between the upper bound in (4.2) and
${\mathbb{E}}\\{\chi(A_{u})\\}$ is also super-exponentially small. The proofs
for Theorems 3.2 and 3.3 follow the same ideas, aiming to establish super-
exponential smallness for the terms involved in the lower bounds, as well as
for the difference between the upper bound and EEC.

##  5 Estimation of super-exponential smallness for terms in the lower bound

###  5.1 Factorial moments

We first state the following result, which is a modified version (restricted
on a face $K$) of Lemma 4 in Piterbarg  [7], characterizing the decaying rate
for factorial moments of the number of critical points exceeding a high level
for Gaussian fields.

######  Lemma 5.1.

Assume $({\bf H}1)$ and $({\bf H}2)$. Then there exists a positive constant
$C$ such that for any $\varepsilon>0$ one can find a number
$\varepsilon_{1}>0$ such that for any $K\in\partial_{k}T$,

$\begin{split}{\mathbb{E}}\\{M_{u}(K)(M_{u}(K)-1)\\}\leq
Cu^{2k+1}\exp\bigg{\\{}-\frac{u^{2}}{2\beta_{K}^{2}+\varepsilon}\bigg{\\}}+Cu^{4k+2}\exp\bigg{\\{}-\frac{u^{2}}{2\sigma_{K}^{2}-\varepsilon_{1}}\bigg{\\}},\end{split}$
(5.1)

where

$\beta_{K}^{2}=\sup_{t\in K}\sup_{e\in{\mathbb{S}}^{k-1}}{\rm Var}(X(t)|\nabla
X_{|K}(t),\nabla^{2}X_{|K}(t)e),\quad\sigma_{K}^{2}=\sup_{t\in K}{\rm
Var}(X(t)).$

The following result shows that the factorial moments in (4.3) are super-
exponentially small under our assumptions.

######  Proposition 5.2.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and $({\bf H}3)$. Then there exists $\alpha>0$ such
that as $u\to\infty$,

$\sum_{k=0}^{N}\sum_{K\in\partial_{k}T}{\mathbb{E}}\\{M_{u}(K)(M_{u}(K)-1)\\}=o\left(e^{-u^{2}/(2\sigma_{T}^{2})-\alpha
u^{2}}\right).$ (5.2)

###### Proof.

Due to Lemma 5.1, it suffices to show that for each $K\in\partial_{k}T$,
$\beta_{K}^{2}<\sigma_{T}^{2}$, which is equivalent to ${\rm Var}(X(t)|\nabla
X_{|K}(t),\nabla^{2}X_{|K}(t)e)<\sigma_{T}^{2}$ for all
$t\in\bar{K}=K\cup\partial K$ and $e\in{\mathbb{S}}^{k-1}$. Suppose ${\rm
Var}(X(t)|\nabla X_{|K}(t),\nabla^{2}X_{|K}(t)e)=\sigma_{T}^{2}$ for some
$t\in K$, then

$\sigma_{T}^{2}={\rm Var}(X(t)|\nabla X_{|K}(t),\nabla^{2}X_{|K}(t)e)\leq{\rm
Var}(X(t)|\nabla^{2}X_{|K}(t)e)\leq{\rm Var}(X(t))\leq\sigma_{T}^{2}.$

Note that

$\begin{split}\quad{\rm Var}(X(t)|\nabla^{2}X_{|K}(t)e)={\rm
Var}(X(t))\Leftrightarrow{\mathbb{E}}\\{X(t)(\nabla^{2}X_{|K}(t)e)\\}=0\Leftrightarrow\Sigma_{K}(t)e=0.\end{split}$

But $t$ is a point with $\nu(t)=\sigma_{T}^{2}$, thus $\Sigma_{K}(t)\prec 0$
by Proposition 2.1, implying $\Sigma_{K}(t)e\neq 0$ for all
$e\in{\mathbb{S}}^{k-1}$ and causing a contradiction.

On the other hand, suppose ${\rm Var}(X(t)|\nabla
X_{|K}(t),\nabla^{2}X_{|K}(t)e)=\sigma_{T}^{2}$ for some $t\in\partial K$,
then ${\rm Var}(X(t)|\nabla X_{|K}(t))=\sigma_{T}^{2}$ and hence
$\nu_{i}(t)=0$ for all $i\in\tau(K)$, implying $\Sigma_{K}(t)\prec 0$ by
Proposition 2.1. Similarly to the previous arguments, this will lead to a
contradiction. The proof is completed. ∎

###  5.2 Non-adjacent faces

For two sets $D,D^{\prime}\subset{\mathbb{R}}^{N}$, let
$d(D,D^{\prime})=\inf\\{\|t-t^{\prime}\|:t\in D,t^{\prime}\in D^{\prime}\\}$
denote their distance. The following result demonstrates that the last two
sums involving the joint moment of two non-adjacent faces in (4.3) are super-
exponentially small.

######  Proposition 5.3.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$ and $({\bf H}2)$. Then there exists $\alpha>0$ such that as
$u\to\infty$,

$\begin{split}{\mathbb{E}}\\{M_{u}(K)M_{u}(K^{\prime})\\}&=o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right),\end{split}$ (5.3)

where $K$ and $K^{\prime}$ are different faces of $T$ with
$d(K,K^{\prime})>0$.

###### Proof.

Consider first the case where ${\rm dim}(K)=k\geq 1$ and ${\rm
dim}(K^{\prime})=k^{\prime}\geq 1$. By the Kac-Rice formula for high moments
[2], we have

$\begin{split}&\quad{\mathbb{E}}\\{M_{u}(K)M_{u}(K^{\prime})\\}\\\
&=\int_{K}dt\int_{K^{\prime}}dt^{\prime}\,{\mathbb{E}}\big{\\{}|{\rm
det}\nabla^{2}X_{|K}(t)||{\rm
det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\mathbbm{1}_{\\{X(t)\geq
u,X(t^{\prime})\geq u\\}}\\\
&\quad\times\mathbbm{1}_{\\{\nabla^{2}X_{|K}(t)\prec
0,\,\nabla^{2}X_{|K^{\prime}}(t^{\prime})\prec 0\\}}\big{|}\nabla
X_{|K}(t)=0,\nabla X_{|K^{\prime}}(t^{\prime})=0\big{\\}}p_{\nabla
X_{|K}(t),\nabla X_{|K^{\prime}}(t^{\prime})}(0,0)\\\
&\leq\int_{K}dt\int_{K^{\prime}}dt^{\prime}\int_{u}^{\infty}dx\int_{u}^{\infty}dx^{\prime}\,p_{X(t),X(t^{\prime})}(x,x^{\prime})p_{\nabla
X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime})}(0,0|X(t)=x,X(t^{\prime})=x^{\prime})\\\
&\quad\times{\mathbb{E}}\big{\\{}|{\rm det}\nabla^{2}X_{|K}(t)||{\rm
det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\big{|}X(t)=x,X(t^{\prime})=x^{\prime},\nabla
X_{|K}(t)=0,\nabla X_{|K^{\prime}}(t^{\prime})=0\\}.\end{split}$ (5.4)

Notice that the following two inequalities hold: for constants $a_{i_{1}}$ and
$b_{i_{2}}$,

$\begin{split}&\prod_{i_{1}=1}^{k}|a_{i_{1}}|\prod_{i_{2}=1}^{k^{\prime}}|b_{i_{2}}|\leq\frac{\sum_{i_{1}=1}^{k}|a_{i_{1}}|^{k+k^{\prime}}+\sum_{i_{2}=1}^{k^{\prime}}|b_{i_{2}}|^{k+k^{\prime}}}{k+k^{\prime}};\end{split}$

and for any Gaussian variable $\xi$ and positive integer $m$, by Jensen’s
inequality,

$\begin{split}{\mathbb{E}}|\xi|^{m}\leq{\mathbb{E}}(|{\mathbb{E}}\xi|+|\xi-{\mathbb{E}}\xi|)^{m}&\leq
2^{m-1}(|{\mathbb{E}}\xi|^{m}+{\mathbb{E}}|\xi-{\mathbb{E}}\xi|^{m})=2^{m-1}(|{\mathbb{E}}\xi|^{m}+B_{m}({\rm
Var}(\xi))^{m/2}),\end{split}$

where $B_{m}$ is some constant depending only on $m$. Combining these two
inequalities with the well-known conditional formula for Gaussian variables,
we obtain that there exist positive constants $C_{1}$ and $N_{1}$ such that
for sufficiently large $x$ and $x^{\prime}$,

$\begin{split}\sup_{t\in K,t^{\prime}\in
K^{\prime}}&{\mathbb{E}}\big{\\{}|{\rm det}\nabla^{2}X_{|K}(t)||{\rm
det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\big{|}X(t)=x,X(t^{\prime})=x^{\prime},\nabla
X_{|K}(t)=0,\nabla X_{|K^{\prime}}(t^{\prime})=0\big{\\}}\\\ &\leq
C_{1}+(xx^{\prime})^{N_{1}}.\end{split}$ (5.5)

Further, there exists $C_{2}>0$ such that

$\begin{split}&\quad\sup_{t\in K,t^{\prime}\in K^{\prime}}p_{\nabla
X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime})}(0,0|X(t)=x,X(t^{\prime})=x^{\prime})\\\
&\leq\sup_{t\in K,t^{\prime}\in K^{\prime}}(2\pi)^{-(k+k^{\prime})/2}[{\rm
detCov}(\nabla X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime})|X(t)=x,X(t^{\prime})=x^{\prime})]^{-1/2}\\\ &\leq
C_{2}.\end{split}$ (5.6)

Plugging (5.5) and (5.6) into (5.4), we obtain that there exists $C_{3}$ such
that, for $u$ large enough,

$\begin{split}{\mathbb{E}}\\{M_{u}(K)M_{u}(K^{\prime})\\}&\leq C_{3}\sup_{t\in
K,t^{\prime}\in
K^{\prime}}{\mathbb{E}}\\{(C_{1}+|X(t)X(t^{\prime})|^{N_{1}})\mathbbm{1}_{\\{X(t)\geq
u,X(t^{\prime})\geq u\\}}\\}\\\ &\leq C_{3}\sup_{t\in K,t^{\prime}\in
K^{\prime}}{\mathbb{E}}\\{(C_{1}+(X(t)+X(t^{\prime}))^{2N_{1}})\mathbbm{1}_{\\{[X(t)+X(t^{\prime})]/2\geq
u\\}}\\}\\\ &\leq
C_{3}\exp\left(-\frac{u^{2}}{(1+\rho)\sigma_{T}^{2}}+\varepsilon
u^{2}\right),\end{split}$ (5.7)

where $\varepsilon$ is any positive number and $\rho=\sup_{t\in
K,t^{\prime}\in K^{\prime}}{\rm Corr}[X(t),X^{\prime}(t)]<1$ due to $({\bf
H}2)$. The case when one of the dimensions of $K$ and $K^{\prime}$ is zero can
be proved similarly. ∎

###  5.3 Adjacent faces

The following result shows that the last two sums involving the joint moment
of two adjacent faces in (4.3) are super-exponentially small.

