# High-level moving excursions for spatiotemporal Gaussian random fields with
long range dependence

###### Abstract

This paper analyzes the limit distribution of spatiotemporal sojourn measures
of Gaussian subordinated random fields. The family of Gaussian spatiotemporal
random fields studied displays long–range dependence. The approach presented
is based on Hermite expansion, and reduction theorems. Two new ingredients are
incorporated to this analysis. Specifically, on the one hand, the limiting
distribution is computed under increasing domain asymptotics in space and/or
time, and, on the other hand, the involved non–linear transformation in the
Gaussian subordination is time dependent. In particular, a central limit
theorem is obtained for first Minkowski functional providing the
spatiotemporal volume of excursion sets of spatiotemporal Gaussian random
fields under moving levels. These results are also applied to the case where
subordination to a spherical Gaussian random field, defined by restriction to
the sphere, is performed.

N.N. Leonenko and M.D. Ruiz–Medina

_MSC2020_. 60G10; 60G12; 60G18; 60G20; 60G22 (Primary). 60G60 (60G60).

_Keywords_. Central limit results; Gaussian subordinated random fields;
Long–range dependence; Moving levels; Reduction theorems; Spatiotemporal
increasing domain asymptotics; Strong dependence.

##  1 Introduction

Sojourn functionals were extensively analyzed since the nighties in the
context of weak–dependent random fields (see, e.g., Bulinski _et al._ [6] and
Ivanov and Leonenko [14]). Special attention has been paid to the long–range
dependent (LRD) random field case (see Berman [5]; Leonenko [17]; Leonenko and
Olenko [18]; Makogin and Spodarev [24]; Marinucci, Rossi and Vidotto [26],
among others). Limit theorems for level functionals of stationary Gaussian
processes and fields constitute a major topic in this literature (see, e.g.,
Auffinger and Ben Arous [2]; Azäis and Wschebor [3]; Cabaña [7]; Estrade and
León [10]; Iribarren [13]; Kratz and León [15]; [16]; Marinucci and Vadlamani
[27]; Slud [32]). These papers contribute, in particular, to the
characterization of geometrical properties of random fields. That is the case
of the Euler characteristic of an excursion set, the number of up–crossings at
level $u$ on a bounded closed cube of $\mathbb{R}^{d},$ the probability of the
maximum to be greater than a given threshold $u,$ the analysis of anisotropy
based on the line integral with respect to the level curve at any threshold
$u,$ just to mention a few related stochastic geometrical topics.

In the Gaussian case, Leonenko and Ruiz–Medina [20] have recently extended the
above referred results to the context of subordination to LRD spatiotemporal
random fields under increasing domain asymptotic in time. Some central and
non–central limit results can be obtained as particular cases. A challenging
topic in the application of these results arises when subordination to a
Gaussian random field with non–separable covariance function is considered.
See also recent results in Marinucci, Rossi and Vidotto [27] on limit theorems
of sojourn functionals given by subordination to Short–Range Dependence (SRD)
and Long–Range Dependence (LRD) sphere cross–time Gaussian random fields,
under increasing domain asymptotic in time. Limit theorems for Minkowski
functionals subordinated to Gaussian spatiotemporal random fields restricted
to the sphere are obtained as special cases of the results derived in the
present paper.

This paper addresses the problem of determining the asymptotic probability
distribution, via reduction theorems, of integral functionals of nonlinear
transformations of LRD spatiotemporal Gaussian random fields. The asymptotic
analysis achieved is based on increasing domain asymptotics in space and time.
The case of subordination via nonlinear transformations depending on the size
of the temporal domain is also addressed. As particular case, first Minkowski
functionals involving moving levels subordinated to spatiotemporal Gaussian
random fields arise. Namely, we restrict our attention to the case where time
domain is a temporal interval $[0,T],$ and $\Lambda(T)=T^{\gamma},$
$\gamma\geq 0,$ defines the scale factor of the homothetic transformation of a
convex compact set $K\subset\mathbb{R}^{d},$ with center the point
$\mathbf{0}\in K.$ The case $\gamma=0$ was addressed in Leonenko and
Ruiz–Medina [20] (see also Marinucci, Rossi and Vidotto [27], where
alternative conditions were formulated in a Hilbert space framework, under a
parametric modelling of spherical scale dependent LRD). When $\gamma=1,$ our
Theorem 1 extends Theorem 1 in Leonenko, Ruiz–Medina and Taqqu [21] to the
spatiotemporal context (see also Leonenko, Ruiz–Medina and Taqqu [22]).

