#  Asymptotic formula for the tail of the maximum of smooth Gaussian fields on
non locally convex sets

Jean-Marc Azaïs and Viet-Hung Pham
Institut de Mathématiques de Toulouse
Université Paul Sabatier (Toulouse III), France

###### Abstract

In this paper we consider the distribution of the maximum of a Gaussian field
defined on non locally convex sets. Adler and Taylor or Azaïs and Wschebor
give the expansions in the locally convex case. The present paper generalizes
their results to the non locally convex case by giving a full expansion in
dimension 2 and some generalizations in higher dimension. For a given class of
sets, a Steiner formula is established and the correspondence between this
formula and the tail of the maximum is proved. The main tool is a recent
result of Azaïs and Wschebor that shows that under some conditions the
excursion set is close to a ball with a random radius. Examples are given in
dimension 2 and higher.

Key-words: Stochastic processes, Gaussian fields, Rice formula, distribution
of the maximum, non locally convex indexed set.
Classifications: 60G15, 60G60, 60G70.

##  1 Introduction

Let $\mathcal{X}=\\{X(t):t\in S\subset{\mathbb{R}}^{n}\\}$ be a random field
with real values and let $M_{S}$ be its maximum (or supremum) on $S$.
Computing the distribution of the maximum is a very important issue from the
theoretical point of view and also has a great impact on applications,
especially in spatial statistics. This problem has therefore received a great
deal of attention from many authors.

However an exact result is known only in very few cases, (see Azaïs and
Wschebor [5]). In other cases, the only available results are asymptotic
expansions or bounds mainly in the case of stationary Gaussian random fields.

One of the most well-known and quite general methods is the "double-sum
method", first proposed by Pickands [11] and extended by Piterbarg [12], [13].
The main idea of this method is to use the inclusion-exclusion principle and
the Bonferroni inequality after dividing the parameter set into suitable
smaller subsets. It was first proposed in dimension 1 : $n=1$ (in this case we
use the classical terminology of "random processes" instead of "random
field"). More precisely, for some particular processes, i.e., the "$\alpha$
processes", Pickands proposed an equivalent for the tail of the maximum.
However, the result depends on some unknown constants,referred to as Pickands’
constants and just gives an equivalent.

Another method is the "tube method" proposed by Sun [16]. She observed that if
the Karhunen-Loève expansion of the field is finite in the sense that there
exist a finite number of random variables
$\xi_{1},\ldots,\xi_{k}\overset{i.i.d}{\sim}\mathcal{N}(0,1)$ such that at
every point $t$ in the parameter set, the value of the field at this point
$X(t)$ can be expressed as

$X(t)=a_{t}^{1}\xi_{1}+\ldots+a_{t}^{k}\xi_{k},$

where the vector $(a_{t}^{1},\ldots,a_{t}^{k})$ has unit norm since
$\textrm{Var}(X(t))=1$, then the original parameter set can be transformed
into a subset of the unit sphere $\mathcal{S}^{k-1}$. She then used Weyl’s
tube formula to compute the polynomial expansion of the volume of the tube
around a subset of the unit sphere and derived the asymptotic formula of the
tail of the distribution of the maximum from this expansion. She is the first
one who realizes the strong connection between the geometric functionals of
the parameter set (the coefficients of the polynomial expansion) and the tail
of the distribution. When the Karhunen-Loève expansion is not finite, she uses
a truncation argument to derive an asymptotic formula with two terms. Later
on, this method was extended by Takemura and Akimichi [17], [18].

In the 1940s, in his pioneering work, Rice [14] considered a stationary
process $\mathcal{X}$ with $\mathcal{C}^{1}$ paths defined on the compact
interval $[0,T]$. He observed that for every level $u$:

$\begin{array}[]{rl}\displaystyle\mathbb{P}\left(\underset{t\in[0,T]}{\max}X(t)\geq
u\right)&\displaystyle\leq\mathbb{P}(X(0)\geq u)+\mathbb{P}\left(\exists
t\in[0,T]:\,X(t)=u,X^{\prime}(t)\geq 0\right)\\\
&\displaystyle\leq\mathbb{P}(X(0)\geq
u)+\mathbb{E}\left(\textrm{card}\left\\{t\in[0,T]:\,X(t)=u,X^{\prime}(t)\geq
0\right\\}\right),\\\ \end{array}$

where the last expectation can be evaluated by the famous Rice-Kac formula.
This upper bound was later proved to be sharp by Piterbarg [15]. This Rice-Kac
formula is the starting point of the following methods dealing with the random
fields: the "Rice method" by Azaïs and Delmas [3], [8], the "direct method" by
Azaïs and Wschebor [5] and the "Euler characteristic method" by Adler and
Taylor [1]. These methods use a multidimensional Rice-Kac formula :
Generalized Rice formula (Azaïs and Wschebor) or Metatheorem (Adler and
Taylor).

In the direct method, Azaïs and Wschebor used some results from the random
matrix theory to compute the expectation of the absolute value of the
determinant of the Hessian that appears in the Rice formula. They obtained an
upper bound for the tail of the distribution depending on some geometric
functionals of the parameter set. This upper bound is also sharp.

Adler and Taylor combined differential and integral geometry to find the
"Euler characteristic method" that gives one of most frequently used results
in this area. They considered stratified sets, i.e. locally convex Whitney
stratified manifolds. First, they used the Metatheorem to compute the
expectation of the Euler characteristic of the excursion set (see Theorem
12.4.1) and, second, they proved that the difference between the above
expectation and the excursion probability (the tail of the distribution) is
super exponentially smaller (see Theorem 14.3.3). Note that the geometric
functionals of the parameter set appear in the expectation of the Euler
characteristic under the name of Lipschitz-Killing curvatures.

We recall an important example when the parameter set $S$ is a convex body in
${\mathbb{R}}^{2}$ (compact, convex, with non-empty interior) and
$\mathcal{X}$ is an isotropic centered Gaussian field defined on some
neighborhood of $S$ and satisfying ${\textnormal{V}ar}(X(t)=1$ and
$\textrm{Var}(X^{\prime}(t))=I_{n}$, where $I_{n}$ is the identity matrix of
size $n$ . Let us denote:

$M_{S}=\max_{t\in S}X(t).$

Then, under some regularity conditions, the Euler characteristic method gives:

$\mathbb{P}(M_{S}\geq u)=\overline{\Phi}(u)+\frac{\sigma_{1}(\partial
S)}{2\sqrt{2\pi}}\varphi(u)+\frac{\sigma_{2}(S)}{2\pi}u\varphi(u)+o\left(\varphi\left((1+\alpha)u\right)\right),$
(1)

for some $\alpha>0$, where $\overline{\Phi}(u)$ and $\varphi(u)$ are the tail
distribution and the density of a standard normal variable, $\sigma_{2}(S)$ is
the area of $S$ and $\sigma_{1}(S)$ is the perimeter of $S$ . Note that the
coefficient $1$ of the term $\overline{\Phi}(u)$ can be interpreted as the
Euler characteristic of $S$.

Adler and Taylor use the local convexity that can be defined as the fact that
for every point $t\in S$, the support cone $C_{t}$ generated by the set of
directions

$\Biggl{\\{}\lambda\in{\mathbb{R}}^{2}:\;\|\lambda\|=1,\,\exists s_{n}\in
S\;\textnormal{such\; that}\;s_{n}\rightarrow
t\;\textnormal{and}\;\frac{s_{n}-t}{\|s_{n}-t\|}\rightarrow\lambda\Biggr{\\}},$
(2)

is convex, plus some regularity conditions (see, for example [1, Section 8.2])
($\|.\|$ is the Euclidean norm). Similarly, Azaïs and Wschebor [6, p. 231] use
the condition:

$\kappa(S)=\underset{t\in S}{\sup}\underset{s\in S,\;s\neq
t}{\sup}\frac{\textnormal{dist}(s-t,C_{t})}{\|s-t\|^{2}}<\infty$ (3)

where dist is the Euclidean distance.

However none of these methods is able to provide a full expansion for the
asymptotic formula in the non-locally convex cases, even the very simple case
of $S$ being "the angle" that is the union of two segments with the angle
$\beta\in(0,\pi)$, see Figure 1,

$S_{1}$$S_{2}$$\beta$ Figure 1: The angle, an example of non-local convexity.

which is presented in [1, Section 14.4.4]. By a full expansion, we mean a
formula of the type (1) with three terms in dimension 2 and n+1 in the general
case.

We are therefore interested in the following question:

"Can we find some full expansions for the tail of the maximum in some non-
locally convex cases in dimension $2$ and higher?"

In a previous article [4], we gave an upper bound for the tail of the
distribution for quite general parameter sets $S$. More precisely, if $S$ is
the Hausdorff limit of connected polygons $S_{n}$, if $\mathcal{X}$ is a
stationary centered Gaussian field with variance $1$ and
$\textrm{Var}(X^{\prime}(t))=I_{n}$ defined on a neighborhood of $S$ then for
every level $u$:

$\mathbb{P}\\{M_{S}\geq
u\\}\leq\overline{\Phi}(u)+\frac{\liminf_{n}\sigma_{1}(S_{n})\varphi(u)}{2\sqrt{2\pi}}+\frac{\sigma_{2}(S)}{2\pi}\left[c\varphi(u\textrm{/}c)+u\Phi(u\textrm{/c})\right]\varphi(u),$
(4)

where $c=\sqrt{\textnormal{Var}(X^{\prime\prime}_{11}(t))-1}$,
$X^{\prime\prime}_{11}(t)=\frac{\partial^{2}X(t)}{\partial t^{2}_{1}}$
$\sigma_{2}$ is the area and $\sigma_{1}$ is the perimeter.

Note that (4) can be applied to polygons taking the simpler form:

$\mathbb{P}\\{M_{S}\geq
u\\}\leq\overline{\Phi}(u)+\frac{\sigma_{1}(S)\varphi(u)}{2\sqrt{2\pi}}+\frac{\sigma_{2}(S)}{2\pi}\left[c\varphi(u\textrm{/}c)+u\Phi(u\textrm{/c})\right]\varphi(u),$
(5)

When the polygon is convex, we can check that (5) is sharp by comparing (1)
and (5). However we do not have such information in the non-convex case.

Recently Azaïs and Wschebor [7] proposed a new method, still based on the
generalized Rice formula, to derive the asymptotic formula when the parameter
set $S$ is fractal. They also gave an asymptotic expansion with two terms in
the case of a parameter set with a finite perimeter (defined as an outer
Minkowsky content). However, for example in dimension 2, this result does not
give the coefficient of $\overline{\Phi}(u)$, which is the third term by order
of importance.

Section 2 is devoted to dimension 2. We define a quite general class of
parameter sets in ${\mathbb{R}}^{2}$ (see Definition 1) and derive the
asymptotic formula for the tail of the maximum of the random fields defined on
these parameter sets. This is our main result (Theorem 1). It shows that the
coefficient corresponding to $\overline{\Phi}(u)$ is not always equal to the
Euler characteristic of the parameter set and, in fact, it is derived from the
Steiner formula that gives the volume (area) of the tube around $S$. Here
again, we emphasize the strong connection between the tube formula of the
parameter set and the tail of the maximum.

In Section 3, we examine this connection by considering some examples. We use
elementary geometry to compute the tube formula, obtain the geometric
functionals, and then immediately obtain the asymptotic expansion of the tail
distribution. All the examples correspond to new results. In particular, the
examples in Subsection 3.5 and 3.6 could shed new light on this problem. We
also conjecture that the strong connection still occurs in dimensions higher
than $2$ and 3, and even in fractal dimension.