######  Proposition 5.4.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and $({\bf H}3)$. Then there exists $\alpha>0$ such
that as $u\to\infty$,

$\begin{split}{\mathbb{E}}\\{M_{u}^{E}(K)M_{u}^{E}(K^{\prime})\\}&=o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right),\end{split}$ (5.8)

where $K$ and $K^{\prime}$ are different faces of $T$ with
$d(K,K^{\prime})=0$.

###### Proof.

Let $I:=\bar{K}\cap\bar{K^{\prime}}$, which is nonempty since
$d(K,K^{\prime})=0$. To simplify notation, let us assume without loss of
generality:

$\begin{split}\tau(K)&=\\{1,\ldots,m,m+1,\ldots,k\\},\\\
\tau(K^{\prime})&=\\{1,\ldots,m,k+1,\ldots,k+k^{\prime}-m\\},\end{split}$

where $0\leq m\leq k\leq k^{\prime}\leq N$ and $k^{\prime}\geq 1$. If $k=0$,
we conventionally consider $\tau(K)=\emptyset$. Under this assumption,
$K\in\partial_{k}T$, $K^{\prime}\in\partial_{k^{\prime}}T$, $\text{dim}(I)=m$,
and all elements in $\varepsilon(K)$ and $\varepsilon(K^{\prime})$ are 1.

We first consider the case when $k\geq 1$ and $l\geq 1$. By the Kac-Rice
formula,

$\begin{split}&\quad{\mathbb{E}}\\{M_{u}^{E}(K)M_{u}^{E}(K^{\prime})\\}\\\
&\leq\int_{K}dt\int_{K^{\prime}}dt^{\prime}\int_{u}^{\infty}dx\int_{u}^{\infty}dx^{\prime}\int_{0}^{\infty}dz_{k+1}\cdots\int_{0}^{\infty}dz_{k+k^{\prime}-m}\int_{0}^{\infty}dw_{m+1}\cdots\int_{0}^{\infty}dw_{k}\\\
&\quad{\mathbb{E}}\big{\\{}|\text{det}\nabla^{2}X_{|K}(t)||\text{det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\big{|}X(t)=x,X(t^{\prime})=x^{\prime},\nabla
X_{|K}(t)=0,X_{k+1}(t)=z_{k+1},\\\
&\quad\ldots,X_{k+k^{\prime}-m}(t)=z_{k+k^{\prime}-m},\nabla
X_{|K^{\prime}}(t^{\prime})=0,X_{m+1}(t^{\prime})=w_{m+1},\ldots,X_{k}(t^{\prime})=w_{k}\big{\\}}\\\
&\quad\times
p_{t,t^{\prime}}(x,x^{\prime},0,z_{k+1},\ldots,z_{k+k^{\prime}-m},0,w_{m+1},\ldots,w_{k})\\\
&:=\int\int\int_{K\times
K^{\prime}}A(t,t^{\prime},u)\,dtdt^{\prime},\end{split}$ (5.9)

where
$p_{t,t^{\prime}}(x,x^{\prime},0,z_{k+1},\ldots,z_{k+k^{\prime}-m},0,w_{m+1},\ldots,w_{k})$
is the density of the joint distribution of the variables involved in the
given condition. We define

$\begin{split}\mathcal{M}_{0}:=\\{t\in
I:\,\nu(t)=\sigma_{T}^{2},\,\nu_{i}(t)=0,\ \forall
i=1,\ldots,k+k^{\prime}-m\\},\end{split}$ (5.10)

and consider two cases for $\mathcal{M}_{0}$.

Case (i): $\bm{\mathcal{M}_{0}=\emptyset}$. Under this case, since $I$ is a
compact set, by the uniform continuity of conditional variance, there exist
constants $\varepsilon_{1},\delta_{1}>0$ such that

$\begin{split}&\sup_{t\in B(I,\delta_{1}),\,t^{\prime}\in
B^{\prime}(I,\delta_{1})}{\rm Var}(X(t)|\nabla X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime}))\leq\sigma_{T}^{2}-\varepsilon_{1},\end{split}$
(5.11)

where $B(I,\delta_{1})=\\{t\in K:d(t,I)\leq\delta_{1}\\}$ and
$B^{\prime}(I,\delta_{1})=\\{t^{\prime}\in
K^{\prime}:d(t^{\prime},I)\leq\delta_{1}\\}$. By partitioning $K\times
K^{\prime}$ into $B(I,\delta_{1})\times B^{\prime}(I,\delta_{1})$ and
$(K\times K^{\prime})\backslash(B(I,\delta_{1})\times
B^{\prime}(I,\delta_{1}))$ and applying the Kac-Rice formula, we obtain

$\begin{split}&\quad{\mathbb{E}}\\{M_{u}(K)M_{u}(K^{\prime})\\}\\\
&\leq\int_{(K\times K^{\prime})\backslash(B(I,\delta_{1})\times
B^{\prime}(I,\delta_{1}))}dtdt^{\prime}\,p_{\nabla X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime})}(0,0)\\\ &\quad\times{\mathbb{E}}\big{\\{}|{\rm
det}\nabla^{2}X_{|K}(t)||{\rm
det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\mathbbm{1}_{\\{X(t)\geq
u,X(t^{\prime})\geq u\\}}\big{|}\nabla X_{|K}(t)=0,\nabla
X_{|K^{\prime}}(t^{\prime})=0\big{\\}}\\\ &+\int_{B(I,\delta_{1})\times
B^{\prime}(I,\delta_{1})}dtdt^{\prime}\,p_{\nabla X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime})}(0,0)\\\ &\quad\times{\mathbb{E}}\big{\\{}|{\rm
det}\nabla^{2}X_{|K}(t)||{\rm
det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\mathbbm{1}_{\\{X(t)\geq
u,X(t^{\prime})\geq u\\}}\big{|}\nabla X_{|K}(t)=0,\nabla
X_{|K^{\prime}}(t^{\prime})=0\big{\\}}\\\ &:=I_{1}(u)+I_{2}(u).\end{split}$
(5.12)

Note that

$\begin{split}(K\times K^{\prime})\backslash(B(I,\delta_{1})\times
B^{\prime}(I,\delta_{1}))&=\Big{(}(K\backslash B(I,\delta_{1}))\times
B^{\prime}(I,\delta_{1})\Big{)}\bigcup\Big{(}B(I,\delta_{1})\times(K\backslash
B(I,\delta_{1}))\Big{)}\\\ &\quad\bigcup\Big{(}(K\backslash
B(I,\delta_{1}))\times(K\backslash B(I,\delta_{1}))\Big{)},\end{split}$

where each product on the right hand side consists of two sets with a positive
distance. It then follows from Proposition 5.3 that $I_{1}(u)$ is super-
exponentially small. On the other hand, since $\mathbbm{1}_{\\{X(t)\geq
u,X(t^{\prime})\geq u\\}}\leq\mathbbm{1}_{\\{[X(t)+X(t^{\prime})]/2\geq
u\\}}$, one has

$\begin{split}I_{2}(u)&\leq\int_{B(I,\delta_{1})\times
B^{\prime}(I,\delta_{1})}dtdt^{\prime}\int_{u}^{\infty}dx\,p_{\frac{X(t)+X(t^{\prime})}{2}}(x|\nabla
X_{|K}(t)=0,\nabla X_{|K^{\prime}}(t^{\prime})=0)\\\
&\quad\times{\mathbb{E}}\big{\\{}|{\rm det}\nabla^{2}X_{|K}(t)||{\rm
det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\big{|}[X(t)+X(t^{\prime})]/2=x,\\\
&\qquad\nabla X_{|K}(t)=0,\nabla
X_{|K^{\prime}}(t^{\prime})=0\big{\\}}p_{\nabla X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime})}(0,0).\end{split}$ (5.13)

Combining this with (5.11), we conclude that $I_{2}(u)$ and hence
${\mathbb{E}}\\{M_{u}^{E}(X,K)M_{u}^{E}(X,K^{\prime})\\}$ are super-
exponentially small.

Case (ii): $\bm{\mathcal{M}_{0}\neq\emptyset}$. Let

$\begin{split}B(\mathcal{M}_{0},\delta_{2}):=\\{(t,t^{\prime})\in K\times
K^{\prime}:d(t,\mathcal{M}_{0})\vee
d(t^{\prime},\mathcal{M}_{0})\leq\delta_{2}\\},\end{split}$

where $\delta_{2}$ is a small positive number to be specified. Note that, by
the definitions of $\mathcal{M}_{0}$ and $B(\mathcal{M}_{0},\delta_{2})$,
there exists $\varepsilon_{2}>0$ such that

$\begin{split}\sup_{(t,t^{\prime})\in(K\times K^{\prime})\backslash
B(\mathcal{M}_{0},\delta_{2})}&{\rm Var}([X(t)+X(t^{\prime})]/2|\nabla
X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime}))\leq\sigma_{T}^{2}-\varepsilon_{2}.\end{split}$
(5.14)

Similarly to (5.13), we obtain that $\int_{(K\times K^{\prime})\backslash
B(\mathcal{M}_{0},\delta_{2})}A(t,t^{\prime},u)dtdt^{\prime}$ is super-
exponentially small. It suffices to show below that
$\int_{B(\mathcal{M}_{0},\delta_{2})}A(t,t^{\prime},u)\,dtdt^{\prime}$ is
super-exponentially small.