A general reduction principle is derived in Theorem 1, under increasing domain
asymptotics in space and time. This principle was first discovered by Taqqu
[33] (see also Dobrushin and Major [9]; Leonenko, Ruiz–Medina and Taqqu [21];
[22]; Taqqu [34]). The limiting distributions of properly normalised integrals
of non-linear transformations of spatiotemporal Gaussian random fields are
proved here to be equivalent to the asymptotic distribution of Hermite
polynomial type functionals of such Gaussian random fields. The methodology
adopted in the proof is based on Hermite expansion of spatiotemporal local
functionals of Gaussian subordinated random fields, but it is standard, and
can be applied to the Laguerre chaos as well, covering the case of
subordination to gamma–correlated random fields (see, e.g., Leonenko,
Ruiz–Medina and Taqqu [21]). As particular case, Minkowski functionals,
providing the spatiotemporal volume of excursion sets, can be analyzed. This
paper goes beyond the regularly varying condition on the spatiotemporal
covariance function, covering a larger class of spatiotemporal covariance
functions, including Gneiting class (see Gneiting [11]). This class of
covariance functions is popular in many applications, including Meteorology or
Earth sciences, among others. See also Section 4 below.

In the second place, this paper addresses a general reduction principle under
$T$-varying nonlinear transformations of Gaussian random fields, with, as
before, $T$ denoting the size of the time interval (see Theorem 3). The
novelty of this second part relies on methodological approach adopted for
proofs. Although, the Gaussian subordination is required. In the particular
case of first Minkowski functionals, involving moving levels, Theorem 4
obtains the limiting distribution of these functionals by a suitable
normalization, under increasing domain asymptotic in time and in space.
General conditions on the decaying of the covariance function to zero in time
are assumed, leading to the case of LRD random fields. The divergence rate of
the moving threshold parameter must be controlled at the logarithm scale by
the increasing of the size $T$ of the temporal interval.

The outline of the paper is as follows. Some results on geometrical
probabilities are first reviewed in Section 2. Section 3 derives the
conditions for a general reduction principle under increasing domain
asymptotics in time and in space (see Theorem 1). In Section 4, some examples
are provided by considering subordination to spatiotemporal Gaussian random
fields with nonseparable covariance function. The case where local means are
computed from a $T$–dependent nonlinear transformation of a Gaussian random
field is studied in Section 5 (see Theorem 3). In Section 6, the asymptotic
distribution of suitable normalized first Minkowski functionals involving
moving levels, under increasing domain asymptotics in space–time, is derived
in Theorem 4. The obtained results are applied in Section 7 to the case of
$\gamma=0,$ when subordination is achieved to the restriction to the sphere of
a Gaussian spatiotemporal random field.

##  2 Preliminaries

Variance components characterizing variability of the Wiener chaos of the
underlying spatiotemporal stationary Gaussian random field, normalizing the
integral functionals arising in the paper in reduction theorems, are computed
from geometrical probabilities under spatial isotropy (see, e.g., Lord [23];
Ivanov and Leonenko [14]). Some extended results can also be found in Ahronyan
and Khalatyan [1], and the references therein. We now summarize them and
introduce the corresponding notation.

Let $\nu_{d}(\cdot)$ be the Lebesgue measure in $\mathbb{R}^{d},$ $d\geq 2,$
and $K$ be a convex body in $\mathbb{R}^{d},$ i.e., a compact convex set with
non–empty interior with center at the point $0\in K.$ Let
$\mathcal{D}(K)=\max\left\\{\|x-y\|,\ x,y\in K\right\\}$ be the diameter of
$K.$ Let also $\nu_{d}(K)=|K|$ be the volume of $K,$ and $\nu_{d-1}(\delta
K)=U_{d-1}(K)$ be the surface area of $K,$ where $\delta K$ is the boundary of
$K.$ Note that for the case where $K=\mathcal{B}(1)=\\{x\in\mathbb{R}^{d};\
\|x\|\leq 1\\}$ is the unit ball, and $\delta
K=S_{d-1}=\\{x\in\mathbb{R}^{d};\ \|x\|=1\\}$ is the unit sphere, then
$\mathcal{D}(\mathcal{B}(1))=2,$ and