### Hypotheses and notation

We will use the following assumption on the random field $\mathcal{X}$
througout this paper:

Assumption $A$: $\mathcal{X}$ is a random field defined on a ball
$B\subset{\mathbb{R}}^{n}$ satisfying:

  * i.

$\mathcal{X}$ is a stationary centered Gaussian field.

  * ii.

Almost surely the paths of $X(t)$ are of class $\mathcal{C}^{3}$.

  * iii.

$\textnormal{Var}(X(t))=1$ and $\textnormal{Var}(X^{\prime}(t))$ is the
identity matrix.

  * iv.

For all $s\neq t\in B$, the distribution of
$(X(s),X(t),X^{\prime}(s),X^{\prime}(t))$ does not degenerate.

  * v.

For all $t\in B,\;\gamma\in\mathcal{S}^{n-1}$, the distribution of
$(X(t),X^{\prime}(t),X^{\prime\prime}(t)\gamma)$ does not degenerate.

We use the following additional notation and hypotheses.

  * •

$\mathcal{S}^{n-1}$ is the unit sphere in ${\mathbb{R}}^{n}$.

  * •

$S$ is a compact subset of $B$ at a positive distance from the boundary
$\partial B$ and satisfies some regularity properties (see Definition 1).

  * •

$B(t,r)$ is the ball of radius $r$ centered at $t$.

  * •

$M_{Z}$ is the maximum of $X(t)$ on the set $Z\subset{\mathbb{R}}^{n}$.

  * •

$S^{+\epsilon}$ is the tube around $S$ defined as:

$S^{+\epsilon}=\left\\{t\in{\mathbb{R}}^{n}:\;\textnormal{dist}(t,S)\leq\epsilon\right\\}.$

##  2 Main results

Firstly, we define, the class of parameter sets $S$ that will be considered in
dimension 2.

######  Definition 1 (Two dimensional sets with piecewise-$\mathcal{C}^{2}$
boundary).

We assume that the compact set $S$ consists of a finite number of connected
components of the same nature. We describe in detail the case where $S$ has
only one connected component. $S$ contains three parts:

  * •

The interior $\overset{\circ}{S}$.

  * •

The non isolated curves: they form the boundary of $\overset{\circ}{S}$, and
consist of a finite number of closed continuous piecewise-$\mathcal{C}^{2}$
curves. One of these curves is the exterior boundary and the others are the
boundaries of the holes inside $S$. We assume that these curves are simple and
do not intersect each other. Each of these curves is parameterized by its arc
length using the positive orientation.

  * •

The isolated curves: they consist of a finite number of disjoint self-avoiding
continuous and piecewise-$\mathcal{C}^{2}$ open curves. If
$\overset{\circ}{S}$ is not empty, each curve has exactly one extremity that
belongs to $\overline{\overset{\circ}{S}}$. If $\overset{\circ}{S}$ is empty,
then $S$ consists of only one isolated curve.

Irregular points are the points where the parametrization of the curve is no
longer $\mathcal{C}^{2}$. They divide the curves above into a finite number of
edges. An edge of a non-isolated curve will be referred to as non-isolated
edge, and similarly for an isolated edge. To limit the number of
configurations, we assume that an irregular point belongs to one of four
following categories:

  * •

Convex binary points: the intersection of two non-isolated edges and the
support cone defined by (2) is convex.

  * •

Concave binary points: as above but the support cone is not convex. Denote
$\beta\in[0,\pi)$ as the discontinuity of the angle of the tangent.

  * •

Angle points: they are the intersection of two edges belonging to the same
isolated curve. Denote $\beta\in[0,\pi)$ by the discontinuity of the angle
(the orientation does not matter) as in Figure 1.

  * •

Concave ternary points: the intersection of two non isolated edges
$E_{1},E_{2}$ and one isolated one $E_{3}$. In the main result, these points
will be considered with multiplicity two. We associate two concave angles to
each of these points:

\- $\beta_{1}$: the discontinuity of the angle of the tangent when we pass
from $E_{1}$ to $E_{3}$.

\- $\beta_{2}$: the discontinuity of the angle of the tangent when we pass
from $E_{3}$ to $E_{2}$.

To obtain a rather simple result, we only consider the concave ternary points
such that $\beta_{1}+\beta_{2}\leq\pi$, and we exclude more complicated
situations such as point of order four or the existence of handles, for
example.

Finally, the $\beta$’s described above will be referred to as concave angles.

$\beta$$E_{1}$$E_{2}$$E_{3}$$\beta_{1}$$\beta_{2}$ Figure 2: Convex, concave
binary and concave ternary points, respectively.

Remark. It should be observed that the sets with piecewise-$\mathcal{C}^{2}$
boundary considered here are Whitney stratified manifolds in the sense of [1,
Section 8.1] with some additional restrictions. We refer readers to this book
for more details.

Our proof will be hereditary proof. We will start from a set without concave
points and use the result recalled in the Appendix to establish an expansion
for such a set. We will then proceed by union to extend the result to the
general class of sets of Definition 1. To do so, we need a definition of the
property that will be extended by union. This is the object of the following
definition.

######  Definition 2 (Steiner formula heuristic property).

A compact subset $S$ of ${\mathbb{R}}^{2}$ is said to satisfy the Steiner
formula heuristic (SFH) if it satisfies the following conditions:

  * •

There exist two non-negative constants $L_{1}(S)$ and $L_{0}(S)$ such that, as
$\epsilon$ tends to $0$,

$\sigma_{2}(S^{+\epsilon})=\sigma_{2}(S)+\epsilon
L_{1}(S)+\pi\epsilon^{2}L_{0}(S)+o(\epsilon^{2}).$ (6)

  * •

For all processes $X(t)$ satisfying Assumption $A$,

$\mathbb{P}(M_{S}\geq
u)=L_{0}(S)\overline{\Phi}(u)+L_{1}(S)\frac{\varphi(u)}{2\sqrt{2\pi}}+\sigma_{2}(S)\frac{u\varphi(u)}{2\pi}+o\left(u^{-1}\varphi(u)\right),$
(7)

as $u\to\infty$.

Remarks.

  * 1.

There exist some generalizations of the Steiner formula that hold true for
every closed set, see [10]. The present form is more restrictive.

  * 2.

If $S$ is a convex body, then (6) will take the form : for all $\epsilon>0$

$\sigma_{2}(S^{+\epsilon})=\sigma_{2}(S)+\epsilon
L_{1}(S)+\pi\epsilon^{2}L_{0}(S).$ (8)

$L_{1}(S)$ is just the perimeter $\sigma_{1}(S)$ and $L_{0}(S)$ is the Euler
characteristic of $S$ which is equal to $1$.

If, in addition, the number of irregular points of $S$ (points where the
support cone is not a half space) is finite, then on the basis of the result
of Adler and Taylor, (7) follows. Thus a convex body with a finite number of
irregular points satisfies the SFH property.

  * 3.

If $S$ has a positive reach in the sense that there exists a positive constant
$r$ such that for all $t\in S^{+r}$, $t$ has only one projection on $S$, then
(8) is true for all $\epsilon<r$ (see [2], [9]). Moreover, if $S$ is a set
with a piecewise-$\mathcal{C}^{2}$ boundary in the sense of Definition 1 and
satisfies $\kappa(S)<\infty$ where $\kappa(S)$ is defined in (3), then (7)
still holds true (see Appendix).

  * 4.

In the most general cases, the constant $L_{1}(S)$ is the outer Minkowski
content of $S$ ($\textnormal{OMC}(S)$), which is defined, when it exists, by:

$\sigma_{2}(S^{+\epsilon})=\sigma_{2}(S)+\epsilon\textnormal{OMC}(S)+o(\epsilon).$

For more details, see [2]. It corresponds to the definition of the perimeter
of a curve in convex geometry. It can differ from the length of the boundary
of $S$, for example in the case of "the square with whiskers" (see Figure 3).

Figure 3: The square with whiskers.

In this case, the length of the boundary is equal to the perimeter of the
square plus the length of the whiskers, while $\textnormal{OMC}(S)$ is equal
to the perimeter of the square plus two times the length of the whiskers. In
addition it should be noticed that $L_{0}(S)$ is not always equal to the Euler
characteristic (see Subsection 3.4).

We are now able to state our main result.

######  Theorem 1.

Let $S$ be a compact subset in ${\mathbb{R}}^{2}$ with a
piecewise-$\mathcal{C}^{2}$ boundary and with concave angles
$\beta_{1},\ldots,\beta_{k}$ as defined in Definition 1. Let $\mathcal{X}$ be
a random field satisfying Assumption $A$. Let $M_{S}$ be the maximum of $X(t)$
on $S$.

Then $S$ satisfies the SFH and:

$\mathbb{P}(M_{S}\geq
u)=\left[\chi(S)+\frac{1}{\pi}\sum_{i=1}^{k}\left(\frac{\beta_{i}}{2}-\tan\frac{\beta_{i}}{2}\right)\right]\overline{\Phi}(u)+\frac{\textnormal{OMC}(S)}{2\sqrt{2\pi}}\varphi(u)+\frac{\sigma_{2}(S)}{2\pi}u\varphi(u)+o\left(u^{-1}\varphi(u)\right),$
(9)

where $\chi(S)$ is the Euler characteristic of $S$ that is equal to the number
of connected components minus the number of holes.

In addition, the outer Minkowski content $\textnormal{OMC}(S)$ is equal to the
length of the non-isolated edges plus twice the length of the isolated edges.

For an illustration of this theorem, see the examples in Section 3. In order
to prove the main theorem, we need some auxiliary lemmas. Firstly, we recall a
well-known result on Gaussian processes [6].

######  Lemma 1 (Borel-Sudakov-Tsirelson inequality).

Let $\mathcal{X}$ be a centered Gaussian field defined on a parameter $Z$ such
that it is almost surely bounded. Then $\mathbb{E}(M_{Z})<\infty$, and, for
all $u>0$,

$\mathbb{P}\left(M_{Z}-\mathbb{E}(M_{Z})\geq
u\right)\leq\exp(-u^{2}/(2\sigma^{2}_{Z})),$

where $\displaystyle\sigma^{2}_{Z}=\underset{t\in
Z}{\sup}\mathbb{E}(X^{2}(t))$.

An easy consequence of the BST inequality is that, for each $\epsilon>0$,
there exists a constant $C_{\epsilon}>0$ such that for all $u>0$:

$\mathbb{P}(M_{Z}\geq u)\leq
C_{\epsilon}\exp\left(\frac{-u^{2}}{2(\sigma^{2}_{Z}+\epsilon)}\right).$ (10)

We will use the above observation to prove the following lemma.