Due to $({\bf H}3)$ and Proposition 2.1, we can choose $\delta_{2}$ small
enough such that for all $(t,t^{\prime})\in B(\mathcal{M}_{0},\delta_{2})$,

$\begin{split}\Lambda_{K\cup
K^{\prime}}(t)&:=-{\mathbb{E}}\\{X(t)\nabla^{2}X_{|K\cup
K^{\prime}}(t)\\}=-({\mathbb{E}}\\{X(t)X_{ij}(t)\\})_{i,j=1,\ldots,k+k^{\prime}-m}\end{split}$

are positive definite. Let $\\{e_{1},e_{2},\ldots,e_{N}\\}$ be the standard
orthonormal basis of ${\mathbb{R}}^{N}$. For $t\in K$ and $t^{\prime}\in
K^{\prime}$, let $e_{t,t^{\prime}}=(t^{\prime}-t)/\|t^{\prime}-t\|$ and
$\alpha_{i}(t,t^{\prime})={\langle}e_{i},\Lambda_{K\cup
K^{\prime}}(t)e_{t,t^{\prime}}\rangle$. Then

$\Lambda_{K\cup
K^{\prime}}(t)e_{t,t^{\prime}}=\sum_{i=1}^{N}{\langle}e_{i},\Lambda_{K\cup
K^{\prime}}(t)e_{t,t^{\prime}}\rangle
e_{i}=\sum_{i=1}^{N}\alpha_{i}(t,t^{\prime})e_{i}$ (5.15)

and there exists $\alpha_{0}>0$ such that for all $(t,t^{\prime})\in
B(\mathcal{M}_{0},\delta_{2})$,

${\langle}e_{t,t^{\prime}},\Lambda_{K\cup
K^{\prime}}(t)e_{t,t^{\prime}}\rangle\geq\alpha_{0}.$ (5.16)

Since all elements in $\varepsilon(K)$ and $\varepsilon(K^{\prime})$ are 1, we
may write

$\begin{split}t&=(t_{1},\ldots,t_{m},t_{m+1},\ldots,t_{k},b_{k+1},\ldots,b_{k+k^{\prime}-m},0,\ldots,0),\\\
t^{\prime}&=(t^{\prime}_{1},\ldots,t^{\prime}_{m},b_{m+1},\ldots,b_{k},t^{\prime}_{k+1},\ldots,t^{\prime}_{k+k^{\prime}-m},0,\ldots,0),\end{split}$

where $t_{i}\in(a_{i},b_{i})$ for $i\in\tau(K)$ and
$t^{\prime}_{j}\in(a_{j},b_{j})$ for $j\in\tau(K^{\prime})$. Therefore,

$\begin{split}{\langle}e_{i},e_{t,t^{\prime}}\rangle&\geq 0,\quad\forall\
m+1\leq i\leq k,\\\ {\langle}e_{i},e_{t,t^{\prime}}\rangle&\leq
0,\quad\forall\ k+1\leq i\leq k+k^{\prime}-m,\\\
{\langle}e_{i},e_{t,t^{\prime}}\rangle&=0,\quad\forall\ k+k^{\prime}-m<i\leq
N.\end{split}$ (5.17)

Let

$\begin{split}D_{i}&=\\{(t,t^{\prime})\in
B(\mathcal{M}_{0},\delta_{2}):\alpha_{i}(t,t^{\prime})\geq\beta_{i}\\},\quad\text{if}\
m+1\leq i\leq k,\\\ D_{i}&=\\{(t,t^{\prime})\in
B(\mathcal{M}_{0},\delta_{2}):\alpha_{i}(t,t^{\prime})\leq-\beta_{i}\\},\quad\text{if}\
k+1\leq i\leq k+k^{\prime}-m,\\\ D_{0}&=\bigg{\\{}(t,t^{\prime})\in
B(\mathcal{M}_{0},\delta_{2}):\sum_{i=1}^{m}\alpha_{i}(t,t^{\prime}){\langle}e_{i},e_{t,t^{\prime}}\rangle\geq\beta_{0}\bigg{\\}},\end{split}$
(5.18)

where $\beta_{0},\beta_{1},\ldots,\beta_{k+k^{\prime}-m}$ are positive
constants such that
$\beta_{0}+\sum_{i=m+1}^{k+k^{\prime}-m}\beta_{i}<\alpha_{0}$. It follows from
(5.17) and (5.18) that, if $(t,s)$ does not belong to any of
$D_{0},D_{m+1},\ldots,D_{k+k^{\prime}-m}$, then by (5.15),

${\langle}\Lambda_{K\cup
K^{\prime}}(t)e_{t,t^{\prime}},e_{t,t^{\prime}}\rangle=\sum_{i=1}^{N}\alpha_{i}(t,t^{\prime}){\langle}e_{i},e_{t,t^{\prime}}\rangle\leq\beta_{0}+\sum_{i=m+1}^{k+k^{\prime}-m}\beta_{i}<\alpha_{0},$

which contradicts (5.16). Thus $D_{0}\cup\cup_{i=m+1}^{k+k^{\prime}-m}D_{i}$
is a covering of $B(\mathcal{M}_{0},\delta_{2})$. By (5.9),

$\begin{split}{\mathbb{E}}&\\{M_{u}^{E}(K)M_{u}^{E}(K^{\prime})\\}\leq\int_{D_{0}}A(t,t^{\prime},u)\,dtdt^{\prime}+\sum_{i=m+1}^{k+k^{\prime}-m}\int_{D_{i}}A(t,t^{\prime},u)\,dtdt^{\prime}.\end{split}$

By the Kac-Rice metatheorem and the fact $\mathbbm{1}_{\\{X(t)\geq u,Y(s)\geq
u\\}}\leq\mathbbm{1}_{\\{X(t)\geq u\\}}$, we obtain

$\begin{split}&\quad\int_{D_{0}}A(t,t^{\prime},u)\,dtdt^{\prime}\\\
&\leq\int_{D_{0}}dtdt^{\prime}\int_{u}^{\infty}dx\,p_{\nabla X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime})}(0,0)p_{X(t)}(x|\nabla X_{|K}(t)=0,\nabla
X_{|K^{\prime}}(t^{\prime})=0)\\\
&\quad\times{\mathbb{E}}\big{\\{}|\text{det}\nabla^{2}X_{|K}(t)||\text{det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\big{|}X(t)=x,\nabla
X_{|K}(t)=0,\nabla X_{|K^{\prime}}(t^{\prime})=0\big{\\}},\end{split}$ (5.19)

and that for $i=m+1,\ldots,k$,

$\begin{split}&\quad\int_{D_{i}}A(t,t^{\prime},u)\,dtdt^{\prime}\\\
&\leq\int_{D_{i}}dtdt^{\prime}\int_{u}^{\infty}dx\int_{0}^{\infty}dw_{i}\,p_{X(t),\nabla
X_{|K}(t),X_{i}(t^{\prime}),\nabla
X_{|K^{\prime}}(t^{\prime})}(x,0,w_{i},0)\\\
&\quad\times{\mathbb{E}}\big{\\{}|\text{det}\nabla^{2}X_{|K}(t)||\text{det}\nabla^{2}X_{|K^{\prime}}(t^{\prime})|\big{|}X(t)=x,\nabla
X_{|K}(t)=0,X_{i}(t^{\prime})=w_{i},\nabla
X_{|K^{\prime}}(t^{\prime})=0\big{\\}}.\end{split}$ (5.20)

Comparing (5.19) and (5.20) with Eqs. (4.33) and (4.36) respectively in the
proof of Theorem 4.8 in Cheng and Xiao  [5], one can employ the same reasoning
therein to show that ${\rm Var}(X(t)|\nabla X_{|K}(t),\nabla
X_{|K^{\prime}}(t^{\prime}))<\sigma_{T}^{2}$ uniformly on $D_{0}$ and
${\mathbb{P}}(X(t)>u,X_{i}(t^{\prime})>0|\nabla X_{|K}(t)=0,\nabla
X_{|K^{\prime}}(t^{\prime})=0)=o(e^{-u^{2}/(2\sigma_{T}^{2})-\alpha u^{2}})$
uniformly on $D_{i}$, and deduce that
$\int_{D_{0}}A(t,t^{\prime},u)\,dtdt^{\prime}$ and
$\int_{D_{i}}A(t,t^{\prime},u)\,dtdt^{\prime}$ ($i=m+1,\ldots,k$) are super-
exponentially small.

It is similar to show that $\int_{D_{i}}A(t,t^{\prime},u)\,dtdt^{\prime}$ are
super-exponentially small for $i=k+1,\ldots,k+k^{\prime}-m$. For the case
$k=0$ or $l=0$, the argument is even simpler when applying the Kac-Rice
formula; the details are omitted here. The proof is finished. ∎

In the proof of Proposition 5.4, we have shown in (5.12) that, if
$\mathcal{M}_{0}=\emptyset$, then the moment
${\mathbb{E}}\\{M_{u}(X,K)M_{u}(X,K^{\prime})\\}$ is super-exponentially
small. It is important to note that, the boundary condition (3.3) implies (and
generalizes) the condition $\mathcal{M}_{0}=\emptyset$, yielding the following
result.

######  Proposition 5.5.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and the boundary condition (3.3). Then there exists
$\alpha>0$ such that as $u\to\infty$,

$\begin{split}{\mathbb{E}}\\{M_{u}(K)M_{u}(K^{\prime})\\}&=o\Big{(}\exp\Big{\\{}-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\Big{\\}}\Big{)},\end{split}$

where $K$ and $K^{\prime}$ are adjacent faces of $T$.

##  6 Estimation of the difference between EEC and the upper bound

In this section, we demonstrate that the difference between
${\mathbb{E}}\\{\chi(A_{u})\\}$ and the expected number of extended outward
local maxima, i.e. the upper bound in (4.2), is super-exponentially small.