$|\mathcal{B}(1)|=\frac{\pi^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)},\
U_{d-1}(\mathcal{B}(1))=|S_{d-1}|=\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}.$
(1)

Let $\Lambda(T)K$ be a homothetic transformation of body $K$ with center $0\in
K,$ and coefficient or scaling factor $\Lambda(T)>0.$ We assume that
$\Lambda(T)=T^{\gamma},$ $\gamma\geq 0$. Then, $|\Lambda(T)K|=|K|T^{\gamma
d}.$ Following the approach presented in Lord [23] (see also Ivanov and
Leonenko [14]), the probability density $\psi_{\Lambda(T),\mathcal{B}(1)}(z)$
of the random variable $Z=\rho_{\mathcal{B}(1)}=\|P_{1}-P_{2}\|,$ with $P_{1}$
and $P_{2}$ being two independent random points with uniform distribution in
$\mathcal{B}(1),$ is given by, for $0\leq z\leq 2T^{\gamma},$

$\displaystyle\psi_{\Lambda(T),\mathcal{B}(1)}(z)$ $\displaystyle=$
$\displaystyle\frac{d}{[\Lambda(T)]^{d}}z^{d-1}I_{1-\left(\frac{z}{2\Lambda(T)}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right),$
(2)

in terms of the incomplete beta–function

$I_{\mu}(p,q)=\frac{\Gamma(p+q)}{\Gamma(p)\Gamma(q)}\int_{0}^{\mu}t^{p-1}(1-t)^{q-1}dt,\quad\mu\in(0,1].$
(3)

In formula (2.6) in Ahoronyan and Khalatyan [1], an extended version of the
probability density of the distance between two independent uniformly
distributed points is derived for a convex body $K$ in $\mathbb{R}^{d},$
applying an alternative methodology to Lord [23] for the case of hyperspheres.
Specifically, let $\mathcal{J}$ be the space of all straight lines in
$\mathbb{R}^{d},$ and $d\gamma$ is an element of a locally finite measure in
the space $\mathcal{J},$ which is invariant, with respect to the group $M$ of
all Euclidean motions in the space $\mathbb{R}^{d}$ (the uniform measure on
$\mathcal{J}$). Let also $F_{K}(v)$ be the chord length distribution function
of body $K,$ defined as

$F_{K}(v)=\frac{2(d-1)}{|S_{d-2}|}\int_{\chi(\gamma)\leq v}d\gamma,$ (4)

where $\chi(\gamma)=\gamma\cap K$ is a chorf in $K.$ Then, for $0\leq
z\leq\mathcal{D}(K),$

$\displaystyle\psi_{\rho_{K}}(z)=\frac{1}{|K|^{2}}\left[z^{d-1}|S_{d-1}||K|-z^{d-1}|S_{d-2}|\frac{U_{d-1}(K)}{d-1}\int_{0}^{z}\left(1-F_{K}(v)\right)dv\right].$
(5)

From (5), we also have, for $0\leq z\leq\mathcal{D}(\Lambda(T)K),$

$\displaystyle\psi_{\Lambda(T),K}(z)=\psi_{\rho_{\Lambda(T)K}}(z)=\frac{1}{[\left(|K|[\Lambda(T)]^{d}\right)]^{2}}\left[\left(|K|[\Lambda(T)]^{d}\right)z^{d-1}|S_{d-1}|\right.$
$\displaystyle\hskip
99.58464pt\left.-|S_{d-2}|z^{d-1}\frac{U_{d-1}(\Lambda(T)K)}{d-1}\int_{0}^{z}\left(1-F_{\Lambda(T)K}(v)\right)dv\right].$

##  3 Reduction theorems for spatiotemporal random fields with LRD

We consider the spatiotemporal random field
$Z:\left(\Omega\times\mathbb{R}^{d}\times\mathbb{R}\right)\longrightarrow\mathbb{R},$
with $(\Omega,\mathcal{A},P)$ denoting the basic probability space.