######  Lemma 2.

Let $\mathcal{X}$ be a random field satisfying Assumption $A$. Let
$Z_{1},\ldots,Z_{k}$ be some compact subsets of $B$ such that:

$Z_{1}\cap\ldots\cap Z_{k}=\emptyset.$

Then there exist two constants $\theta>1$ and $C$ such that for all $u>0$,

$\mathbb{P}\left(M_{Z_{1}}\geq u,\,\ldots,\,M_{Z_{k}}\geq u\right)\leq
C.\exp(-\theta u^{2}/2).$

###### Proof.

On the set $Z:=Z_{1}\times\ldots\times Z_{k}$, we consider the Gaussian field
$Y$ defined by:

$Y(t_{1},\ldots,t_{k})=X(t_{1})+\ldots+X(t_{k}).$

Then

$\mathbb{P}\left(M_{Z_{1}}\geq u,\,\ldots,\,M_{Z_{k}}\geq
u\right)\leq\mathbb{P}\left(\underset{t\in Z}{\sup}Y(t)\geq k.u\right).$

Applying (10) to the Gaussian field $Y$, we see that for each $\epsilon>0$,
there exists a constant $C_{\epsilon}>0$ such that for all $u>0$:

$\mathbb{P}(\underset{t\in Z}{\sup}Y(t)\geq k.u)\leq
C_{\epsilon}\exp\left(\frac{-k^{2}u^{2}}{2(\sigma^{2}_{Z}+\epsilon)}\right),$

where

$\sigma^{2}_{Z}=\underset{t\in Z}{\sup}\mathbb{E}(Y^{2}(t))=\underset{t\in
Z}{\sup}\mathbb{E}\left[(X(t_{1})+\ldots+X(t_{k}))^{2}\right].$

Since $\mathbb{E}(X^{2}(t_{i}))=1$,
$\mathbb{E}\left(X(t_{i})X(t_{j})\right)<1$ if $t_{i}\neq t_{j}$ and $Z$ is
compact, we have $\sigma^{2}_{Z}<k^{2}$. By choosing $\epsilon>0$ such that
$k^{2}>\sigma^{2}_{Z}+\epsilon$, the result follows. ∎

Since we look at a result of the type (9), every event with proba
$o\left(u^{-1}\varphi(u)\right)$ can be neglected and will be called
"negligible". Lemma 2 shows that the event $(M_{Z_{1}}\geq
u,\,\ldots,\,M_{Z_{k}}\geq u)$ is negligible as $u\to+\infty$.

The following lemma is a recent result of Azaïs and Wschebor [7].

######  Lemma 3.

Let $X$ be a random field satisfying Assumption $A$ and $\alpha$ be a given
real number $0<\alpha<1$. Then the following events are negligible:

  * $A_{1}=\left\\{\exists\;\mbox{a local maximum in}\;B\mbox{ with value}\;\geq u+1\right\\}$.

  * $A_{2}=\left\\{\exists\;\mbox{two or more local maxima in}\;\overset{\circ}{B}\mbox{ with value}\;\geq u\right\\}$.

  * $\begin{array}[]{rl}A_{3}=&\left\\{\exists\,\mbox{a local maximum}\;t\in\overset{\circ}{B}\right.\\\ &\left.\displaystyle\mbox{such that}\;u<X(t)<u+1,\;\min\left\\{\gamma^{T}X^{\prime\prime}(s)\gamma:\,s\in B(t,u^{-\beta}),\,\gamma=\frac{s-t}{\|s-t\|}\right\\}\leq-X(t)-u^{\alpha}\right\\},\end{array}$

where $\alpha$ and $\beta$ are some positive constants in $(0,1)$, satisfying
$\beta>(1-\alpha)/2$.

  * $\begin{array}[]{rl}A_{4}=&\bigg{\\{}\exists\,\mbox{a local maximum}\;t\in B\\\ &\left.\displaystyle\mbox{such that}\;u<X(t)<u+1,\;\max\left\\{\gamma^{T}X^{\prime\prime}(s)\gamma:\,s\in B(t,u^{-\beta}),\,\gamma=\frac{s-t}{\|s-t\|}\right\\}\geq-X(t)+u^{\alpha}\right\\}.\end{array}$

Let us comment on Lemma 3. Consider the event $\\{M>u\\}\cap
A_{1}^{c}\cap\cdots\cap A_{4}^{c}$ that differs from the event of interest
$\\{M>u\\}$ by a negligible probability.

Because we are in $A_{2}^{c}$, there exists at most in $B$ one local maximum
with a value larger than $u$. This implies that the excursion set:

$K_{u}:=\\{s\in B:\,X(s)\geq u\\}$

consists of one connected component. Moreover , because we are in
$A_{1}^{c}\cap A_{4}^{c}$, and thanks to a Taylor expansion:

$X(s)=X(t)+\frac{1}{2}\|s-t\|^{2}\gamma^{T}X^{\prime\prime}(\eta)\gamma,$

this component is included in:

$B(t,\overline{r})\ \ ;\mbox{ with }\ \
\overline{r}=\sqrt{2\frac{X(t)-u}{u-u^{\alpha}}}$ (11)

and where $t$ is the location of the local maximum.

This implies that, for $u$ large enough, $t$ lies in $\overset{\circ}{B}$.
Using the fact that we are in $A_{3}^{c}$ , we obtain, in the same manner,

$B(t,r)\subset K_{u}$

with

$\underline{r}=\sqrt{2\frac{X(t)-u}{X(t)+u^{\alpha}}}$

Eventually, we obtain:

$\mathbb{P}(\exists\,\mbox{a local maxima}\;t\in\overset{\circ}{B}:\,t\in
S^{+\underline{r}})+o(u^{-1}\varphi(u))\leq\mathbb{P}(M_{S}\geq
u)\leq\mathbb{P}(\exists\,\mbox{a local maxima}\;t\in\overset{\circ}{B}:\,t\in
S^{+\overline{r}})+o(u^{-1}\varphi(u)).$

On the basis of this observation, Azaïs and Wschebor derived an asymptotic
formula for the excursion distribution $\mathbb{P}(M_{S}\geq u)$. For more
details see [7].

Our starting point in this paper is the following lemma that extend the ideas
of Azaïs and Wschebor by considering several sets.

######  Lemma 4.

Let $\mathcal{X}$ be a random field satisfying Assumption $A$ and
$S_{1},\ldots,S_{m}$ be $m$ subsets of $B$ at a positive distance from
$\partial B$. Assume that there exist two constants $C>0$ and $0\leq d<n$ such
that:

$\lambda_{n}\left(S_{1}^{+\epsilon}\cap\ldots\cap
S_{m}^{+\epsilon}\right)=\left(C+o(1)\right)\epsilon^{n-d}\;as\;\epsilon\rightarrow
0.$ (12)

Then, as $u\rightarrow+\infty$,

$\mathbb{P}\left(\forall i=1\ldots m:\;M_{S_{i}}\geq
u\right)=u^{d-1}\varphi(u)\left(\frac{C}{2^{d/2}\pi^{n/2}}\Gamma\left(1+(n-d)/2\right)+o(1)\right),$
(13)

where $\Gamma$ is the Gamma function.

###### Proof.

Using Lemma 3 we have the upper-bound

$\begin{array}[]{rl}&\displaystyle\mathbb{P}\left(\forall i=1\ldots
m:\;M_{S_{i}}\geq u\right)\\\
\leq&o\left(u^{-1}\varphi(u)\right)+\mathbb{P}\left(\exists
t\in\overset{\circ}{B}:\;X(.)\textnormal{ \;has\; a\; local\; maximum\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}}\right)\\\
\leq&o\left(u^{-1}\varphi(u)\right)+\mathbb{E}\left(\textnormal{card}\left\\{t\in\overset{\circ}{B}:\;X(.)\textnormal{
\;has\; a\; local\; maximum\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}}\right\\}\right).\end{array}$

Applying the Rice formula (see [6, Chapter 6]),

$\displaystyle E:=$
$\displaystyle\mathbb{E}\left(\textnormal{card}\left\\{t\in\overset{\circ}{B}:\;X(.)\textnormal{
\;has\; a\; local\; maxima\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}}\right\\}\right)$
$\displaystyle=$
$\displaystyle\int_{u}^{+\infty}dx\int_{\overset{\circ}{B}}\mathbb{E}\left(|\det(X^{\prime\prime}(t))|\mathbb{I}_{\\{X^{\prime\prime}(t)\preceq
0\\}}\mathbb{I}_{\\{t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}}\\}}\mid
X(t)=x,\;X^{\prime}(t)=0\right)p_{X(t),X^{\prime}(t)}(x,0)\sigma_{n}(dt)$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{n/2}}\int_{u}^{+\infty}\sigma_{n}(\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}^{*}})\mathbb{E}\left(|\det(X^{\prime\prime}(0))|\mathbb{I}_{\\{X^{\prime\prime}(0)\preceq
0\\}}\mid X(0)=x,X^{\prime}(0)=0\right)\varphi(x)\,dx,$

where $X^{\prime\prime}(0)\preceq 0$ means that the matrix
$X^{\prime\prime}(0)$ is semi definite negative, $p_{X(t),X^{\prime}(t)}(x,0)$
is the value of the joint density function of the random vector
$(X(t),X^{\prime}(t))$ at the point $(x,0)$, and $\overline{r}^{*}$ is the
value of $\overline{r}$ given by (11) when $X(t)=x$. We use the stationary
property of the field here and the fact that $X(t)$ and $X^{\prime}(t)$ are
two independent Gaussian vectors.

Using the following result (see Azaïs and Delmas [3]):

$\mathbb{E}\left(|\det(X^{\prime\prime}(0))|\mathbb{I}_{\\{X^{\prime\prime}(0)\preceq
0\\}}\mid
X(0)=x,X^{\prime}(0)=0\right)=x^{n}+O\left(x^{n-2}\right)\;\textnormal{as}\;x\rightarrow\infty,$

and hypothesis (12), we have, since we are in $A_{1}^{c}$:

$\begin{array}[]{rl}E=&\displaystyle\frac{1}{(2\pi)^{n/2}}\int_{u}^{u+1}x^{n}\varphi(x)C\left[2\frac{x-u}{u-u^{\alpha}}\right]^{(n-d)/2}dx+o\left(u^{d-1}\varphi(u)\right)\\\
=&\displaystyle\frac{Cu^{(n+d)/2}}{2^{d/2}\pi^{n/2}}\int_{u}^{u+1}\varphi(x)(x-u)^{(n-d)/2}dx+o\left(u^{d-1}\varphi(u)\right).\\\
\end{array}$

By the change of variable $x=u+y/u$,

$\begin{array}[]{rl}E=&\displaystyle\frac{C}{2^{d/2}\pi^{n/2}}u^{d-1}\varphi(u)\int_{0}^{u}\exp\left(-y-\frac{y^{2}}{2u^{2}}\right)y^{(n-d)/2}dy+o\left(u^{d-1}\varphi(u)\right)\\\
=&\displaystyle
u^{d-1}\varphi(u)\left(\frac{C}{2^{d/2}\pi^{n/2}}\Gamma(1+(n-d)/2)+o(1)\right).\end{array}$

We then obtain the upper bound as above.