######  Proposition 6.1.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and $({\bf H}3)$. Then there exists $\alpha>0$ such
that for any $K\in\partial_{k}T$ with $k\geq 0$, as $u\to\infty$,

$\begin{split}{\mathbb{E}}\\{M_{u}^{E}(K)\\}&=(-1)^{k}\int_{K}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\mathbbm{1}_{\\{X(t)\geq u,\
\varepsilon^{*}_{\ell}X_{\ell}(t)\geq 0\ {\rm for\ all}\
\ell\notin\tau(K)\\}}\big{|}\nabla X_{|K}(t)=0\big{\\}}\\\ &\quad\times
p_{\nabla
X_{|K}(t)}(0)dt+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right)\\\
&=(-1)^{k}{\mathbb{E}}\bigg{\\{}\bigg{(}\sum^{k}_{i=0}(-1)^{i}\mu_{i}(K)\bigg{)}\bigg{\\}}+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$ (6.1)

###### Proof.

The second equality in (6.1) arises from the application of the Kac-Rice
formula:

$\begin{split}&\quad{\mathbb{E}}\bigg{\\{}\bigg{(}\sum^{k}_{i=0}(-1)^{i}\mu_{i}(K)\bigg{)}\bigg{\\}}\\\
&=\sum^{k}_{i=0}(-1)^{i}\int_{K}{\mathbb{E}}\big{\\{}|{\rm
det}\nabla^{2}X_{|K}(t)|\mathbbm{1}_{\\{\text{index}(\nabla^{2}X_{|K}(t))=i\\}}\\\
&\quad\times\mathbbm{1}_{\\{X(t)\geq u,\ \varepsilon^{*}_{\ell}X_{\ell}(t)\geq
0\ {\rm for\ all}\ \ell\notin\tau(K)\\}}\big{|}\nabla
X_{|K}(t)=0\big{\\}}p_{\nabla X_{|K}(t)}(0)\,dt\\\
&=\int_{K}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\mathbbm{1}_{\\{X(t)\geq u,\
\varepsilon^{*}_{\ell}X_{\ell}(t)\geq 0\ {\rm for\ all}\
\ell\notin\tau(K)\\}}\big{|}\nabla X_{|K}(t)=0\big{\\}}p_{\nabla
X_{|K}(t)}(0)\,dt.\end{split}$

To prove the first approximation in (6.1) and convey the main idea, we start
with the case when the face $K$ represents the interior of $T$.

Case (i): $\bm{k=N}$. By the Kac-Rice formula, we have

$\begin{split}&\quad{\mathbb{E}}\\{M_{u}^{E}(K)\\}\\\ &=\int_{K}p_{\nabla
X(t)}(0)dt\int_{u}^{\infty}dx\,p_{X(t)}(x|\nabla
X(t)=0){\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X(t)\mathbbm{1}_{\\{\nabla^{2}X(t)\prec
0\\}}\big{|}X(t)=x,\nabla X(t)=0\big{\\}}\\\ &:=\int_{K}p_{\nabla
X(t)}(0)dt\int_{u}^{\infty}A(t,x)dx.\end{split}$

Let

$\begin{split}\mathcal{M}_{1}&=\\{t\in\bar{K}=T:\nu(t)=\sigma_{T}^{2},\
\nabla\nu(t)=2{\mathbb{E}}\\{X(t)\nabla X(t)\\}=0\\},\\\
B(\mathcal{M}_{1},\delta_{1})&=\left\\{t\in
K:d\left(t,\mathcal{M}_{1}\right)\leq\delta_{1}\right\\},\end{split}$ (6.2)

where $\delta_{1}$ is a small positive number to be specified. Then, we only
need to estimate

$\begin{split}\int_{B(\mathcal{M}_{1},\delta_{1})}p_{\nabla
X(t)}(0)dt\int_{u}^{\infty}A(t,x)dx,\end{split}$ (6.3)

since the integral above with $B(\mathcal{M}_{1},\delta_{1})$ replaced by
$K\backslash B(\mathcal{M}_{1},\delta_{1})$ becomes super-exponentially small
due to the fact

$\sup_{t\in K\backslash B(\mathcal{M}_{1},\delta_{1})}{\rm Var}(X(t)|\nabla
X(t)=0)<\sigma_{T}^{2}.$

Notice that, by Proposition 2.1, ${\mathbb{E}}\\{X(t)\nabla^{2}X(t)\\}\prec 0$
for all $t\in\mathcal{M}_{1}$. Thus there exists $\delta_{1}$ small enough
such that ${\mathbb{E}}\\{X(t)\nabla^{2}X(t)\\}\prec 0$ for all $t\in
B(\mathcal{M}_{1},\delta_{1})$. In particular, let $\lambda_{0}$ be the
largest eigenvalue of ${\mathbb{E}}\\{X(t)\nabla^{2}X(t)\\}$ over
$B(\mathcal{M}_{1},\delta_{1})$, then $\lambda_{0}<0$ by the uniform
continuity. Also note that ${\mathbb{E}}\\{X(t)\nabla X(t)\\}$ tends to 0 as
$\delta_{1}\to 0$. Therefore, as $\delta_{1}\to 0$,

$\begin{split}&\quad{\mathbb{E}}\\{X_{ij}(t)|X(t)=x,\nabla X(t)=0\\}\\\
&=({\mathbb{E}}\\{X_{ij}(t)X(t)\\},{\mathbb{E}}\\{X_{ij}(t)X_{1}(t)\\},\ldots,{\mathbb{E}}\\{X_{ij}(t)X_{N}(t)\\})\left[{\rm
Cov}(X(t),\nabla X(t))\right]^{-1}(x,0,\ldots,0)^{T}\\\
&=\frac{{\mathbb{E}}\\{X_{ij}(t)X(t)\\}x}{\sigma_{T}^{2}}(1+o(1)).\end{split}$

Thus, for all $x\geq u$ and $t\in B(\mathcal{M}_{1},\delta_{1})$ with
$\delta_{1}$ small enough,

$\begin{split}\Sigma_{1}(t,x)&:={\mathbb{E}}\\{\nabla^{2}X(t)|X(t)=x,\nabla
X(t)=0\\}\prec 0.\end{split}$

Let $\Delta_{1}(t,x)=\nabla^{2}X(t)-\Sigma_{1}(t,x)$. We have

$\begin{split}\int_{u}^{\infty}A(t,x)dx&=\int_{u}^{\infty}p_{X(t)}(x|\nabla
X(t)=0){\mathbb{E}}\big{\\{}{\rm det}(\Delta_{1}(t,x)+\Sigma_{1}(t,x))\\\
&\quad\times\mathbbm{1}_{\\{\Delta_{1}(t,x)+\Sigma_{1}(t,x)\prec
0\\}}\big{|}X(t)=x,\nabla X(t)=0\big{\\}}\,dx\\\
&:=\int_{u}^{\infty}p_{X(t)}(x|\nabla X(t)=0)E(t,x)\,dx.\end{split}$ (6.4)

Note that the following is a centered Gaussian random matrix not depending on
$x$:

$\begin{split}\Omega(t)&=(\Omega_{ij}(t))_{1\leq i,j\leq
N}=(\Delta_{1}(t,x)|X(t)=x,\nabla X(t)=0).\end{split}$

Let $h_{t}(v)$ denote the density of the Gaussian vector
$((\Omega_{ij}^{X}(t))_{1\leq i\leq j\leq N}$ with $v=(v_{ij})_{1\leq i\leq
j\leq N}\in{\mathbb{R}}^{N(N+1)/2}$. Then

$\begin{split}E(t,x)&={\mathbb{E}}\big{\\{}{\rm
det}(\Omega(t)+\Sigma_{1}(t,x))\mathbbm{1}_{\\{\Omega(t)+\Sigma_{1}(t,x)\prec
0\\}}\big{\\}}\\\ &=\int_{v:\,(v_{ij})+\Sigma_{1}(t,x)\prec 0}{\rm
det}((v_{ij})+\Sigma_{1}(t,x))h_{t}(v)dv,\end{split}$ (6.5)

where $(v_{ij})$ is the abbreviations of the matrix $v=(v_{ij})_{1\leq i,j\leq
N}$. There exists a constant $c>0$ such that for $\delta_{1}$ small enough and
all $t\in B(\mathcal{M}_{1},\delta_{1})$, and $x\geq u$, we have

$(v_{ij})+\Sigma_{1}(t,x)\prec
0,\quad\forall\|(v_{ij})\|:=\Big{(}\sum_{i,j=1}^{N}v_{ij}^{2}\Big{)}^{1/2}<cu.$

This implies $\\{v:\,(v_{ij})+\Sigma_{1}(t,x)\not\prec
0\\}\subset\\{v:\,\|(v_{ij})\|\geq cu\\}$. Consequently, the integral in (6.5)
with the domain of integration replaced by
$\\{v:\,(v_{ij})+\Sigma_{1}(t,x)\not\prec 0\\}$ is
$o(e^{-\alpha^{\prime}u^{2}})$ uniformly for all $t\in
B(\mathcal{M}_{1},\delta_{1})$, where $\alpha^{\prime}$ is a positive
constant. As a result, we conclude that, uniformly for all $t\in
B(\mathcal{M}_{1},\delta_{1})$ and $x\geq u$,

$\begin{split}E(t,x)=\int_{{\mathbb{R}}^{N(N+1)/2}}{\rm
det}((v_{ij})+\Sigma_{1}(t,x))h_{t}(v)dv+o(e^{-\alpha^{\prime}u^{2}}).\end{split}$

By substituting this result into (6.4), we observe that the indicator function
$\mathbbm{1}_{\\{\nabla^{2}X(t)\prec 0\\}}$ in (6.3) can be eliminated,
causing only a super-exponentially small error. Thus, for sufficiently large
$u$, there exists $\alpha>0$ such that

$\begin{split}{\mathbb{E}}\\{M_{u}^{E}(K)\\}&=\int_{K}p_{\nabla
X(t)}(0)dt\int_{u}^{\infty}{\mathbb{E}}\\{{\rm
det}\nabla^{2}X(t)|X(t)=x,\nabla X(t)=0\\}\\\ &\quad\times p_{X(t)}(x|\nabla
X(t)=0)dx+o\Big{(}\exp\Big{\\{}-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\Big{\\}}\Big{)}.\end{split}$

Case (ii): $\bm{k,l\geq 0}$. It is worth noting that when $k=0$, the terms in
(6.1) related to the Hessian will vanish, simplifying the proof. Therefore,
without loss of generality, let $k\geq 1$, $\tau(K)=\\{1,\cdots,k\\}$ and
assume all the elements in $\varepsilon(K)$ are 1. By the Kac-Rice formula,

$\begin{split}{\mathbb{E}}\\{M_{u}^{E}(K)\\}&=(-1)^{k}\int_{K}p_{\nabla
X_{|K}(t)}(0)dt\int_{u}^{\infty}p_{X(t)}\big{(}x\big{|}\nabla
X_{|K}(t)=0\big{)}{\mathbb{E}}\big{\\{}{\rm det}\nabla^{2}X_{|K}(t)\\\
&\quad\times\mathbbm{1}_{\\{\nabla^{2}X_{|K}(t)\prec
0\\}}\mathbbm{1}_{\\{X_{k+1}(t)>0,\ldots,X_{N}(t)>0\\}}\big{|}X(t)=x,\nabla
X_{|K}(t)=0\big{\\}}dx\\\ &:=(-1)^{k}\int_{K}p_{\nabla
X_{|K}(t)}(0)dt\int_{u}^{\infty}A^{\prime}(t,x)dx.\end{split}$