Condition 1. Assume that $Z$ is a measurable mean–square continuous
homogeneous and isotropic in space and stationary in time Gaussian random
field with $\mathbb{E}[Z(x,t)]=0,$ $\mathbb{E}[Z^{2}(x,t)]=1,$ and
$\widetilde{C}(\|x-y\|,|t-s|)=\mathbb{E}\left[Z(x,t)Z(y,s)\right]\geq 0.$

Under Condition 1, the following notation will be considered
$C(z,\tau)=\widetilde{C}(\|x-y\|,|t-s|),\quad z=\|x-y\|\geq 0,$ for
$\tau=|t-s|\geq 0.$

Let $Z\sim\mathcal{N}(0,1)$ be a standard Gaussian random variable with
density

$\phi(w)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{w^{2}}{2}\right),\
w\in\mathbb{R},\ \Phi(u)=\int_{-\infty}^{u}\phi(w)dw.$

Let $G(z)$ be a real–valued Borel function satisfying

Condition 2. $\mathbb{E}G^{2}(Z(x,t))<\infty.$

Function $G$ then admits an expansion with respect to the normalized Hermite
polynomials in the Hilbert space $L_{2}(\mathbb{R},\phi(u)du):$

$G(z)=\sum_{n=0}^{\infty}\frac{\mathcal{J}_{n}}{n!}H_{n}(z),\
\mathcal{J}_{n}=\int_{\mathbb{R}}G(z)H_{n}(z)\phi(z)dz,$

where the Hermite polynomial of order $n,$ denoted by $H_{n}(z)$ is defined by
the equation

$\frac{d^{n}}{dz^{n}}\phi(z)=(-1)^{n}H_{n}(z)\phi(z).$

Note that $H_{0}=1,$ $H_{1}(z)=z,$ and $H_{2}(z)=z^{2}-1,\dots$

Condition 3. The Hermite rank of the function $G$ is $m\geq 1.$ Hence, for
$m=1,$ $\mathcal{J}_{1}\neq 0,$ or for $m\geq 2,$
$\mathcal{J}_{1}=\cdots=\mathcal{J}_{m-1}=0,$ $\mathcal{J}_{m}\neq 0.$

We define the following spatiotemporal local functional of $Z(x,t),$ that is,
in the Hilbert space $L_{2}(\Omega,\mathcal{A},P):$

$\displaystyle A_{T}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Lambda(T)K}G(Z(x,t))dxdt=\sum_{n=0}^{\infty}\frac{\mathcal{J}_{n}}{n!}\xi_{n,T}$
$\displaystyle E[A_{T}]$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Lambda(T)K}\mathcal{J}_{0}dxdt=\mathcal{J}_{0}|T||K|[\Lambda(T)]^{d}$
$\displaystyle\xi_{n,T}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Lambda(T)K}H_{n}(Z(x,t))dxdt$
$\displaystyle\mathbb{E}[\xi_{n,T}]$ $\displaystyle=$ $\displaystyle 0,\
\mathbb{E}[\xi_{n,T}\xi_{l,T}]=0,\ n\neq l,\ n,l\geq m,$ (7)

where $m$ denotes the Hermite rank of function $G.$

Thus, for $n\geq m,$

$\displaystyle\sigma_{n,K}^{2}(T)=\mbox{Var}(\xi_{n,T})$ $\displaystyle=$
$\displaystyle
2n!T\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{K\Lambda(T)\times
K\Lambda(T)}\widetilde{C}^{n}(\|\mathbf{x}-\mathbf{y}\|,\tau)d\tau
d\mathbf{x}d\mathbf{y}$ $\displaystyle=$ $\displaystyle
2n!T|K\Lambda(T)|^{2}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\mathbb{E}\left[\widetilde{C}^{n}\left(\|P_{1}-P_{2}\|,\tau\right)\right]d\tau$
$\displaystyle=2n!T|K|^{2}T^{2\gamma
d}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{\mathcal{D}(K\Lambda(T))}\psi_{\Lambda(T),K}(z)C^{n}(z,\tau)dzd\tau.$

Here, $\psi_{\Lambda(T),K}(z),$ denotes, as before, the probability density of
the random variable $Z=\rho_{K}=\|P_{1}-P_{2}\|,$ where $P_{1}$ and $P_{2}$
are two independent random points with uniform distribution in $\Lambda(T)K.$