For the lower bound, we see that

$\begin{array}[]{rl}&\displaystyle\mathbb{P}\left(\forall i=1\ldots
m:\;M_{S_{i}}\geq u\right)\\\
\geq&o\left(u^{-1}\varphi(u)\right)+\mathbb{P}\left(\exists
t\in\overset{\circ}{B}:\;X(.)\textnormal{ \;has\; a\; local\; maximum\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\underline{r}}\right).\\\
\end{array}$

Set

$\mathcal{M}^{\underline{r}}=\textnormal{card}\left\\{t\in\overset{\circ}{B}:\;X(.)\textnormal{
\;has\; a\; local\; maximum\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\underline{r}}\right\\}.$

It is proven in [13] or [7] that

$0\leq\mathbb{E}(\mathcal{M}^{\underline{r}})-\mathbb{P}(\mathcal{M}^{\underline{r}}\geq
1)\leq\mathbb{E}(\mathcal{M}^{\underline{r}}(\mathcal{M}^{\underline{r}}-1))/2\leq\mathbb{E}(\mathcal{M}_{u}(\mathcal{M}_{u}-1))/2=o\left(u^{-1}\varphi(u)\right),$

where

$\mathcal{M}_{u}=\textnormal{card}\left\\{t\in\overset{\circ}{B}:\;X(.)\textnormal{
\;has\; a\; local\; maximum\; at}\;t,\;X(t)>u\right\\}.$

Then

$\mathbb{P}\left(\min_{i}\\{M_{S_{i}}\\}\geq u\right)\geq
o\left(u^{-1}\varphi(u)\right)+\mathbb{E}(\mathcal{M}^{\underline{r}}).$

Here, using the Rice formula again and by the same arguments, we obtain the
same equivalent formula for both the upper and lower bounds. The result then
follows. ∎

The main idea is to use the inclusion-exclusion principle to compute the
probability of the union of events $\\{M_{S_{i}}>u\\}$ through Lemma 4 that
gives probability of the intersection of some of them. Let us give a simple
introductory example. Suppose that $S=S_{1}\cup S_{2}$ with $S_{1}$ and
$S_{2}$ satisfy the SFH as in Definition 2. Suppose, in addition, that the
condition (12) is met, i.e.,

$\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon})=\left(C+o(1)\right)\epsilon^{2}.$

Then, using Lemma 4, we have an expansion of $\mathbb{P}\left(M_{S_{1}}\geq
u,\,M_{S_{2}}\geq u\right)$ and, by consequence, an expansion of
$\mathbb{P}(M_{S}\geq u)$ with an error of $o(u^{-1}\varphi(u))$.

However, in general, we need to decompose $S$ into three or even four sets.
The next lemma is the basis of our method. It shows that the Steiner formula
heuristic property (SFH) is heredity in the sense that: if we start from some
subsets in ${\mathbb{R}}^{2}$ satisfying the SFH, then under some conditions,
the union of these subsets also satisfies the SFH. Therefore, to prove the
main theorem, we just prove that the considered parameter set can be expressed
as the union of the subsets satisfying the SFH.

######  Lemma 5.

Let $S_{1},\;S_{2},\;S_{3}$ and $S_{4}$ be four compact subsets in
${\mathbb{R}}^{2}$ such that:

  * 1.)

For every $i=1,2,3,4$, $S_{i}$ satisfies the SFH .

  * 2.)

$S_{1}\cup S_{2},\;S_{2}\cup S_{3},\;S_{3}\cup S_{4},\;$ and $S_{4}\cup S_{1}$
satisfy the SFH.

  * 3.)

$S_{2}\cap S_{4}=\emptyset$ and $S_{1}\cap S_{3}\cap S_{4}=\emptyset$.

  * 4.)

As $\epsilon$ tends to $0$, there exist two positive constants $C_{13}$ and
$C_{123}$ such that

$\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{3}^{+\epsilon})=\left(C_{13}+o(1)\right)\epsilon^{2}\;\;\textnormal{and}\;\;\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon}\cap
S_{3}^{+\epsilon})=\left(C_{123}+o(1)\right)\epsilon^{2}.$ (14)

Then $S=S_{1}\cup S_{2}\cup S_{3}\cup S_{4}$ also satisfies the SFH and:

$\begin{array}[]{rl}-\indent L_{1}(S)=&\displaystyle L_{1}(S_{1}\cup
S_{2})+L_{1}(S_{2}\cup S_{3})+L_{1}(S_{3}\cup S_{4})+L_{1}(S_{4}\cup
S_{1})-\sum_{i=1}^{4}L_{1}(S_{i}),\\\ -\indent L_{0}(S)=&\displaystyle
L_{0}(S_{1}\cup S_{2})+L_{0}(S_{2}\cup S_{3})+L_{0}(S_{3}\cup
S_{4})+L_{0}(S_{4}\cup
S_{1})-\sum_{i=1}^{4}L_{0}(S_{i})+\frac{C_{123}-C_{13}}{\pi}.\\\ \end{array}$

Note that in many cases, Lemma 5 will be used with $S_{4}=\emptyset$ and will
take a simpler form.

###### Proof.

In order to prove that $S$ satisfies the SFH, we need to shows the
correspondence between the tube formula of $S$ as in (6) and the asymptotic
expansion for the tail of the distribution as in (7).

$\bullet$ First, we consider the tube formula of $S$. We prove the following
equality for the volume of $S^{+\epsilon}$ for a sufficiently small
$\epsilon$:

$\displaystyle A:=$
$\displaystyle\sigma_{2}(S^{+\epsilon})=B:=\sigma_{2}((S_{1}\cup
S_{2})^{+\epsilon})+\sigma_{2}((S_{2}\cup
S_{3})^{+\epsilon})+\sigma_{2}((S_{3}\cup
S_{4})^{+\epsilon})+\sigma_{2}((S_{4}\cup S_{1})^{+\epsilon})$
$\displaystyle-\sigma_{2}(S_{1}^{+\epsilon})-\sigma_{2}(S_{2}^{+\epsilon})-\sigma_{2}(S_{3}^{+\epsilon})-\sigma_{2}(S_{4}^{+\epsilon})+\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon}\cap S_{3}^{+\epsilon})-\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{3}^{+\epsilon}).$ (15)

Concerning $A$, we can observe that $A=\sigma_{2}(S_{1}^{+\epsilon}\cup
S_{2}^{+\epsilon}\cup S_{3}^{+\epsilon}\cup S_{4}^{+\epsilon})$ and use the
inclusion-exclusion principle to obtain a full expansion. Doing the same on
$B$ we see that the following quantity is missing:

$-\\{2,4\\}+\\{1,2,4\\}+\\{2,3,4\\}+\\{1,3,4\\}-\\{1,2,3,4\\},$

where, for example, $\\{2,4\\}=\sigma_{2}(S_{2}^{+\epsilon}\cap
S_{4}^{+\epsilon})$. Our hypotheses shows that these quantities vanish as soon
as $\epsilon$ is small enough. This proves (15) and implies that:

$\sigma_{2}(S^{+\epsilon})=\sigma_{2}(S)+\epsilon
L_{1}(S)+\pi\epsilon^{2}L_{0}(S)+o(\epsilon^{2}),$

where

$\begin{array}[]{rl}-\indent L_{1}(S)=&\displaystyle L_{1}(S_{1}\cup
S_{2})+L_{1}(S_{2}\cup S_{3})+L_{1}(S_{3}\cup S_{4})+L_{1}(S_{4}\cup
S_{1})-\sum_{i=1}^{4}L_{1}(S_{i}),\\\ -\indent L_{0}(S)=&\displaystyle
L_{0}(S_{1}\cup S_{2})+L_{0}(S_{2}\cup S_{3})+L_{0}(S_{3}\cup
S_{4})+L_{0}(S_{4}\cup
S_{1})-\sum_{i=1}^{4}L_{0}(S_{i})+\frac{C_{123}-C_{13}}{\pi}.\\\ \end{array}$

$\bullet$ For the excursion probability on $S$, using the inclusion-exclusion
principle once again,

$\begin{array}[]{rl}\mathbb{P}(M_{S}\geq u)=&\mathbb{P}(M_{S_{1}\cup S_{2}\cup
S_{3}\cup S_{4}}\geq u)\\\
=&\displaystyle\sum_{i=1}^{4}\mathbb{P}(M_{S_{i}}\geq u)-\sum_{1\leq i<j\leq
4}\mathbb{P}(M_{S_{i}}\geq u,\,M_{S_{j}}\geq u)\\\ &\displaystyle+\sum_{1\leq
i<j<k\leq 4}\mathbb{P}(M_{S_{i}}\geq u,\,M_{S_{j}}\geq u,\,M_{S_{k}}\geq
u)-\mathbb{P}(M_{S_{i}}\geq u,\,\forall i=1,2,3,4).\end{array}$

On the basis of Lemma 2, it is easy to see that the events $\\{M_{S_{2}}\geq
u,\,M_{S_{4}}\geq u\\}$ and $\\{M_{S_{1}}\geq u,\,M_{S_{3}}\geq
u,\,M_{S_{4}}\geq u\\}$ have negligible probabilities ($o(u^{-1}\varphi(u))$),
yielding:

$\displaystyle\mathbb{P}(M_{S}\geq u)=$
$\displaystyle\sum_{i=1}^{4}\mathbb{P}(M_{S_{i}}\geq
u)-\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u)-\mathbb{P}(M_{S_{2}}\geq
u,\,M_{S_{3}}\geq u)$ $\displaystyle\displaystyle-\mathbb{P}(M_{S_{3}}\geq
u,\,M_{S_{4}}\geq u)-\mathbb{P}(M_{S_{4}}\geq u,\,M_{S_{1}}\geq
u)-\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{3}}\geq u)$
$\displaystyle\displaystyle+\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq
u,\,M_{S_{3}}\geq u)+o\left(u^{-1}\varphi(u)\right)$ $\displaystyle=$
$\displaystyle\displaystyle\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq
u)+\mathbb{P}(M_{S_{2}}\geq u,\,M_{S_{3}}\geq u)+\mathbb{P}(M_{S_{3}}\geq
u,\,M_{S_{4}}\geq u)$ $\displaystyle+\mathbb{P}(M_{S_{4}}\geq
u,\,M_{S_{1}}\geq u)-\displaystyle\sum_{i=1}^{4}\mathbb{P}(M_{S_{i}}\geq
u)-\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{3}}\geq u)$
$\displaystyle\displaystyle+\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq
u,\,M_{S_{3}}\geq u)+o\left(u^{-1}\varphi(u)\right).$

Now, using the SFH property in the first and second conditions and applying
Lemma 4 for two probabilities $\mathbb{P}(M_{S_{1}}\geq u,\;M_{S_{3}}\geq u)$
and $\mathbb{P}(M_{S_{1}}\geq u,\;M_{S_{2}}\geq u,\;M_{S_{3}}\geq u)$, we can
deduce that:

$\mathbb{P}(M_{S}\geq
u)=L_{0}(S)\overline{\Phi}(u)+L_{1}(S)\frac{\varphi(u)}{2\sqrt{2\pi}}+\sigma_{2}(S)\frac{u\varphi(u)}{2\pi}+o\left(u^{-1}\varphi(u)\right),$

where the constants $L_{0}(S)$ and $L_{1}(S)$ are defined as in the statement.

Since a correspondence exists between the two formulas obtained, we have
proved the SFH property of $S$. ∎

An introductory example to understand the method

To introduce our method, we consider the case of the simplest non-convex
polygon shown in Figure 4. Note that in this case, we have exactly one concave
binary point with concave angle $\beta$.

$S_{1}$$S_{2}$$S_{3}$ Figure 4: Non-convex polygon with concave binary
irregular point.

$S$ is decomposed into three polygons $S_{1},S_{2}$ and $S_{3}$ with a zero
measure intersection, as indicated in Figure 4. These polygons are convex so
they satisfy the SFH as well as $S_{1}\cup S_{2}$ and $S_{2}\cup S_{3}$.

To apply Lemma 5, with $S_{4}=\emptyset$, it remains to compute the areas of
$(S_{1}^{+\epsilon}\cap S_{3}^{+\epsilon})$ and $(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon}\cap S_{3}^{+\epsilon})$. Elementary geometry shows that
$(S_{1}^{+\epsilon}\cap S_{3}^{+\epsilon})$ consists of two sections of a disc
with angle $(\pi-\beta)$ and two quadrilaterals of area
$\epsilon^{2}\tan(\beta/2)$ each, whereas in $(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon}\cap S_{3}^{+\epsilon})$, one quadrilateral is replaced by a
section of a disc of angle $\beta$ (see Figure 5).