Let

$\begin{split}\mathcal{M}_{2}&=\\{t\in\bar{K}:\nu(t)=\sigma_{T}^{2},\
\nabla\nu_{|K}(t)=2{\mathbb{E}}\\{X(t)\nabla X_{|K}(t)\\}=0\\},\\\
B(\mathcal{M}_{2},\delta_{2})&=\left\\{t\in
K:d\left(t,\mathcal{M}_{2}\right)\leq\delta_{2}\right\\},\end{split}$ (6.6)

where $\delta_{2}$ is another small positive number to be specified. Here, we
only need to estimate

$\begin{split}\int_{B(\mathcal{M}_{2},\delta_{2})}p_{\nabla
X_{|K}(t)}(0)dt\int_{u}^{\infty}A^{\prime}(t,x)dx,\end{split}$ (6.7)

since the integral above with $B(\mathcal{M}_{2},\delta_{2})$ replaced by
$K\backslash B(\mathcal{M}_{2},\delta_{2})$ is super-exponentially small due
to the fact

$\sup_{t\in K\backslash B(\mathcal{M}_{2},\delta_{2})}{\rm Var}(X(t)|\nabla
X(t)=0)<\sigma_{T}^{2}.$

On the other hand, following similar arguments in the proof for Case (i), we
have that removing the indicator functions
$\mathbbm{1}_{\\{\nabla^{2}X_{|K}(t)\prec 0\\}}$ in (6.7) will only cause a
super-exponentially small error. Combining these results, we conclude that the
first approximation in (6.1) holds, thus completing the proof. ∎

From the proof of Proposition 6.1, it is evident that the same line of
reasoning and arguments can be readily extended to
${\mathbb{E}}\\{M_{u}(X,K)\\}$, leading to the following result.

######  Proposition 6.2.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and $({\bf H}3)$. Then there exists a constant
$\alpha>0$ such that for any $K\in\partial_{k}T$, as $u\to\infty$,

$\begin{split}{\mathbb{E}}\\{M_{u}(K)\\}&=(-1)^{k}\int_{K}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\mathbbm{1}_{\\{X(t)\geq u\\}}\big{|}\nabla
X_{|K}(t)=0\big{\\}}p_{\nabla X_{|K}(t)}(0)dt\\\
&\quad+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$

##  7 Proofs of the main results

###### Proof of Theorem 3.1.

By Propositions 5.2, 5.3 and 5.4, together with the fact $M_{u}^{E}(K)\leq
M_{u}(K)$, we obtain that the factorial moments and the last two sums in (4.3)
are super-exponentially small. Therefore, from (4.2) and (4.3), it follows
that there exists a constant $\alpha>0$ such that as $u\to\infty$,

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}=\sum_{k=0}^{N}\sum_{K\in\partial_{k}T}{\mathbb{E}}\\{M_{u}^{E}(K)\\}+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$

This desired result follows as an immediate consequence of Proposition 6.1. ∎

###### Proof of Theorem 3.2.

Remark 4 indicates that both inequalities (4.2) and (4.3) hold with
$M_{u}^{E}(\cdot)$ replaced by $M_{u}(\cdot)$. Therefore, the corresponding
factorial moments and the last two sums in (4.3) with $M_{u}^{E}(\cdot)$
replaced by $M_{u}(\cdot)$ are super-exponentially small by Propositions 5.2,
5.3 and 5.5. Consequently, there exists a constant $\alpha>0$ such that as
$u\to\infty$,

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}=\sum_{k=0}^{N}\sum_{K\in\partial_{k}T}{\mathbb{E}}\\{M_{u}(K)\\}+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$

The desired result follows directly from Proposition 6.2. ∎

###### Proof of Theorem 3.3.

Note that, in the proof of Theorem 3.1, we have seen that the points in
$\mathcal{M}_{2}$ defined in (6.6) make major contribution to the excursion
probability. That is, with up to a super-exponentially small error, we can
focus only on those faces, say $J$, whose closure $\bar{J}$ contains the
unique point $t^{*}$ with $\nu(t^{*})=\sigma_{T}^{2}$ and satisfying
$\tau(J)\subset\mathcal{I}(t^{*})$ (i.e., the partial derivatives of $\nu$ are
0 at $t^{*}$ restricted on $J$). To formalize this concept, we define a set of
faces $T^{*}$ as follows:

$\begin{split}T^{*}&=\\{J\in\partial_{k}T:t^{*}\in\bar{J},\,\tau(J)\subset\mathcal{I}(t^{*}),\,k=0,\ldots,N\\}.\end{split}$

For each $J\in T^{*}$, let

$\begin{split}M_{u}^{E^{*}}(J)&:=\\#\\{t\in J:X(t)\geq u,\nabla
X_{|J}(t)=0,\nabla^{2}X_{|J}(t)\prec 0,\\\
&\qquad\qquad\qquad\varepsilon^{*}_{j}X_{j}(t)\geq 0\ {\rm for\ all}\
j\in\mathcal{I}(t^{*})\setminus\tau(J)\\}.\end{split}$

Note that, both inequalities (4.2) and (4.3) remain valid when we replace
$M_{u}^{E}(J)$ with $M_{u}^{E^{*}}(J)$ for faces $J$ belonging to $T^{*}$, and
replace $M_{u}^{E}(J)$ with $M_{u}(J)$ otherwise. Employing analogous
reasoning as used in the derivation of Theorems 3.1 and 3.2, we obtain that,
there exists $\alpha>0$ such that as $u\to\infty$,

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}=\sum_{J\in
T^{*}}{\mathbb{E}}\\{M_{u}^{E^{*}}(J)\\}+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$

This desired result is then deduced from Proposition 6.1. ∎

##  8 Gaussian fields with a unique maximum point of the variance

In this section, we delve deeper into EEC approximations when the variance
function $\nu(t)$ reaches its maximum value $\sigma_{T}^{2}$ at a solitary
point $t^{*}$. While Theorem 3.3 provides an implicit formula for such
scenarios, our objective here is to obtain explicit formulae by employing
integral approximation techniques based on the Kac-Rice formula. To facilitate
this process, we begin by presenting some auxiliary results related to the
Laplace method for integral approximations.

###  8.1 Auxiliary lemmas on Laplace approximation

The following two formulas state the results on the Laplace approximation
method. Lemma 8.1 can be found in many books on the approximations of
integrals; here we refer to Wong [15]. Lemma 8.2 can be derived by following
similar arguments in the proof of the Laplace method for the case of boundary
points in [15].

######  Lemma 8.1.

[Laplace method for interior points] Let $t_{0}$ be an interior point of $T$.
Suppose the following conditions hold: (i) $g(t)\in C(T)$ and $g(t_{0})\neq
0$; (ii) $h(t)\in C^{2}(T)$ and attains its minimum only at $t_{0}$; and (iii)
$\nabla^{2}h(t_{0})$ is positive definite. Then as $u\to\infty$,

$\int_{T}g(t)e^{-uh(t)}dt=\frac{(2\pi)^{N/2}}{u^{N/2}({\rm
det}\nabla^{2}h(t_{0}))^{1/2}}g(t_{0})e^{-uh(t_{0})}(1+o(1)).$

######  Lemma 8.2.

[Laplace method for boundary points] Let $t_{0}\in K\in\partial_{k}T$ with
$0\leq k\leq N-1$. Suppose that conditions (i), (ii) and (iii) in Lemma 8.1
hold, and additionally $\nabla h(t_{0})=0$. Then as $u\to\infty$,

$\begin{split}\int_{T}g(t)e^{-uh(t)}dt&=\frac{(2\pi)^{N/2}{\mathbb{P}}\\{Z_{i_{\ell}}\varepsilon_{i_{\ell}}^{*}>0,\forall
i_{\ell}\notin\tau(K)\\}}{u^{N/2}({\rm
det}\nabla^{2}h(t_{0}))^{1/2}}g(t_{0})e^{-uh(t_{0})}(1+o(1)),\end{split}$

where $(Z_{i_{1}},\ldots,Z_{i_{N-k}})$ is a centered $(N-k)$-dimensional
Gaussian vector with covariance matrix
$(h_{i_{\ell}i_{\ell^{\prime}}}(t_{0}))_{i_{\ell},i_{\ell^{\prime}}\notin\tau(K)}$
and $\tau(K)$ and $\varepsilon_{i_{\ell}}^{*}$ are defined in Section 2.

###  8.2 Gaussian fields satisfying the boundary condition (3.3)

For $t\in T$, we define the following notation for conditional variances:

$\begin{split}\tilde{\nu}_{|K}(t)={\rm Var}(X(t)|\nabla
X_{|K}(t)=0),\quad\tilde{\nu}(t)={\rm Var}(X(t)|\nabla
X_{|K}(t)=0).\end{split}$ (8.1)

The following result provides explicit approximations to the excursion
probabilities when the maximum of the variance is reached only at a single
point and the boundary condition (3.3) is satisfied.

######  Theorem 8.3.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$ and $({\bf H}2)$. Suppose $\nu$ attains its maximum
$\sigma_{T}^{2}$ only at $t^{*}\in K\in\partial_{k}T$, $\nu_{i}(t^{*})\neq 0$
for all $i\notin\tau(K)$, and $\nabla^{2}\nu_{|K}(t^{*})\prec 0$. Then, as
$u\to\infty$,

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}&=\Psi\left(\frac{u}{\sigma_{T}}\right)+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right)\text{\rm for some $\alpha>0$,}\quad\text{ if }k=0,\\\
{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}&=\sqrt{\frac{{\rm
det}(\Sigma_{K}(t^{*}))}{{\rm
det}(\Lambda_{K}(t^{*})+\Sigma_{K}(t^{*}))}}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)),\quad\text{
if }k\geq 1,\end{split}$ (8.2)

where $\Lambda_{K}(t^{*})$ and $\Sigma_{K}(t^{*})$ are defined in (2.3).