In particular, for $K=\mathcal{B}(1),$ for $0\leq z\leq 2T^{\gamma},$

$\displaystyle\psi_{\Lambda(T),\mathcal{B}(1)}(z)$ $\displaystyle=$
$\displaystyle\frac{d}{T^{\gamma
d}}z^{d-1}I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right),$
(9)

in terms of the incomplete beta function (3). Thus, from equation (1),

$\displaystyle\sigma_{m,\mathcal{B}(1)}^{2}(T)$ $\displaystyle=$
$\displaystyle\mbox{Var}\left(\int_{0}^{T}\int_{\Lambda(T)\mathcal{B}(1)}H_{m}\left(Z(x,t)\right)\right)dxdt$
(10) $\displaystyle=$ $\displaystyle
2m!T|\mathcal{B}(1)|^{2}[\Lambda(T)]^{2d}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{\mathcal{D}(K\Lambda(T))}C^{m}(z,\tau)$
$\displaystyle\hskip
28.45274pt\times\left[\frac{d}{[\Lambda(T)]^{d}}z^{d-1}I_{1-\left(\frac{z}{2\Lambda(T)}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)\right]dzd\tau$
$\displaystyle=$ $\displaystyle 2m!|\mathcal{B}(1)|^{2}dT^{\gamma
d+1}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)$ $\displaystyle\hskip
56.9055pt\times\int_{0}^{2T^{\gamma}}z^{d-1}C^{m}(z,\tau)I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)dzd\tau$
$\displaystyle=$ $\displaystyle
8m!\frac{\pi^{d}}{d\Gamma^{2}\left(\frac{d}{2}\right)}T^{\gamma
d+1}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2T^{\gamma}}z^{d-1}C^{m}(z,\tau)$
$\displaystyle\hskip 99.58464pt\times
I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)dzd\tau,$

###### Remark 1

In the following, we will apply Lord (1954) results on the derivation of the
probability density of the distance between two independent uniform points in
hypersheres. Consider the positive constants $F_{1}$ and $F_{2}$ respectively
defining the supremum and infimum of the radius of the balls such that the
following inclusions hold

$F_{1}\mathcal{B}\left(1\right)\subseteq K\subseteq
F_{2}\mathcal{B}\left(1\right).$ (11)

From equations (4)–(6) in Lord (1954),

$C_{1}\psi_{T^{\gamma},\mathcal{B}\left(S_{1}\right)}\leq\psi_{T^{\gamma},K}\leq
C_{2}\psi_{T^{\gamma},\mathcal{B}\left(S_{2}\right)},\ 0\leq C_{1}\leq C_{2}.$
(12)

Here, $\psi_{T^{\gamma},K}(z)$ denotes the probability density of the random
variable $Z=\rho_{KT^{\gamma}}=\|P_{1}-P_{2}\|,$ with $P_{1}$ and $P_{2}$
being two independent random points with uniform distribution in
$KT^{\gamma}.$ As before, $K$ denotes a compact convex set with non–empty
interior, and with center at the point $0\in K.$ In particular,
$\mathcal{B}\left(S_{i}\right)$ denotes the ball with center
$0\in\mathcal{B}\left(S_{i}\right),$ and radius $S_{i},$ $i=1,2,$ according to
constants $F_{i},$ $i=1,2,$ in equation (11).

The following additional condition will be assumed.

Condition 4.

  * (i)

$C(z,\tau)\rightarrow 0,$ if $\max\\{z,\tau\\}\rightarrow\infty.$

  * (ii)

For some fixed $m\in\\{1,2,3\dots\\},$ there exist $\delta_{1}\in(0,1)$ and
$\delta_{2}\in(0,1)$ such that for

$\Lambda(T)=T^{\gamma},\ \mbox{for\ certain\ fixed}\ \gamma\geq 0,T\geq 1,$

and

$\displaystyle\lim_{T\rightarrow\infty}\frac{\sigma_{m,K}^{2}(T)}{T^{1+\delta_{1}}T^{\gamma
d(1+\delta_{2})}}=\infty,$

for certain $\delta_{1}\in(0,1)$ and $\delta_{2}\in(0,1).$

From Remark 1, Condition 4(ii) holds if there exist $\delta_{1}\in(0,1)$ and
$\delta_{2}\in(0,1)$ such that as $T\to\infty,$

$\displaystyle\frac{1}{T^{\delta_{1}+\gamma
d\delta_{2}}}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2T^{\gamma}}z^{d-1}C^{m}(z,\tau)I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)dzd\tau\to\infty.$
(13)

Thus, from equations (3), (10) and (12), in the next results of this paper,
Condition 4(ii) will be equivalently tested from equation (13).