$S_{1}$$S_{2}$$S_{3}$$\beta$ Figure 5: Intersection of $\epsilon$-neighborhood
sets.

Thus,

$\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{3}^{+\epsilon})=\left[(\pi-\beta)+2\tan\frac{\beta}{2}\right]\epsilon^{2},$
$\sigma_{2}(S_{1}^{+\epsilon}\cap S_{2}^{+\epsilon}\cap
S_{3}^{+\epsilon})=\left[(\pi-\beta)+\frac{\beta}{2}+\tan\frac{\beta}{2}\right]\epsilon^{2}.$

Then using (14), we can define the constants $C_{123}$ and $C_{13}$, and
compute that:

$C_{123}-C_{13}=\frac{\beta}{2}-\tan\frac{\beta}{2}.$

This quantity measures the non convexity of the concave binary point. Since
the $L_{0}-$ constants of $S_{2},\,S_{1}\cup S_{2}$ and $S_{2}\cup S_{3}$ are
both equal to $1$ in this case, an application of Lemma 5 shows that the
coefficient of $\overline{\Phi}(u)$ in the expansion of the tail of $M_{S}$ is
now $\displaystyle 1+\frac{\beta/2-\tan(\beta/2)}{\pi}$.

### Proof of the main theorem

Using the above lemmas, we are able now to prove the main theorem. If the
parameter set $S$ consists of several disjoint connected components, then by
using Lemma 2 the tail distribution of the maxima defined on these components
can be added with an error of $o\left(u^{-1}\varphi(u)\right)$, and the right-
hand side of (9) is also additive, we can assume in the sequel that $S$ is
connected.

Our proof is based on induction on the number of concave points of $S$. It
should be recalled here that there are three types of concave points: binary,
angle and ternary (see Definition 1).

$\bullet$ Suppose that $S$ has no concave point. Two cases will be considered:
depending on whether $\overset{\circ}{S}$ is empty or not.

If $\overset{\circ}{S}$ is empty, then $S$ consists of only one isolated edge.
Using the parametrization of the unique edge, we see that $M_{S}$ is just the
maximum of a smooth random process (with parameter of dimension 1). By using
the Rice method for the number of up-crossings, Piterbarg [12] or Rychlik [15]
showed that $S$ satisfies the SFH.

If $\overset{\circ}{S}$ is not empty, then $S$ cannot have isolated edges and
$S$ has a positive reach in the sense of Federer [9] because the curvature on
the compact edges is bounded. Therefore,

$\sigma_{2}(S^{+\epsilon})=\chi(S)\pi\epsilon^{2}+\textnormal{OMC}(S)\epsilon+\sigma_{2}(S),$
(16)

for small enough $\epsilon$. On the other hand, on the basis of Theorem 8.12
of Azaïs and Wschebor [6], it can deduced that the SFH applies (see Appendix
for details).

$\bullet$ Suppose $S$ has at least one concave point. We will decompose $S$
into the subsets whose number of concave points is strictly smaller than that
of $S$. The induction hypothesis then ensures that these subsets will satisfy
the SFH. We can therefore use Lemma 5 to "glue" them together and to show that
the union $S$ also satisfies the SFH. In fact, our method is based on the
"destruction" of concave points as in the introductory example. More
precisely, there are four possibilities regarding $P$:

  * 1.

Concave binary point on the exterior boundary of $S$. We decompose $S$ into
three compact subsets $S_{1},\,S_{2}$ and $S_{3}$ as in Figure 4. By
decomposition we mean an essential partition, i.e. that $S=S_{1}\cup S_{2}\cup
S_{3}$ and that $S_{1}$, $S_{2}$ and $S_{3}$ have disjoint interiors. The
decomposition is as follows: at $P$ we prolong the two tangents inward and
construct two $\mathcal{C}^{2}$ paths that avoid all the holes and end at one
regular point on the exterior boundary such that these two paths have no
intersection other than point $P$. This is always possible because the
connected open set $\overset{\circ}{S}$ is path connected. We then define
$S_{1}$, $S_{2}$ and $S_{3}$ as in Figure 6. To apply Lemma 5 with
$S_{4}=\emptyset$, we need to verify all the required conditions.

To compute $\sigma_{2}(S_{1}^{+\epsilon}\cap S_{3}^{+\epsilon})$ and
$\sigma_{2}(S_{1}^{+\epsilon}\cap S_{2}^{+\epsilon}\cap S_{3}^{+\epsilon})$,
we can locally replace the edges starting from $P$ by their tangents with an
error of $O(\epsilon^{3})$ and thus $o(\epsilon^{2})$. In that case the
computation of these areas is exactly the same as in the introductory example.

$S_{1}$$S_{3}$$S_{2}$$P$ Figure 6: Decomposition of $S$ at a binary concave
point on the exterior boundary.

On the other hand, let us consider the number of concave points of
$S_{1},\,S_{2},\,S_{3},\,S_{1}\cup S_{2}$ and $S_{2}\cup S_{3}$. Due to the
way we have constructed these sets, we have destroyed the concavity of $P$ in
the sense that with respect to these subsets, $P$ becomes a convex binary
point or a regular one. We can see that an irregular point of these subsets is
also an irregular set of $S$ unless it is $P$ or one of the other endpoints on
the exterior boundary of the two prolonged paths. However, we have proved that
$P$ is no longer a concave point with respect to these subsets. Moreover,
since the other endpoint is chosen to be a regular point on the exterior
boundary of $S$ and since the support cones at this point with respect to
these subsets (as defined in (2)) are included in the one with respect to $S$,
these support cones are convex; then this endpoint is therefore not a concave
point. Hence, a concave point of the constructed subsets is also a concave
point of $S$ and $P$ is a concave point only with respect to $S$. These
subsets therefore have a number of concave points equal at most to the one of
$S$ minus 1. They therefore satisfy the SFH by induction.

Since all the required conditions are met, on the basis of Lemma 5, $S$
satisfies the SFH with the desired constants.

  * 2.

Concave binary point on the boundary of a hole inside $S$. Drawing two
prolonged paths as above, we simply decompose $S$ into two subsets $S_{2}$ and
$S^{\prime}$. We then divide $S^{\prime}$ into three subsets as follows: we
also choose two regular points on the boundary of the hole and two
corresponding regular points on the exterior boundary of $S$ and construct two
smooth curves so that they connect one regular point on the boundary of the
hole with the corresponding one on the exterior boundary, and do not intersect
themselves or two curves from the irregular point or additional holes. Again,
this is possible because $\overset{\circ}{S}$ is path-connected.
$S_{1},S_{2},S_{3},S_{4}$ are then constructed as in Figure 7.

$S_{1}$$S_{3}$$S_{2}$$S_{4}$$P$ Figure 7: Decomposition at a concave point on
the interior boundary.

We will use Lemma 5 to prove that $S$ satisfies the SFH. Indeed, the
computation of the areas $\sigma_{2}(S_{1}^{+\epsilon}\cap S_{3}^{+\epsilon})$
and $\sigma_{2}(S_{1}^{+\epsilon}\cap S_{2}^{+\epsilon}\cap
S_{3}^{+\epsilon})$ and the arguments to show that the subsets in the first
and second conditions of Lemma 5 satisfy the SFH remain the same as in the
case above. The third condition about the empty intersections is easily
verified from the construction. We can therefore deduce the SFH property of
$S$.

  * 3.

Concave ternary point. We define $S_{1}$ as the isolated edge containing $P$,
$S_{2}=\\{P\\}$ and $S_{3}$ as the closure of the complement of $S_{1}$ (see
Figure 8).

$S_{3}$$S_{2}$$S_{1}$ Figure 8: Decomposition at a concave ternary point.

By the same arguments as in the above cases, we can check that all the
required conditions in Lemma 5 are met. $S$ then satisfies the SFH.

  * 4.

Angle point. We do the same as in the concave ternary point case (see Figure
9).

$S_{1}$$S_{3}$$S_{2}$ Figure 9: At an angle point.

We have proved that $S$ satisfies the SFH. In order to establish (9), we need
to compute the constant $L_{0}(S)$.

Firstly, we have seen that when $S$ contains no concave points:

$L_{0}(S)=\chi(S)$

Secondly, Lemma 5 shows that when we "glue" $S_{1},S_{2},S_{3}$ and $S_{4}$
together, each concave points causes a distortion to the additivity which is
equal to:

$\frac{\beta_{i}/2-\tan(\beta_{i}/2)}{\pi}.$

Therefore we eventually have:

$L_{0}(S)=\chi(S)+\sum_{i=1}^{k}\frac{\beta_{i}/2-\tan(\beta_{i}/2)}{\pi}$

and we are done.

##  3 Examples

In this section, we give some examples that are direct applications or direct
generalizations of Theorem 1. All these results are new and rather unexpected.
In most two-dimensional cases, the parameter set $S$ has the
piecewise-$\mathcal{C}^{2}$ boundary as in Definition 1, satisfying the SFH.
Therefore, in order to derive the asymptotic formula for the tail of the
maximum, we just use elementary geometry to compute the area of the tube and
consider the corresponding coefficients.

###  3.1 The angle

Let $S$ be the angle as in Figure 1. On the basis of Theorem 1, $S$ satisfies
the SFH. Using elementary geometry, we can compute that for small enough
$\epsilon$,

$\lambda_{2}(S^{+\epsilon})=2\left(\lambda_{1}(S_{1})+\lambda_{1}(S_{2})\right)\epsilon+\left(\pi+\beta/2-\tan(\beta/2)\right)\epsilon^{2},$

where $\lambda_{1}(.)$ is simply the length of the segment. We then have:

$\mathbb{P}(M_{S}\geq
u)=\left(1+\frac{\beta/2-\tan(\beta/2)}{\pi}\right)\overline{\Phi}(u)+\frac{\lambda_{1}(S_{1})+\lambda_{1}(S_{2})}{\sqrt{2\pi}}\varphi(u)+o\left(u^{-1}\varphi(u)\right).$

###  3.2 The multi-angle

This is an extension of the angle case. Let $S$ be a self-avoiding continuous
curve that is the union of $k+1$ curves with concave angles
$\\{\beta_{1},\ldots,\beta_{k}\\}$. In this case, the induction process in the
proof of the main theorem can be seen as the induction on the number of
segments, i.e., we add one more segment into the union each time. Here again
using elementary geometry, for small enough $\epsilon$:

$\lambda_{2}(S^{+\epsilon})=2\lambda_{1}(S)\epsilon+\left(\pi+\underset{i=1}{\overset{k}{\sum}}\left(\beta_{i}/2-\tan(\beta_{i}/2)\right)\right)\epsilon^{2},$

where $\lambda_{1}(S)$ is the length of the curve that is equal to the sum of
the lengths of the segments. Hence we immediately have the asymptotic formula:

$\mathbb{P}(M_{S}\geq
u)=\left(1+\frac{\underset{i=1}{\overset{k}{\sum}}\left(\beta_{i}/2-\tan(\beta_{i}/2)\right)}{\pi}\right)\overline{\Phi}(u)+\frac{\sigma_{1}(S)}{\sqrt{2\pi}}\varphi(u)+o\left(u^{-1}\varphi(u)\right).$

###  3.3 The empty square

Let $S$ be the empty square, i.e. the boundary of a square in
${\mathbb{R}}^{2}$. This case is very similar to the multi-angle case, but the
curves are no longer self-avoiding. In this case, the induction process on the
number of segments still works. We can therefore deduce that $S$ satisfies the
SFH. The elementary geometry shows that for small enough $\epsilon$:

$\lambda_{2}(S^{+\epsilon})=2\lambda_{1}(S)\epsilon+(\pi-4)\epsilon^{2};$

then, as a consequence,

$\mathbb{P}(M_{S}\geq
u)=\frac{\pi-4}{\pi}\overline{\Phi}(u)+\frac{\sigma_{1}(S)}{\sqrt{2\pi}}\varphi(u)+o\left(u^{-1}\varphi(u)\right).$

###  3.4 The full square with whiskers

We consider "the square with whiskers" as in Figure 3. In this case, $S$ has
two concave ternary points. From the main theorem, we know that $S$ satisfies
the SFH. Therefore, since we can compute the area of the tube as

$\lambda_{2}(S^{+\epsilon})=\lambda_{2}(S)+\textnormal{OMC}(S)\epsilon+(2\pi-4)\epsilon^{2},$

for small enough $\epsilon$ ; we have the expansion:

$\mathbb{P}(M_{S}\geq
u)=\frac{2\pi-4}{\pi}\overline{\Phi}(u)+\frac{\textnormal{OMC}(S)}{2\sqrt{2\pi}}\varphi(u)+\frac{\sigma_{2}(S)}{2\pi}u\varphi(u)+o\left(u^{-1}\varphi(u)\right).$

###  3.5 An irregular locally convex set

In this subsection, we consider a strange and interesting example. We consider
$S$ as the union of two tangent curves as in Figure 10.