###### Proof.

If $k=0$, then $\nu_{i}(t^{*})\neq 0$ for all $i\geq 1$, and hence
$\mathcal{I}(t^{*})=\emptyset$. The first line of (8.2) follows from Theorem
3.3 that

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}={\mathbb{P}}\\{X(t^{*})\geq
u\\}+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$

Now, let us consider the case when $k\geq 1$. Note that the assumption on
partial derivatives of $\nu(t)$ implies $\mathcal{I}(t^{*})=\tau(K)$. By
Theorem 3.3, we have

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}=(-1)^{k}I(u,K)+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right),\end{split}$ (8.3)

where

$\begin{split}I(u,K)&=\int_{K}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\mathbbm{1}_{\\{X(t)\geq u\\}}\big{|}\nabla
X_{|K}(t)=0\big{\\}}p_{\nabla X_{|K}(t)}(0)dt\\\
&=\int_{u}^{\infty}\int_{K}\frac{{(2\pi)^{-(k+1)/2}}}{\sqrt{\tilde{\nu}_{|K}(t){\rm
det}\left(\Lambda_{K}(t)\right)}}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\big{|}X(t)=x,\nabla
X_{|K}(t)=0\big{\\}}e^{-\frac{x^{2}}{2\tilde{\nu}(t)}}dtdx.\end{split}$

Applying the Laplace method in Lemma 8.1 with

$\begin{split}g(t)&=\frac{1}{\sqrt{\tilde{\nu}_{|K}(t){\rm
det}\left(\Lambda_{K}(t)\right)}}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\big{|}X(t)=x,\nabla X_{|K}(t)=0\big{\\}},\\\
h(t)&=\frac{1}{{2\tilde{\nu}_{|K}(t)}},\quad u=x^{2},\end{split}$

and noting that the Hessian matrix of $1/(2\tilde{\nu}_{|K}(t))$ evaluated at
$t^{*}$ is

$-\frac{1}{2\tilde{\nu}_{|K}^{2}(t^{*})}\left(\tilde{\nu}_{ij}(t^{*})\right)_{i,j\in\tau(K)}=-\frac{1}{2\sigma_{T}^{4}}\nabla^{2}\tilde{\nu}_{|K}(t^{*})\succ
0,$ (8.4)

we obtain

$\begin{split}I(u,K)&=\frac{(2\sigma_{T}^{4})^{k/2}}{\sqrt{2\pi\sigma_{T}^{2}{\rm
det}\left(\Lambda_{K}(t^{*})\right)}\sqrt{|{\rm
det}\nabla^{2}\tilde{\nu}_{|K}(t^{*})|}}I(u)(1+o(1)),\end{split}$ (8.5)

where

$\begin{split}I(u)&=\int^{\infty}_{u}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t^{*})\big{|}X(t^{*})=x,\nabla
X_{|K}(t^{*})=0\big{\\}}x^{-k}e^{-\frac{x^{2}}{2\sigma_{T}^{2}}}\,dx\\\ &={\rm
det}(\Sigma_{K}(t^{*}))\int^{\infty}_{u}{\mathbb{E}}\big{\\{}{\rm
det}(Q\nabla^{2}X_{|K}(t^{*})Q)\big{|}X(t^{*})=x,\nabla
X_{|K}(t^{*})=0\big{\\}}x^{-k}e^{-\frac{x^{2}}{2\sigma_{T}^{2}}}dx.\end{split}$
(8.6)

Here, noting that
$\Sigma_{K}(t^{*})={\mathbb{E}}\\{X(t)\nabla^{2}X_{|K}(t^{*})\\}\prec 0$ by
Proposition 2.1, we let $Q$ in (8.6) be a $k\times k$ positive definite matrix
such that $Q(-\Sigma_{K}(t^{*}))Q=I_{k}$, where $I_{k}$ is the size-$k$
identity matrix. Then

${\mathbb{E}}\\{X(t)(Q\nabla^{2}X_{|K}(t^{*})Q)\\}=Q\Sigma_{K}(t^{*})Q=-I_{k}.$

Notice that ${\mathbb{E}}\\{X(t^{*})\nabla X_{|K}(t^{*})\\}=0$ due to
$\nu_{|K}(t^{*})=0$, we have

$\begin{split}{\mathbb{E}}\big{\\{}Q\nabla^{2}X_{|K}(t^{*})Q\big{|}X(t^{*})=x,\nabla
X_{|K}(t^{*})=0\big{\\}}=-\frac{x}{\sigma_{T}^{2}}I_{k}.\end{split}$

One can write

$\begin{split}{\mathbb{E}}\big{\\{}{\rm
det}(Q\nabla^{2}X_{|K}(t^{*})Q)\big{|}X(t^{*})=x,\nabla
X_{|K}(t^{*})=0\big{\\}}={\mathbb{E}}\\{{\rm
det}(\Delta(t^{*})-(x/\sigma_{T}^{2})I_{k})\\},\end{split}$

where $\Delta(t^{*})$ is a centered Gaussian random matrix with covariance
independent of $x$. According to the Laplace expansion of determinant,
${\mathbb{E}}\\{{\rm det}(\Delta(t^{*})-(x/\sigma_{T}^{2})I_{k})\\}$ is a
polynomial in $x$ with the highest-order term being
$(-1)^{k}\sigma_{T}^{-2k}x^{k}$. Plugging this into (8.6) and (8.5), we obtain

$I(u,K)=\frac{(-1)^{k}2^{k/2}|{\rm det}(\Sigma_{K}(t^{*}))|}{\sqrt{{\rm
det}(\Lambda_{K}(t^{*}))}\sqrt{|{\rm
det}\left(\nabla^{2}\tilde{\nu}_{|K}(t^{*})\right)|}}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)).$

Finally, note that

$\tilde{\nu}_{|K}(t)={\mathbb{E}}\\{X(t)^{2}\\}-{\mathbb{E}}\\{X(t)\nabla
X_{|K}(t)\\}^{T}\Lambda_{K}^{-1}(t){\mathbb{E}}\\{X(t)\nabla X_{|K}(t)\\},$

we have

$\begin{split}\nabla^{2}\tilde{\nu}_{|K}(t^{*})&=2[\Lambda_{K}(t^{*})+\Sigma_{K}(t^{*})]-2[\Lambda_{K}(t^{*})+\Sigma_{K}(t^{*})]\Lambda_{K}^{-1}(t^{*})[\Lambda_{K}(t^{*})+\Sigma_{K}(t^{*})]\\\
&=-2\Sigma_{K}(t^{*})[I_{k}+\Lambda_{K}^{-1}(t^{*})\Sigma_{K}(t^{*})].\end{split}$
(8.7)

Therefore,

$I(u,K)=(-1)^{k}\sqrt{\frac{{\rm det}(\Sigma_{K}(t^{*}))}{{\rm
det}(\Lambda_{K}(t^{*})+\Sigma_{K}(t^{*}))}}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)),$

where $\Sigma_{K}(t^{*})\prec 0$ by Proposition 2.1 and
$\Lambda_{K}(t^{*})+\Sigma_{K}(t^{*})=\nabla^{2}\nu_{|K}(t^{*})/2\prec 0$ by
assumption. Plugging this into (8.3) yields the desired result. ∎

Now we apply Theorem 8.3 to the 1D case when $T=[a,b]$. If $t^{*}=a$ or
$t^{*}=b$, then it is a direct application of the first line in (8.2). If
$t^{*}\in(a,b)$, then it follows from (8.2) that

${\mathbb{P}}\left\\{\sup_{t\in[a,b]}X(t)\geq
u\right\\}=\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X^{\prime\prime}(t^{*})\\}}{{\rm
Var}(X^{\prime}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X^{\prime\prime}(t^{*})\\}}}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)).$

###  8.3 Gaussian fields not satisfying the boundary condition (3.3)

We consider here the other case when $\nu_{i}(t^{*})\neq 0$ for some
$i\notin\tau(K)$. For a symmetric matrix $B=(B_{ij})_{1\leq i,j\leq N}$, we
call $(B_{ij})_{i,j\in\mathcal{I}}$ the matrix $B$ with indices restricted on
$\mathcal{I}$.

######  Theorem 8.4.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$ and $({\bf H}2)$. Suppose $\nu$ attains its maximum
$\sigma_{T}^{2}$ only at $t^{*}\in K\in\partial_{k}T$ such that
$\mathcal{I}(t^{*})\setminus\tau(K)$ contains $m\geq 1$ indices and
$(\nu_{ii^{\prime}}(t^{*}))_{i,i^{\prime}\in\mathcal{I}(t^{*})}\prec 0$. Then,
as $u\to\infty$,

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}\\\
&=\sum_{J}\sqrt{\frac{{\rm det}(\Sigma_{J}(t^{*}))}{{\rm
det}(\Lambda_{J}(t^{*})+\Sigma_{J}(t^{*}))}}{\mathbb{P}}\\{(Z_{J_{1}^{\prime}},\ldots,Z_{J_{j-k}^{\prime}})\in
E^{\prime}(J)\\}\\\
&\quad\times{\mathbb{P}}\big{\\{}(X_{J_{1}}(t^{*}),\ldots,X_{J_{k+m-j}}(t^{*}))\in
E(J)\big{|}\nabla
X_{|J}(t^{*})=0\big{\\}}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)),\end{split}$
(8.8)

where the sum is taken over all faces $J$ such that $t^{*}\in\bar{J}$ and
$\tau(J)\subset\mathcal{I}(t^{*})$, $j={\rm dim}(J)$,

$\begin{split}&(J_{1},\ldots,J_{k+m-j})=\mathcal{I}(t^{*})\setminus\tau(J),\quad(J_{1}^{\prime},\ldots,J_{j-k}^{\prime})=\tau(J)\setminus\tau(K),\\\
&E(J)=\\{(y_{J_{1}},\ldots,y_{J_{k+m-j}})\in{\mathbb{R}}^{k+m-j}:\varepsilon^{*}_{J_{\ell}}(J)y_{J_{\ell}}\geq
0,\,\forall\ell=1,\ldots,k+m-j\\},\\\
&E^{\prime}(J)=\\{(y_{J_{1}^{\prime}},\ldots,y_{J_{j-k}^{\prime}})\in{\mathbb{R}}^{j-k}:\varepsilon^{*}_{J_{\ell}^{\prime}}(K)y_{J_{\ell}^{\prime}}\geq
0,\,\forall\ell=1,\ldots,j-k\\},\end{split}$