### Separable covariance functions

For the case of separable spatiotemporal covariance functions in the unit
ball,

$\mathsf{C(z,\tau)=C_{Space}(z)C_{Time}(\tau),}$

we have

$\sigma_{m,\mathcal{B}(1)}^{2}(T)=m!b_{1m}(T)b_{2m}(T),$

where

$\displaystyle
b_{1m}(T)=2T\int_{0}^{T}\left(1-\frac{\tau}{T}\right)C_{Time}^{m}(\tau)d\tau$
$\displaystyle b_{2m}(T)=|\mathcal{B}(1)|^{2}dT^{\gamma
d}\int_{0}^{2T^{\gamma}}z^{d-1}C_{Space}^{m}(z)I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)dz.$

Then, in the weak dependence case in time, as T $\rightarrow\infty,$ we obtain

$\displaystyle b_{1m}(T)=2L_{1}T(1+o(1))$ $\displaystyle
L_{1}=\int_{0}^{\infty}C_{Time}^{m}(\tau)d\tau<\infty,\quad\int_{0}^{\infty}C_{Time}^{m}(\tau)d\tau\neq
0.$ (14)

Suppose that for some bounded slowly varying functions at infinity
$\mathcal{L}_{1}(\tau):$

$C_{Time}(\tau)=\frac{\mathcal{L}_{1}(\tau)}{\tau^{A}},\quad A>0,$

then for $A=\frac{1}{m},$ $b_{1m}(T)=2T\log(T)\mathcal{L}_{1}^{m}(T)(1+o(1)),$
while for $0<A<\frac{1}{m}:$

$\displaystyle b_{1m}(T)$ $\displaystyle=$ $\displaystyle
2L_{2}T^{2-mA}\mathcal{L}_{1}^{m}(T)(1+o(1))$ $\displaystyle L_{2}$
$\displaystyle=$
$\displaystyle\left[\int_{0}^{1}(1-\tau)\tau^{-Am}d\tau\right]=(1-mA)(2-Am).$

Similarly, as $T\rightarrow\infty,$ under weak–dependence in space,
$b_{2m}(T)=L_{3}T^{d\gamma}(1+o(1)),$ if
$L_{3}=|\mathcal{B}(1)|^{2}d\int_{0}^{\infty}z^{d-1}C_{Space}^{m}(z)dz<\infty$
and $\int_{0}^{\infty}z^{d-1}C_{Space}^{m}(z)dz\neq 0,$ and if for some
bounded slowly varying functions at infinity $\mathcal{L}(\tau):$

$C_{Space}(z)=\frac{\mathcal{L}(z)}{z^{\widetilde{\alpha}}},\quad\widetilde{\alpha}>0,$

then for $\widetilde{\alpha}=\frac{d}{m},$ $b_{2m}(T)=L_{4}T^{\gamma
d}\log(T^{\gamma})\mathcal{L}^{m}(T^{\gamma})(1+o(1)),$ with
$L_{4}=\frac{4\pi^{d}}{d\Gamma^{2}(d/2)}.$ For
$0<\widetilde{\alpha}<\frac{d}{m},$
$b_{2m}(T)=L_{5}T^{\gamma(2d-m\widetilde{\alpha})}\mathcal{L}^{m}(T^{\gamma})(1+o(1)),$
where
$L_{5}=\frac{2^{d-m\widetilde{\alpha}+1}\pi^{d-\frac{1}{2}}\Gamma\left(\frac{d-m\widetilde{\alpha}+1}{2}\right)}{(d-m\widetilde{\alpha})\Gamma\left(\frac{d}{2}\right)\Gamma\left(\frac{2d-m\widetilde{\alpha}+2}{2}\right)}$
(see Lemma 2.1.3 of Ivanov and Leonenko [14]) .