Figure 10: Two tangent edges.

Suppose that $S_{1}$ is a section of a circle of radius $R$ and $S_{2}$ is a
segment tangent to that circle. For small enough $\epsilon$ , the area of the
intersection between two tubes is:

$\frac{\pi}{2}\epsilon^{2}+\frac{(R+\epsilon)^{2}}{2}\arcsin\frac{2\sqrt{R\epsilon}}{R+\epsilon}-(R-\epsilon)\sqrt{R\epsilon}=\frac{\pi}{2}\epsilon^{2}+\frac{8}{3}\sqrt{R}\epsilon^{3/2}+O(\epsilon^{5/2}).$

In the above equation, we used the fact that for small enough $x$ ,

$\arcsin x=x+\frac{1}{2}\frac{x^{3}}{3}+\frac{1\cdot 3}{2\cdot
4}\frac{x^{5}}{5}+\ldots\;.$

It is clear that the order of the area of the intersection is not of $2$ as in
Condition (14), so we cannot apply Lemma 5 directly. In this example, the area
of the intersection contains two order: $2$ and $3/2$.

The asymptotic formulas for the tail of the maximum of the random fields
defined on $S_{1}$ and $S_{2}$ are well understood since they are one-
dimensional cases. Then by the inclusion-exclusion principle,

$\mathbb{P}(M_{S}\geq u)=\mathbb{P}(M_{S_{1}}\geq u)+\mathbb{P}(M_{S_{2}}\geq
u)-\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u\right).$

To compute $\mathbb{P}(M_{S}\geq u)$, we need to derive an expansion for
$\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u\right)$.

By carefully examining in the proof of Lemma 4, we can choose $\alpha$ such
that the difference between the upper and the lower bounds of the probability
$\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u)$ is negligible. Indeed, as in
Lemma 4, after substituting the area of the intersection of the tubes into the
expectation, we obtain the upper bound:

$\begin{array}[]{rl}&\displaystyle\frac{1}{2\pi}\int_{u}^{u+1}\left(x^{2}+O(1)\right)\varphi(x)\left[\frac{\pi}{2}\frac{2(x-u)}{u-u^{\alpha}}+\frac{8}{3}\sqrt{R}\left(2\frac{x-u}{u-u^{\alpha}}\right)^{3/4}+O\left(\left(2\frac{x-u}{u-u^{\alpha}}\right)^{5/4}\right)\right]dx+o(u^{-1}\varphi(u))\\\
=&\displaystyle\frac{1}{2\pi}\int_{u}^{u+1}x^{2}\varphi(x)\left[\frac{\pi}{2}\frac{2(x-u)}{u-u^{\alpha}}+\frac{8}{3}\sqrt{R}\left(2\frac{x-u}{u-u^{\alpha}}\right)^{3/4}\right]dx+o(u^{-1}\varphi(u)),\end{array}$

and, similarly, the lower one:

$\frac{1}{2\pi}\int_{u}^{u+1}x^{2}\varphi(x)\left[\frac{\pi}{2}\frac{2(x-u)}{x+u^{\alpha}}+\frac{8}{3}\sqrt{R}\left(2\frac{x-u}{x+u^{\alpha}}\right)^{3/4}\right]dx+o(u^{-1}\varphi(u)).$

To control the difference between them, we firstly consider the term:

$\begin{array}[]{rl}D_{1}=&\displaystyle\int_{u}^{u+1}x^{2}\varphi(x)\left[\left(\frac{x-u}{u-u^{\alpha}}\right)^{3/4}-\left(\frac{x-u}{x+u^{\alpha}}\right)^{3/4}\right]dx\\\
=&\displaystyle\int_{u}^{u+1}x^{2}\varphi(x)(x-u)^{3/4}\frac{(x+u^{\alpha})^{3/4}-(u-u^{\alpha})^{3/4}}{(x+u^{\alpha})^{3/4}(u-u^{alpha})^{3/4}}dx.\end{array}$

Since

$a^{3/4}-b^{3/4}=\frac{a^{3}-b^{3}}{\left(a^{3/4}+b^{3/4}\right)\left(a^{3/2}+b^{3/2}\right)}=\frac{(a-b)(a^{2}+ab+b^{2})}{\left(a^{3/4}+b^{3/4}\right)\left(a^{3/2}+b^{3/2}\right)}$

for $a=x+u^{\alpha}$ and $b=u-u^{\alpha}$, and we can replace $x,a,b$ by $u$,
then:

$\begin{array}[]{rl}D_{1}\leq&\displaystyle(const)\int_{u}^{u+1}u^{2}\varphi(x)(x-u)^{3/4}\frac{\left(x-u+2u^{\alpha}\right)u^{2}}{u^{3+3/4}}dx\\\
\leq&\displaystyle(const)u^{\alpha+1/4}\int_{u}^{u+1}\varphi(x)(x-u)^{3/4}dx.\end{array}$

Here using the change of variable $x=u+y/u$ once again,

$D_{1}\leq(const)\frac{u^{\alpha+1/4}}{u^{1+3/4}}\varphi(u)\int_{0}^{u}\exp\left(-y-\frac{y^{2}}{2u^{2}}\right)y^{3/4}dy.$

Therefore, if we choose $\alpha<1/2$ then $D_{1}=o(u^{-1}\varphi(u))$. For the
second term :

$\int_{u}^{u+1}x^{2}\varphi(x)\left[\frac{x-u}{u-u^{\alpha}}-\frac{x-u}{x+u^{\alpha}}\right]dx,$

we can use the same arguments. Note that this case is simpler.

In conclusion, we have proved that if $\alpha<1/2$, then the difference
between the upper and the lower bounds of the probability
$\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u)$ is negligible. As in Lemma 4,
we have the following expansion:

$\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq
u)=\frac{8\sqrt{R}}{2^{1/4}3\pi}\Gamma(7/4)u^{-1/2}\varphi(u)+\frac{\overline{\Phi}(u)}{2}+o(u^{-1}\varphi(u)).$

Thus, we have:

######  Proposition 1.

Using the above notation

$\mathbb{P}(M_{S_{1}\cup S_{2}}\geq
u)=\frac{3\overline{\Phi}(u)}{2}-\frac{8\sqrt{R}}{2^{1/4}3\pi}\Gamma(7/4)u^{-1/2}\varphi(u)+\frac{\sigma_{1}(S_{1})+\sigma_{1}(S_{2})}{\sqrt{2\pi}}\varphi(u)+o\left(u^{-1}\varphi(u)\right).$
(17)

This example is an apparent counter-example to the results of Adler and
Taylor. More precisely, $S$ is clearly a piecewise smooth locally convex
manifold: it is easy to check that at the intersection of the circle and the
straight line, the support cone is limited to one direction and is thus
convex. Thus if the random field $X$ is sufficiently smooth, it seems that
Theorem 14.3.3 of [1] implies the validity of the Euler characteristic
heuristic and Theorem 12.4.2 of [1] gives an expansion of the Euler
characteristic function that should apply. This would be clearly in
contradiction with the term $u^{-1/2}\varphi(u)$ in (17).

In fact, there is no contradiction: Theorem 14.3.3 also demands the manifold
to be regular in the sense of Definition 9.22 of [1] and the present set is
not a cone space in the sense of Definition 8.3.1 of [1]. This shows that the
local convexity itself is not sufficient.

It is surprising to see that in (17), the asymptotic formula contains three
terms corresponding to the powers: $-1$ (in $\overline{\Phi}(u)$), $-1/2$ and
$0$. This is the first time we can see such a combination; in all the well-
known cases before, we only saw a combination of integer powers. We emphasize
that this strange combination comes from the tube formula of the parameter
set.

###  3.6 Some examples in dimension 3

Lemmas 4 and 5 can be applied in higher dimensions. However, in dimension 3,
for example, they do not make it possible to obtain a full Taylor expansion
that would contain, in general, four terms.

In fact, the coefficient of $\overline{\Phi}(u)$ cannot be determined for non-
locally convex sets. We give some examples below.

  * •

$S$ is a dihedral that is the union of two non-coplanar rectangles $S_{1}$ and
$S_{2}$, with a common edge such that the angle of the dihedral is $\alpha$,
see Figure 11.

$\alpha$$S_{1}$$S_{2}$ Figure 11: Example of a dihedral.

Using the inclusion-exclusion principle, we are just concerned with the
probability $\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u\right)$. Using
Lemma 4 in the case where $n=3$ and $d=1$, we obtain the expansion of this
probability with only one term and an error of$o\left(\varphi(u)\right)$.
Then,

$\begin{array}[]{rl}\mathbb{P}(M_{S}\geq
u)=&\quad\displaystyle\frac{\sigma_{1}(\partial S_{1})+\sigma_{1}(\partial
S_{2})-\sigma_{1}(S_{1}\cap
S_{2})((\pi+\alpha)/2+\cot(\alpha/2))/\pi}{2\sqrt{2\pi}}\varphi(u)\\\
&\displaystyle+\;\frac{\sigma_{2}(S_{1})+\sigma_{2}(S_{2})}{2\pi}u\varphi(u)+o\left(\varphi(u)\right).\end{array}$

  * •

$S$ has the $L-$shape, as in Figure 12.

Figure 12: L-shape.

Then, by decomposing $S$ into three hyper-rectangles $S_{1},\;S_{2}$ and
$S_{3}$ that are indicated by the dotted lines with $S_{3}$ between the two
others, we can apply Lemma 5 with a slight modification that since, in this
cas, $n=3$ and $d=1$, then the asymptotic formulas for
$\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u\right)$ and
$\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u,\,M_{S_{3}}\geq u\right)$
are of order $\varphi(u)$ and the error is $o\left(\varphi(u)\right)$ (from
Lemma 4). We then have an expansion with three terms as follows:

$\mathbb{P}(M_{S}\geq
u)=\frac{\varphi(u)L_{1}(S)}{2\sqrt{2\pi}}+\frac{L_{2}(S)u\varphi(u)}{2\pi}+\frac{L_{3}(S)(u^{2}-1)\varphi(u)}{(2\pi)^{3/2}}+o\left(\varphi(u)\right),$
(18)

where the coefficients $\\{L_{i}(S),\;i=1,\ldots,3\\}$ are given by the
Steiner formula and will be defined at the end of this section.