$\varepsilon^{*}_{J_{\ell}}(J)$ and $\varepsilon^{*}_{J_{\ell}^{\prime}}(K)$
are the $\varepsilon^{*}$ numbers for faces $J$ and $K$ respectively,
$(Z_{J_{1}^{\prime}},\ldots,Z_{J_{j-k}^{\prime}})$ is a centered Gaussian
vector having covariance matrix
$\Sigma(t^{*})+\Sigma(t^{*})\Lambda^{-1}(t^{*})\Sigma(t^{*})$ with indices
restricted on $\tau(J)\setminus\tau(K)$, and $\Lambda_{J}(t^{*})$ and
$\Sigma_{J}(t^{*})$ are defined in (2.3). In particular, for $k=0$, the term
inside the sum in (8.8) with $J=K=\\{t^{*}\\}$ is given by

${\mathbb{P}}\\{(X_{J_{1}}(t^{*}),\ldots,X_{J_{m}}(t^{*}))\in
E(J)\\}\Psi\left(\frac{u}{\sigma_{T}}\right).$

###### Proof.

We first prove the case when $k\geq 1$. By Theorem 3.3, we have

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}&=\sum_{J}(-1)^{j}I(u,J)+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right),\end{split}$ (8.9)

where $j={\rm dim}(J)$, the sum is taken over all faces $J$ such that
$t^{*}\in\bar{J}$ and $\tau(J)\subset\mathcal{I}(t^{*})$, and

$\begin{split}I(u,J)&=\int_{J}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|J}(t)\mathbbm{1}_{\\{X(t)\geq
u\\}}\mathbbm{1}_{\\{\varepsilon^{*}_{\ell}X_{\ell}(t)\geq
0,\,\forall\ell\in\mathcal{I}(t^{*})\setminus\tau(J)\\}}\big{|}\nabla
X_{|J}(t)=0\big{\\}}p_{\nabla X_{|J}(t)}(0)dt\\\
&=\int_{u}^{\infty}\int_{J}\frac{{(2\pi)^{-(j+1)/2}}}{\sqrt{\tilde{\nu}_{|J}(t){\rm
det}\left(\Lambda_{J}(t)\right)}}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|K}(t)\mathbbm{1}_{\\{\varepsilon^{*}_{\ell}X_{\ell}(t)\geq
0,\,\forall\ell\in\mathcal{I}(t^{*})\setminus\tau(J)\\}}\big{|}X(t)=x,\\\
&\quad\nabla
X_{|J}(t)=0\big{\\}}e^{-\frac{x^{2}}{2\tilde{\nu}_{|J}(t)}}dtdx.\end{split}$

Applying the Laplace method in Lemma 8.2 with

$\begin{split}g(t)&=\frac{1}{\sqrt{\tilde{\nu}_{|J}(t){\rm
det}\left(\Lambda_{J}(t)\right)}}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|J}(t)\mathbbm{1}_{\\{\varepsilon^{*}_{\ell}X_{\ell}(t)\geq
0,\,\forall\ell\in\mathcal{I}(t^{*})\setminus\tau(J)\\}}\big{|}X(t)=x,\nabla
X_{|J}(t)=0\big{\\}},\\\ h(t)&=\frac{1}{{2\tilde{\nu}_{|J}(t)}},\quad
u=x^{2},\end{split}$

we obtain

$\begin{split}I(u,J)&=\frac{(2\sigma_{T}^{4})^{j/2}{\mathbb{P}}\\{(Z_{J_{1}^{\prime}},\ldots,Z_{J_{j-k}^{\prime}})\in
E^{\prime}(J)\\}}{\sqrt{2\pi\sigma_{T}^{2}{\rm
det}\left(\Lambda_{J}(t^{*})\right)}\sqrt{|{\rm
det}\nabla^{2}\tilde{\nu}_{|J}(t^{*})|}}I(u)(1+o(1)),\end{split}$

where $(Z_{J_{1}^{\prime}},\ldots,Z_{J_{j-k}^{\prime}})$ is a centered
$(j-k)$-dimensional Gaussian vector having covariance matrix
$\nabla^{2}h(t^{*})$ with indices restricted on $\tau(J)\setminus\tau(K)$, and

$\begin{split}I(u)&=\int^{\infty}_{u}{\mathbb{E}}\big{\\{}{\rm
det}\nabla^{2}X_{|J}(t^{*})\mathbbm{1}_{\\{\varepsilon^{*}_{\ell}X_{\ell}(t^{*})\geq
0,\,\forall\ell\in\mathcal{I}(t^{*})\setminus\tau(J)\\}}\big{|}X(t^{*})=x,\nabla
X_{|J}(t^{*})=0\big{\\}}x^{-j}e^{-\frac{x^{2}}{2\sigma_{T}^{2}}}\,dx\\\ &={\rm
det}(\Sigma_{J}(t^{*}))\int^{\infty}_{u}\int_{E(J)}{\mathbb{E}}\big{\\{}{\rm
det}(Q\nabla^{2}X_{|J}(t^{*})Q)\big{|}X(t^{*})=x,\nabla
X_{|J}(t^{*})=0,X_{J_{1}}(t^{*})=y_{J_{1}},\\\
&\quad\ldots,X_{J_{k+m-j}}(t^{*})=y_{J_{k+m-j}}\big{\\}}p(y_{J_{1}},\ldots,y_{J_{k+m-j}}|x,0)x^{-j}e^{-\frac{x^{2}}{2\sigma_{T}^{2}}}dy_{J_{1}}\cdots
dy_{J_{k+m-j}}dx.\end{split}$ (8.10)

Here $p(y_{J_{1}},\ldots,y_{J_{k+m-j}}|x,0)$ is the pdf of
$(X_{J_{1}}(t^{*}),\ldots,X_{J_{k+m-j}}(t^{*})|X(t^{*})=x,\nabla
X_{|J}(t^{*})=0)$, and $Q$ is a $j\times j$ positive definite matrix such that
$Q(-\Sigma_{J}(t^{*}))Q=I_{j}$. Then, similarly to the arguments in the proof
of Theorem 8.3, one can write the last expectation in (8.10) as

$\begin{split}{\mathbb{E}}\\{{\rm
det}(\Delta(t^{*},y_{J_{1}},\ldots,y_{J_{k+m-j}})-(x/\sigma_{T}^{2})I_{k})\\},\end{split}$

where $\Delta(t^{*},y_{J_{1}},\ldots,y_{J_{k+m-j}})$ is a centered Gaussian
random matrix with covariance independent of $x$, and hence the highest-order
term in $x$ is $(-1)^{j}x^{j}/\sigma_{T}^{2j}$. Noting that
${\mathbb{E}}\\{X(t^{*})X_{i}(t^{*})\\}=0$ for all $i\in\mathcal{I}(t^{*})$
and following similar arguments in the proof of Theorem 8.3, we obtain

$\begin{split}I(u,J)&=(-1)^{j}\sqrt{\frac{{\rm det}(\Sigma_{J}(t^{*}))}{{\rm
det}(\Lambda_{J}(t^{*})+\Sigma_{J}(t^{*}))}}{\mathbb{P}}\\{(Z_{J_{1}^{\prime}},\ldots,Z_{J_{j-k}^{\prime}})\in
E^{\prime}(J)\\}\\\
&\quad\times{\mathbb{P}}\\{(X_{J_{1}}(t^{*}),\ldots,X_{J_{k+m-j}}(t^{*}))\in
E(J)|\nabla
X_{|J}(t^{*})=0\\}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)),\end{split}$

which yields the desired result together with (8.9). In particular, by (8.7),
one can treat $(Z_{J_{1}^{\prime}},\ldots,Z_{J_{j-k}^{\prime}})$ having
covariance $\Sigma(t^{*})+\Sigma(t^{*})\Lambda^{-1}(t^{*})\Sigma(t^{*})$ with
indices restricted on $\tau(J)\setminus\tau(K)$ while not changing the
probability that it falls in $E(J)$. Lastly, the case when $k=0$ can be shown
similarly. ∎

Now we apply Theorem 8.4 to the 1D case when $T=[a,b]$. Without loss of
generality, assume $t^{*}=b$ and $\nu^{\prime}(t^{*})=0$. Then it follows from
Theorem 8.4 that

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in[a,b]}X(t)\geq u\right\\}\\\
&=\left({\mathbb{P}}\\{X^{\prime}(t^{*})>0\\}+\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X^{\prime\prime}(t^{*})\\}}{{\rm
Var}(X^{\prime}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X^{\prime\prime}(t^{*})\\}}}{\mathbb{P}}\\{Z>0\\}\right)\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1))\\\
&=\frac{1}{2}\left(1+\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X^{\prime\prime}(t^{*})\\}}{{\rm
Var}(X^{\prime}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X^{\prime\prime}(t^{*})\\}}}\right)\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)),\end{split}$

where $Z$ is a centered Gaussian variable.

Denote by ${\mathbb{R}}_{+}^{n}=(0,\infty)^{n}$. To simplify the statement in
Theorem 8.4, we present below another version with less notations on faces.

######  Corollary 8.5.

Let $\\{X(t),\,t\in T\\}$ be a centered Gaussian random field satisfying
$({\bf H}1)$, $({\bf H}2)$ and $({\bf H}3)$. Suppose $\nu$ attains its maximum
$\sigma_{T}^{2}$ only at $t^{*}\in K\in\partial_{k}T$ with
$\tau(K)=\\{1,\ldots,k\\}$ such that
$\mathcal{I}(t^{*})=\\{1,\ldots,k,k+1,\ldots,k+m\\}$ with $m\geq 1$ and
$\left(\nu_{ii^{\prime}}(t^{*})\right)_{1\leq i,i^{\prime}\leq k+m}\prec 0$.
Then, as $u\to\infty$,

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}\\\
&=\sum_{j=k}^{k+m}\sum_{J\in\partial_{j}T:\,t^{*}\in\bar{J}}\sqrt{\frac{{\rm
det}(\Sigma_{J}(t^{*}))}{{\rm
det}(\Lambda_{J}(t^{*})+\Sigma_{J}(t^{*}))}}{\mathbb{P}}\\{(Z_{1},\ldots,Z_{j-k})\in{\mathbb{R}}_{+}^{j-k}\\}\\\
&\quad\times{\mathbb{P}}\big{\\{}(X_{j+1}(t^{*}),\ldots,X_{k+m}(t^{*}))\in{\mathbb{R}}_{+}^{k+m-j}\big{|}\nabla
X_{|J}(t^{*})=0\big{\\}}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)),\end{split}$
(8.11)

where $(Z_{1},\ldots,Z_{j-k})$ is a centered Gaussian vector having covariance
$\Sigma(t^{*})+\Sigma(t^{*})\Lambda^{-1}(t^{*})\Sigma(t^{*})$ with indices
restricted on $\\{k+1,\ldots,j\\}$, and $\Lambda_{J}(t^{*})$ and
$\Sigma_{J}(t^{*})$ are defined in (2.3). In particular, for $k=0$, the term
inside the sum in (8.11) with $J=K=\\{t^{*}\\}$ is

${\mathbb{P}}\\{(X_{1}(t^{*}),\ldots,X_{m}(t^{*}))\in{\mathbb{R}}_{+}^{m}\\}\Psi\left(\frac{u}{\sigma_{T}}\right).$

##  9 Examples

Throughout this section, we consider a centered Gaussian random field
$\\{X(t),\,t\in T\\}$ satisfying $({\bf H}1)$, $({\bf H}2)$ and $({\bf H}3)$,
where $T=[a_{1},b_{1}]\times[a_{2},b_{2}]\subset{\mathbb{R}}^{2}$.

###  9.1 Examples with a unique maximum point of the variance

Suppose $\nu(t_{1},t_{2})$ attains the maximum $\sigma_{T}^{2}$ only at a
single point $t^{*}=(t_{1}^{*},t_{2}^{*})$; and the assumptions in Theorems
8.3 or 8.4 are satisfied.

Case 1: $\bm{t^{*}=(b_{1},b_{2})}$ and $\bm{\nu_{1}(t^{*})\nu_{2}(t^{*})\neq
0}$. It follows directly from Theorem 8.3 that

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}=\Psi\left(\frac{u}{\sigma_{T}}\right)+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right).\end{split}$

Case 2: $\bm{t^{*}=(b_{1},b_{2})}$, $\bm{\nu_{1}(t^{*})=0}$ and
$\bm{\nu_{2}(t^{*})\neq 0}$. It follows from Corollary 8.5 that

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}\\\
&=\left({\mathbb{P}}\\{X_{1}(t^{*})>0\\}+\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}{{\rm
Var}(X_{1}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}}{\mathbb{P}}\\{Z>0\\}\right)\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1))\\\
&=\frac{1}{2}\left(1+\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}{{\rm
Var}(X_{1}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}}\right)\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)),\end{split}$

where $Z$ is a centered Gaussian variable.

Case 3: $\bm{t^{*}=(b_{1},b_{2})}$ and $\bm{\nu_{1}(t^{*})=\nu_{2}(t^{*})=0}$.
Applying Corollary 8.5 and noting the calculations in Case 2 above, we obtain

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}\\\
&=\Bigg{(}{\mathbb{P}}\\{X_{1}(t^{*})>0,X_{2}(t^{*})>0\\}+\frac{1}{2}\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}{{\rm
Var}(X_{1}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}}\\\
&\qquad+\frac{1}{2}\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X_{22}(t^{*})\\}}{{\rm
Var}(X_{2}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X_{22}(t^{*})\\}}}\\\
&\qquad+{\mathbb{P}}\\{Z_{1}>0,Z_{2}>0\\}\sqrt{\frac{{\rm
det}(\Sigma(t^{*}))}{{\rm
det}(\Lambda(t^{*})+\Sigma(t^{*}))}}\Bigg{)}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)),\end{split}$

where $(Z_{1},Z_{2})$ is a centered Gaussian vector with covariance
$\Sigma(t^{*})+\Sigma(t^{*})\Lambda^{-1}(t^{*})\Sigma(t^{*})$.

Case 4: $\bm{t^{*}=(t_{1}^{*},b_{2})}$, where $\bm{t_{1}^{*}\in(a_{1},b_{1})}$
and $\bm{\nu_{2}(t^{*})\neq 0}$. It follows directly from Theorem 8.3 that

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}=\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}{{\rm
Var}(X_{1}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1).\end{split}$

Case 5: $\bm{t^{*}=(t_{1}^{*},b_{2})}$, where $\bm{t_{1}^{*}\in(a_{1},b_{1})}$
and $\bm{\nu_{2}(t^{*})=0}$. Applying Corollary 8.5 and noting the
calculations in Case 2 above, we obtain

$\begin{split}&\quad{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq u\right\\}\\\
&=\frac{1}{2}\Bigg{(}\sqrt{\frac{{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}{{\rm
Var}(X_{1}(t^{*}))+{\mathbb{E}}\\{X(t^{*})X_{11}(t^{*})\\}}}+\sqrt{\frac{{\rm
det}(\Sigma(t^{*}))}{{\rm
det}(\Lambda(t^{*})+\Sigma(t^{*}))}}\Bigg{)}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1)).\end{split}$

Case 6: $\bm{a_{1}<t_{1}^{*}<b_{1}}$ and $\bm{a_{2}<t_{2}^{*}<b_{2}}$. It
follows directly from Theorem 8.3 that

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}=\sqrt{\frac{{\rm det}(\Sigma(t^{*}))}{{\rm
det}(\Lambda(t^{*})+\Sigma(t^{*}))}}\Psi\left(\frac{u}{\sigma_{T}}\right)(1+o(1).\end{split}$

###  9.2 Examples with the maximum of the variance achieved on a line

Consider the Gaussian random field $X(t)$ defined as:

$X(t)=\xi_{1}\cos t_{1}+\xi_{1}^{\prime}\sin t_{1}+t_{2}(\xi_{2}\cos
t_{2}+\xi_{2}^{\prime}\sin t_{2}),$

where $t=(t_{1},t_{2})\in
T=[a_{1},b_{1}]\times[a_{2},b_{2}]\subset(0,2\pi)^{2}$, and
$\xi_{1},\xi_{1}^{\prime},\xi_{2},\xi_{2}^{\prime}$ are independent standard
Gaussian random variables. This is a Gaussian random field on
${\mathbb{R}}^{2}$ generated from the cosine field, with an additional product
of $t_{2}$ along the vertical direction. The constraint on the parameter space
within $(0,2\pi)^{2}$ is imposed to prevent degeneracy in derivatives. For
this field, we have $\nu(t)=1+t_{2}^{2}$, which reaches the maximum
$\sigma_{T}^{2}=1+b_{2}^{2}$ on the entire real line
$L:=\\{(t_{1},b_{2}):a_{1}\leq t_{1}\leq b_{1}\\}$. Furthermore,

$\nu_{1}(t)|_{t\in L}=0,\quad\nu_{2}(t)|_{t\in L}=2b_{2}>0,\quad\forall t\in
L.$

By employing similar reasoning in the proofs of Theorems 3.1 and 3.2, we see
that, in the EEC approximation ${\mathbb{E}}\\{\chi(A_{u})\\}$, all integrals
(derived from the Kac-Rice formula) over faces not contained within $\bar{L}$
are super-exponentially small. Thus, there exists $\alpha>0$ such that as
$u\to\infty$,

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}&={\mathbb{P}}\\{X(a_{1},b_{2})\geq
u,X_{1}(a_{1},b_{2})<0\\}+{\mathbb{P}}\\{X(b_{1},b_{2})\geq
u,X_{1}(b_{1},b_{2})>0\\}\\\
&\quad+I(u)+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right)\\\
&=\Psi\left(\frac{u}{\sqrt{1+b_{2}^{2}}}\right)+I(u)+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right),\end{split}$ (9.1)

where

$\begin{split}I(u)&=-\int_{a_{1}}^{b_{1}}{\mathbb{E}}\big{\\{}X_{11}(t_{1},b_{2})\mathbbm{1}_{\\{X(t_{1},b_{2})\geq
u\\}}\big{|}X_{1}(t_{1},b_{2})=0\big{\\}}p_{X_{1}(t_{1},b_{2})}(0)dt_{1}.\end{split}$

Since $X_{1}(t_{1},b_{2})=-\xi_{1}\sin t_{1}+\xi_{1}^{\prime}\cos t_{1}$ and
$X_{11}(t_{1},b_{2})=-\xi_{1}\cos t_{1}-\xi_{1}^{\prime}\sin t_{1}$, one has

$\begin{split}{\rm
Cov}(X(t_{1},b_{2}),X_{1}(t_{1},b_{2}),X_{11}(t_{1},b_{2}))&=\begin{pmatrix}1+b_{2}^{2}&0&-1\\\
0&1&0\\\ -1&0&1\end{pmatrix},\end{split}$

which does not depend on $t_{1}$. Particularly, $X_{1}(t_{1},b_{2})$ is
independent of both $X(t_{1},b_{2})$ and $X_{11}(t_{1},b_{2})$. Thus

$\begin{split}I(u)&=-\frac{b_{1}-a_{1}}{\sqrt{2\pi}}{\mathbb{E}}\big{\\{}X_{11}(t_{1},b_{2})\mathbbm{1}_{\\{X(t_{1},b_{2})\geq
u\\}}\big{\\}}\\\
&=-\frac{b_{1}-a_{1}}{\sqrt{2\pi}}\int_{u}^{\infty}{\mathbb{E}}\\{X_{11}(t_{1},b_{2})|X(t_{1},b_{2})=x\\}\phi\left(\frac{x}{\sqrt{1+b_{2}^{2}}}\right)dx\\\
&=\frac{b_{1}-a_{1}}{\sqrt{2\pi}}\int_{u}^{\infty}\frac{x}{1+b_{2}^{2}}\phi\left(\frac{x}{\sqrt{1+b_{2}^{2}}}\right)dx\\\
&=\frac{b_{1}-a_{1}}{\sqrt{2\pi}}\phi\left(\frac{u}{\sqrt{1+b_{2}^{2}}}\right).\end{split}$

Substituting this expression into (9.1), we arrive at the following refined
approximation:

$\begin{split}{\mathbb{P}}\left\\{\sup_{t\in T}X(t)\geq
u\right\\}=\Psi\left(\frac{u}{\sqrt{1+b_{2}^{2}}}\right)+\frac{b_{1}-a_{1}}{\sqrt{2\pi}}\phi\left(\frac{u}{\sqrt{1+b_{2}^{2}}}\right)+o\left(\exp\left\\{-\frac{u^{2}}{2\sigma_{T}^{2}}-\alpha
u^{2}\right\\}\right),\end{split}$

which has a super-exponentially small error.

## Acknowledgments

The author acknowledges support from NSF Grants DMS-1902432 and DMS-2220523,
as well as the Simons Foundation Collaboration Grant 854127.

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> Dan Cheng
> School of Mathematical and Statistical Sciences
> Arizona State University
> 900 S Palm Walk
> Tempe, AZ 85281, USA
> E-mail<EMAIL_ADDRESS>