  * •

In a more complicated case, i.e., non-convex trihedral (see Figure 13).

Figure 13: Example of a non-convex trihedral.

In this case, we have three concave edges in the sense that the angles inside
the trihedral at these edges are strictly greater than $\pi$. We will destroy
this concavity by extending the planes (faces) containing these edges so that
they decompose $S$ into smaller convex subsets $\\{S_{i}\\}$ with disjoint
interiors (see the dotted lines in the figure). Observe that the intersection
between two subsets is of one of four types: empty set, a single point, an
edge or a face. If it is a face, then the union of these two subsets is also
convex. An intersection between three or more subsets is one of three types:
empty set, a single point or an edge. Using the inclusion-exclusion principle,
we need to find the expansion of the probability of the intersection of the
events $M_{S_{i_{k}}}\geq u$ for some $k$. Concerning the intersection of the
$\\{S_{i_{k}}\\}$, we have the following cases:

    * 1.

Empty set. On the basis of Lemma 2, the probability of the intersection of the
events $M_{S_{i_{k}}}\geq u$ is $o(u^{-1}\varphi(u))$.

    * 2.

A single point. By applying Lemma 4 in the case $d=0$, the probability
considered is also $o(u^{-1}\varphi(u))$.

    * 3.

An edge. By applying Lemma 4 in the case $n=3$ and $d=0$, the expansion for
the probability considered is of the order $\varphi(u)$ with the error
$o(\varphi(u))$.

    * 4.

A face. This case just happens when we consider the intersection between two
subsets. Since both of these subsets and their union are convex, the expansion
for the tail distribution of the maxima defined on them is well-known. We can
therefore compute the expansion for the probability considered by the
inclusion-exclusion principle.

We therefore obtain an asymptotic formula for $\mathbb{P}(M_{S}\geq u)$, as in
(18).

In general, by the same arguments and using induction, when $S$ is a polytope,

$\mathbb{P}(M_{S}\geq
u)=\frac{\varphi(u)L_{1}(S)}{2\sqrt{2\pi}}+\frac{L_{2}(S)u\varphi(u)}{2\pi}+\frac{L_{3}(S)(u^{2}-1)\varphi(u)}{(2\pi)^{3/2}}+o\left(\varphi(u)\right),$

where

  * -

$L_{3}(S)$ is the volume of $S$.

  * -

$L_{2}(S)$ is one half of the surface area.

  * -

To compute $L_{1}(S)$, we consider two types of edge: convex and concave. An
internal dihedral angle is associated to each edge $i$. If this angle is less
than or equal to $\pi$, the edge is considered to be convex and the angle is
denoted by $\alpha_{i}$. Let $h$ be the number of such edges. If the angle is
larger than $\pi$ the edge is considered to be "concave" and the angle is
denoted by $\beta_{i}$. Let $k$ be the number of such angles, then:

$L_{1}(S)=\sum_{i=1}^{h}\frac{(\pi-\alpha_{i})}{2\pi}l_{i}+\sum_{i=1}^{k}\frac{\cot(\beta_{i}/2)}{\pi}l_{i},$

where $l_{i}$ is the length of edge $i$.

## Conclusion

In all the examples considered, the Steiner formula for the tube governs the
expansion of the tail of the maximum as if the excursion set was precisely a
unique ball with a random radius. We have not found any counter-example to
that principle and we therefore conjecture that the result is true for a much
wider class of sets than those considered in this paper.

##  4 Appendix

In this appendix, we prove that a compact connected set in ${\mathbb{R}}^{2}$
with piecewise-$\mathcal{C}^{2}$ boundary and without concave irregular points
will satisfy the SFH. This is very similar to the general result of Adler and
Taylor, see Theorem 14.3.3 in [1]. However, these authors just clarified and
specified this theorem in the convex case and we think that there is a need to
provide the following proof.

Firstly, the Steiner formula (16) has already been established. We now
consider the excursion probability. We recall the following definitions

  * •

Let $S_{2}$ be the interior of $S$; $S_{1}$ be the union of the
$\mathcal{C}^{2}$ edges and $S_{0}$ be the union of the convex irregular
points.

  * •

For $t\in S_{j}$, $X^{\prime}_{j}(t)$ and $X^{\prime\prime}_{j}(t)$ are the
first and second derivatives of $X$ along $S_{j}$ respectively;
$X^{\prime}_{j,N}(t)$ denotes the outward normal derivative.

In our case, it is easy to see that:

$\kappa(S)=\underset{t\in S}{\sup}\underset{s\in S,\;s\neq
t}{\sup}\frac{\textnormal{dist}(s-t,C_{t})}{\|s-t\|^{2}}<\infty.$

In order to apply Theorem 8.12 and Corollary 8.13 of Azaïs and Wschebor [6],
we have to check the conditions (A1) to (A5) (see [6, p. 185]). The first
three are regularity conditions that are included in Assumption A. Note that
since the edges are of dimension 1, a direct proof of the Rice formula can be
performed without assuming that they are of class $\mathcal{C}^{3}$ as in
(A1).

  * •

The condition (A4) states that the maximum is attained at a single point. It
can be deduced from the Bulinskaya lemma (Proposition 6.11 in [6]) since for
$s\neq t$, $(X(s),X(t),X^{\prime}(s),X^{\prime}(t))$ has a non-degenerate
distribution.

  * •

The condition (A5) states that there is almost surely no point $t\in S$ such
that $X^{\prime}(t)=0$ and $\det\left(X^{\prime\prime}(t)\right)=0$. It can be
deduced from Proposition 6.5 in [6] applied to the process, $X^{\prime}(t)$,
which is $\mathcal{C}^{2}$.

Since all the required conditions are met, we have:

$\underset{u\rightarrow+\infty}{\liminf}-2u^{-2}\log\big{[}\int_{u}^{\infty}p^{E}(x)dx-\mathbb{P}\\{M_{S}\geq
u\\}\big{]}\geq 1+\underset{t\in
S}{\inf}\frac{1}{\sigma_{t}^{2}+\kappa_{t}^{2}}>1,$ (19)

where

  * •

$p^{E}(x)$ is the approximation of the density of the maximum given by the
Euler characteristic method. More precisely,

$\begin{array}[]{rl}p^{E}(x)=&\displaystyle\sum_{t\in
S_{0}}\mathbb{E}\left(\mathbb{I}_{X^{\prime}_{0}(t)\in\widehat{C}_{t,0}}\mid
X(t)=x\right)\varphi(x)\\\
&+\displaystyle\sum_{j=1}^{2}(-1)^{j}\int_{S_{j}}\mathbb{E}\left(\det\left(X^{\prime\prime}_{j}(t)\right)\mathbb{I}_{X^{\prime}_{j,N}(t)\in\widehat{C}_{t,j}}\mid
X(t)=x,\,X^{\prime}_{j}(t)=0\right)\frac{\varphi(x)}{(2\pi)^{j/2}}dt,\\\
\end{array}$ (20)

where $\widehat{C}_{t,j}$ is the dual cone of the support cone $C_{t}$,

$\widehat{C}_{t,j}=\\{z\in{\mathbb{R}}^{2}:\;\langle z,x\rangle\geq
0,\;\forall\;x\in C_{t}\\}.$

  * •

$\displaystyle\sigma_{t}^{2}=\underset{s\in
S\setminus\\{t\\}}{\sup}\frac{\textnormal{Var}\left(X(s)\mid
X(t),X^{\prime}(t)\right)}{(1-\textnormal{Cov}(X(s),X(t)))^{2}}.$

  * •

$\displaystyle\kappa_{t}=\underset{s\in
S\setminus\\{t\\}}{\sup}\frac{\textnormal{dist}\left(\frac{\partial}{\partial
t}\textnormal{Cov}(X(s),X(t)),C_{t}\right)}{1-\textnormal{Cov}(X(s),X(t))}.$

We compute $p^{E}(x)$ as follows:

  * •

When $j=2$, there is no normal space and $X^{\prime}_{2,N}(t)$ makes no sense.
It is easy to see that (see, for example, Azaïs and Wschebor [6, p. 244])

$\int_{S_{2}}\mathbb{E}\left(\det\left(X^{\prime\prime}_{2}(t)\right)\mid
X(t)=x,\,X^{\prime}_{2}(t)=0\right)dt=\sigma_{2}(S)(x^{2}-1).$

  * •

When $j=0$, $X^{\prime}_{0,N}(t)\\-=X^{\prime}(t)$ and:

$\mathbb{E}\left(\mathbb{I}_{X^{\prime}(t)\in\widehat{C}_{t,0}}\mid
X(t)=x\right)=\frac{\mathcal{A}(\widehat{C}_{t,0})}{2\pi};$

where $\mathcal{A}(\widehat{C}_{t,0})$ is the angle of the cone that is equal
to the discontinuity of the angle of the tangent at the irregular point $t$.

  * •

When $j=1$, we consider a point $t$ on an edge $L$ of the exterior boundary.
Note that in this case, the support cone $C_{t}$ is just a half-plane, so the
event $\\{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}\\}$ can be viewed as
$\\{X^{\prime}_{1,N}(t)\geq 0\\}$.

At the point $t$, the second derivative along the curve can be expressed as:

$X^{\prime\prime}_{1}(t)=X^{\prime\prime}_{T}(t)+C(t)X^{\prime}_{1,N}(t),$

where $X^{\prime\prime}_{T}$ is the second derivative in the tangent direction
and $C(t)$ is the signed curvature at $t$.

It is easy to check that the covariance function of the vector
$(X^{\prime\prime}_{T},X^{\prime}_{1,N},X,X_{1}^{\prime})$ is:

$\left(\begin{array}[]{cccc}\textnormal{Var}(X^{\prime\prime}_{T})&0&-1&0\\\
0&1&0&0\\\ -1&0&1&0\\\ 0&0&0&1\\\ \end{array}\right).$

Therefore, for such an edge $L$,

$\mathbb{E}\left(X^{\prime\prime}_{1}(t)\mathbb{I}_{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}}\mid
X(t)=x,\,X^{\prime}_{1}(t)=0\right)=\mathbb{E}\left(\left(-x+C(t)X^{\prime}_{1,N}(t)\right)\mathbb{I}_{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}}\right)=\frac{-x}{2}+\frac{C(t)}{\sqrt{2\pi}}$

and

$-\int_{L}\mathbb{E}\left(X^{\prime\prime}_{1}(t)\mathbb{I}_{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}}\mid
X(t)=x,\,X_{1}^{\prime}(t)=0\right)\frac{\varphi(x)}{\sqrt{2\pi}}dt=\frac{\sigma_{1}(L)x}{2\sqrt{2\pi}}\varphi(x)-\frac{\varphi(x)}{2\pi}\int_{L}C(t)dt.$

The quantity $-\int_{L}C(t)dt$ can be viewed as the variation of the angle of
the tangent from the beginning to the end of this edge.

Since we complete a whole turn in the positive orientation:

$\sum_{\textnormal{ irregular points of the ext.
boundary}}\mathcal{A}(\widehat{C}_{t})\quad+\sum_{\textnormal{edges of the
ext. boundary}}-\int_{L_{i}}C(t)dt=2\pi.$

For a point $t$ on an edge $L_{i}$ of the interior boundary (holes), the
interpretation of the second derivative changes into:

$X^{\prime\prime}_{1}(t)=X^{\prime\prime}_{T}(t)-C(t)X^{\prime}_{1,N}(t).$

Therefore,

$-\int_{L_{i}}\mathbb{E}\left(X^{\prime\prime}_{1}(t)\mathbb{I}_{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}}\mid
X(t)=x,\,X_{1}^{\prime}(t)=0\right)\frac{\varphi(x)}{\sqrt{2\pi}}dt=\frac{\sigma_{1}(L_{i})x}{2\sqrt{2\pi}}\varphi(x)+\frac{\varphi(x)}{2\pi}\int_{L_{i}}C(t)dt.$

For the boundary of a hole inside $S$,

$\sum_{\textnormal{irregular
points}}\mathcal{A}(\widehat{C}_{t})+\sum_{\textnormal{edges}}\int_{L_{i}}C(t)dt=-2\pi.$

In conclusion, substituting into (20),

$p^{E}(x)=\chi(S)\varphi(x)+\frac{\sigma_{1}(\partial
S)}{2\sqrt{2\pi}}x\varphi(x)+\frac{\sigma_{2}(S)}{2\pi}(x^{2}-1)\varphi(x),$

since the Euler characteristic $\chi(S)$ is equal to $1$ (the number of
connected components) minus the number of the holes.

Integrating $p^{E}(x)$, we obtain the asymptotic expansion:

$\mathbb{P}(M_{S}\geq u)=\chi(S)\overline{\Phi}(u)+\frac{\sigma_{1}(\partial
S)}{2\sqrt{2\pi}}\varphi(u)+\frac{\sigma_{2}(S)}{2\pi}u\varphi(u)+Rest,$

where $Rest$ is super-exponentially smaller in the sense of (19). This implies
a correspondence between the asymptotic expansion and the Steiner formula.

## References

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<EMAIL_ADDRESS>
<EMAIL_ADDRESS>


# Jacob’s ladders and the nonlocal interaction of the function $Z^{2}(t)$ with
the function $\tilde{Z}^{2}(t)$ on the distance $\sim(1-c)\pi(t)$ for the
collections of disconnected sets

Jan Moser  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.

It is shown in this paper that there is a fine correlation of the fourth order
between the functions $Z^{2}[\varphi_{1}(t)]$ and $\tilde{Z}^{2}(t)$,
respectively. This correlation is with respect to two collections of
disconnected sets. Corresponding new asymptotic formulae cannot be obtained
within known theories of Balasubramanian, Heath-Brown and Ivic.

###### Key words and phrases:

Riemann zeta-function

##  1\. The result

In this paper we obtain some new properties of the signal

$Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)$

generated by the Riemann zeta-function, where

(1.1)
$\vartheta(t)=-\frac{t}{2}\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}\left(\frac{1}{t}\right).$

Let (see [3])

(1.2) $\begin{split}&G_{3}(x)=G_{3}(x;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu}\leq
T+U}\\{t:\ g_{2\nu}(-x)\leq t\leq g_{2\nu}(x)\\},\ 0<x\leq\frac{\pi}{2},\\\
&G_{4}(y)=G_{4}(y;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu+1}\leq T+U}\\{t:\
g_{2\nu+1}(-y)\leq t\leq g_{2\nu+1}(y)\\},\ 0<y\leq\frac{\pi}{2},\end{split}$
(1.3) $U=T^{5/12}\ln^{3}T,$

and the collection of sequences $\\{g_{\nu}(\tau)\\}$, $\tau\in[-\pi,\pi]$,
$\nu=1,2,\dots$ is defined by the equation (see [2], [3], (6))

$\vartheta_{1}[g_{\nu}(\tau)]=\frac{\pi}{2}\nu+\frac{\tau}{2};\
g_{\nu}(0)=g_{\nu},$

where (comp. (1.1)

$\vartheta_{1}(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}.$

Let

(1.4) $G_{3}(x)=\varphi_{1}[\mathring{G}_{3}(x)],\
G_{4}(y)=\varphi_{1}[\mathring{G}_{4}(y)],$

where $y=\varphi_{1}(T),\ T\geq T_{0}[\varphi_{1}]$ is the Jacob’s ladder. The
following theorem holds true.

######  Theorem.

(1.5)
$\begin{split}&\int_{\mathring{G}_{3}(x)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\\\ &\frac{x}{\pi}U\ln\frac{T}{2\pi}+\frac{2x}{\pi}\left(c+\frac{\sin
x}{x}\right)U+\mathcal{O}(xT^{5/12}\ln^{2}T),\\\
&\int_{\mathring{G}_{4}(y)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm d}t=\\\
&\frac{y}{\pi}U\ln\frac{T}{2\pi}+\frac{2y}{\pi}\left(c-\frac{\sin
y}{y}\right)U+\mathcal{O}(yT^{5/12}\ln^{2}T),\end{split}$

where

(1.6) $t-\varphi_{1}(t)\sim(1-c)\pi(t),\ t\to\infty,$

and $c$ is the Euler’s constant and $\pi(t)$ is the prime-counting function.

Let (comp. (1.4)) $T=\varphi_{1}(\mathring{T}),\
T+U=\varphi_{1}(\widering{T+U})$. Similarly to [14], (1.8) we have

(1.7) $\rho\\{[T,T+U];[\mathring{T},\widering{T+U}]\\}\sim(1-c)\pi(T),$

where $\rho$ stands for the distance of the corresponding segments.

######  Remark 1.

Some nonlocal interaction of the functions $Z^{2}[\varphi_{1}(t)],\
\tilde{Z}^{2}(t)$ is expressed by eq. (1.5). This interaction is connected
with two collections of disconnected sets unboundedly receding each from other
(see (1.6), (1.7); $\rho\to\infty$ as $T\to\infty$) - like mutually receding
galaxies (the Hubble law).

######  Remark 2.

The asymptotic formulae (1.5) (comp. (1.3)) cannot be received by methods of
Balasubramanian, Heath-Brown and Ivic (comp. [1]).

This paper is a continuation of the series of works [4]-[18].

##  2\. On big asymmetry in the behaviour of the function
$Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t)$ relatively to the sets
$\mathring{G}_{3}(x)$ and $\mathring{G}_{4}(x)$

We obtain from (1.5)

######  Corollary.

(2.1)
$\begin{split}&\int_{\mathring{G}_{3}(x)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t-\int_{\mathring{G}_{4}(x)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\\\ &\frac{4}{\pi}U\sin x+\mathcal{O}(xT^{5/12}\ln^{2}T),\
x\in(0,\pi/2],\end{split}$

especially, in the case $x=\pi/2$, we have

(2.2) $\int_{\mathring{G}_{3}(\pi/2)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t-\int_{\mathring{G}_{4}(\pi/2)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t\sim\frac{4}{\pi}U,$

where
$[\mathring{T},\widering{T+U}]\subset\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$.

######  Remark 3.

The formulae (2.1), (2.2) represent the big difference of the areas (measures)
of the figures which correspond to the functions

$Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t),\ t\in\mathring{G}_{3}(x);\qquad
Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t),\ t\in\mathring{G}_{4}(x).$

The reason for this big asymmetry given by (2.2) is probably the fact that the
zeroes of

$\zeta\left(\frac{1}{2}+it\right),\
t\in\mathring{G}_{3}\left(\frac{\pi}{2}\right)\bigcup\mathring{G}_{4}\left(\frac{\pi}{2}\right)$

lie preferably in the set $\mathring{G}_{4}(\pi/2)$.

##  3\. Proof of the Theorem

### 3.1.

The following lemma holds true (see [6], (2.5); [7], (3.3); [14], (4.1))

######  Lemma.

For every integrable function (in the Lebesgue sense) $f(x),\
x\in[\varphi_{1}(T),\varphi_{1}(T+U)]$ the following is true

(3.1) $\int_{T}^{T+U}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{\varphi_{1}(T)}^{\varphi_{1}(T+U)}f(x){\rm d}x,\
U\in\left(\left.0,\frac{T}{\ln T}\right]\right.,$

where $t-\varphi_{1}(t)\sim(1-c)\pi(t)$, $c$ is the Euler’s constant, $\pi(t)$
is the prime-counting function and

$\begin{split}&\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm d}t},\
\varphi_{1}(t)=\frac{1}{2}\varphi(t),\\\
&\tilde{Z}^{2}(t)=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{Z^{2}(t)}{\left\\{1+\mathcal{O}\left(\frac{\ln\ln
t}{\ln t}\right)\right\\}\ln t},\end{split}$

(see [7], (1.1), (3.1), (3.2)).

######  Remark 4.

The formula (3.1) remains true also in the case when the integral on the
right-hand side of (3.1) is only relatively convergent integral of the second
kind (in the Riemann sense).

In the case (comp. (1.4) $T=\varphi_{1}(\mathring{T}),\
T+U=\varphi_{1}(\widering{T+U})$) we obtain from (3.1)

(3.2)
$\int_{\mathring{T}}^{\widering{T+U}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{T}^{T+U}f(x){\rm d}x.$

### 3.2.

First of all, we have from (3.2), for example

(3.3)
$\int_{\mathring{g}_{2\nu(-x)}}^{\mathring{g}_{2\nu(x)}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{g_{2\nu(-x)}}^{g_{2\nu(x)}}f(t){\rm d}t$

(see (1.4)). Next, in the case $f(t)=Z^{2}(t)$, we have the following
$\tilde{Z}^{2}$-transformation

(3.4)
$\begin{split}&\int_{\mathring{G}_{3}(x)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{G_{3}(x)}Z^{2}(t){\rm d}t,\\\
&\int_{\mathring{G}_{4}(y)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{G_{4}(y)}Z^{2}(t){\rm d}t.\end{split}$

Let us remind that we have proved in the paper [3] the following mean-value
formulae

(3.5) $\begin{split}&\int_{G_{3}(x)}Z^{2}(t){\rm
d}t=\frac{x}{\pi}U\ln\frac{T}{2\pi}+\frac{2x}{\pi}\left(c+\frac{\sin
x}{x}\right)U+\mathcal{O}(xT^{5/12}\ln^{2}T),\\\ &\int_{G_{4}(y)}Z^{2}(t){\rm
d}t=\frac{y}{\pi}U\ln\frac{T}{2\pi}+\frac{2y}{\pi}\left(c-\frac{\sin
y}{y}\right)U+\mathcal{O}(yT^{5/12}\ln^{2}T).\end{split}$

Now, the formula (1.5) follows from (3.4), (3.5).

######  Remark 5.

The formulae (3.5) are the consequences of their discrete form

$\begin{split}&\sum_{T\leq g_{2\nu}\leq T+U}Z^{2}[g_{2\nu}(\tau)]=\\\
&\frac{1}{2\pi}U\ln^{2}\frac{T}{2\pi}+\frac{1}{\pi}(c+\cos\tau)U\ln\frac{T}{2\pi}+\mathcal{O}(T^{5/12}\ln^{3}T),\\\
&\sum_{T\leq g_{2\nu+1}\leq T+U}Z^{2}[g_{2\nu+1}(\tau)]=\\\
&\frac{1}{2\pi}U\ln^{2}\frac{T}{2\pi}+\frac{1}{\pi}(c-\cos\tau)U\ln\frac{T}{2\pi}+\mathcal{O}(T^{5/12}\ln^{3}T)\end{split}$

(see [3], (10)).

I would like to thank Michal Demetrian for helping me with the electronic
version of this work.

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