Summarizing, for the introduced separable covariance function family, when
$K=\mathcal{B}(1),$ Condition 4(ii) holds in the following cases:

For $\gamma=0,$ Condition 4(ii) holds for $0<\delta_{1}<1-mA<1$ (LRD in time
only, $0<A<\frac{1}{m}$). This case has been considered by Leonenko and
Ruiz–Medina (2023). While for $\gamma>0,$ Condition 4(ii) holds for
$0<\delta_{1}<1-mA<1,$ and $0<\delta_{2}<1-(m\widetilde{\alpha})/d<1$ (LRD in
time and in space). In this case for $0<\widetilde{\alpha}<\frac{d}{m},$ and
$0<A<\frac{1}{m},$

$\displaystyle\sigma_{m,\mathcal{B}(1)}^{2}(T)$ $\displaystyle=$
$\displaystyle
m!L_{2}L_{5}T^{2-mA}\mathcal{L}_{1}^{m}(T)T^{\gamma(2d-m\widetilde{\alpha})}\mathcal{L}^{m}(T^{\gamma})(1+o(1)).$

###### Theorem 1

Under Conditions 1,2,3 and 4, the random variables

$\displaystyle Y_{T}$ $\displaystyle=$
$\displaystyle\frac{\left[A_{T}-\mathbb{E}[A_{T}]\right]}{|\mathcal{J}_{m}|\sigma_{m,K}(T)/m!}$
(16) and $\displaystyle
Y_{m,T}=\frac{\mbox{sgn}(\mathcal{J}_{m})\int_{0}^{T}\int_{T^{\gamma}K}H_{m}(Z(x,t))dxdt}{\sigma_{m,K}(T)}$

have the same limiting distributions as $T\rightarrow\infty$ (if one of it
exists).

Proof. We consider the decomposition

$\displaystyle A_{T}-\mathbb{E}[A_{T}]$ $\displaystyle=$ $\displaystyle
S_{1,T}+S_{2,T},$

where

$\displaystyle S_{1,T}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{J}_{m}}{m!}\xi_{m,T}$ $\displaystyle S_{2,T}$
$\displaystyle=$
$\displaystyle\sum_{n=m+1}^{\infty}\frac{\mathcal{J}_{n}}{n!}\xi_{n,T},\
\sum_{n=m}^{\infty}\frac{\mathcal{J}^{2}_{n}}{n!}<\infty.$ (17)

From (7) and (3),

$\mbox{Var}(A_{T})=\mbox{Var}[S_{1,T}]+\mbox{Var}[S_{2,T}],$ (18)

and we will prove that
$\mbox{Var}[S_{2,T}]/\sigma_{m,K\Lambda(T)}^{2}(T)\rightarrow 0,$
$T\rightarrow\infty.$

From Condition 4(i),

$\displaystyle\sup_{(z,\tau)\in
B_{T}^{\delta_{1},\delta_{2}}}C(z,\tau)\rightarrow 0,\quad T\rightarrow\infty$
(19) $\displaystyle B_{T}^{\delta_{1},\delta_{2}}=\\{(z,\tau);\ \tau\geq
T^{\delta_{1}}\ \mbox{or}\ z\geq T^{\gamma\delta_{2}}\\}.$

Futhermore, for the set

$\overline{B}_{T}^{\delta_{1},\delta_{2}}=\left\\{(z,\tau);\ 0\leq\tau\leq
T^{\delta_{1}},\ 0\leq z\leq T^{\gamma\delta_{2}}\right\\},$

one can use the estimate $C^{m+1}(z,\tau)\leq 1.$ Let us consider
$\delta_{1}\in(0,1),\delta_{2}\in(0,1)$ as in Condition 4. Then, from equation
(3),

$\displaystyle\mbox{Var}(S_{2,T})=\sum_{n=m+1}^{\infty}\frac{\mathcal{J}_{n}^{2}}{(n!)^{2}}\sigma_{n,K}^{2}(T)$
$\displaystyle\leq$ $\displaystyle k_{1}\left\\{T^{1+\gamma
2d}\left[\int_{\overline{B}_{T}^{\delta_{1},\delta_{2}}}+\int_{{}^{B_{T}^{\delta_{1},\delta_{2}}}}\right]\left(1-\frac{\tau}{T}\right)C^{m+1}(z,\tau)\psi_{T^{\gamma},K}(z)dzd\tau\right\\},$

for some positive constant $k_{1}>0,$ where we have used
$\sum_{n=m}^{\infty}\frac{\mathcal{J}^{2}_{n}}{n!}<\infty,$ and $C^{m+l}\leq
C^{m+1},$ for $l\geq 2,$ since $C(z,\tau)\leq 1,$ under Condition 1. We then
obtain for some positive constants $k_{2},k_{3},k_{4},$ keeping in mind
equation (3), and Remark 1

$\displaystyle\mbox{Var}(S_{2,T})\leq
k_{2}\left\\{k_{3}T^{1+\delta_{1}}[\Lambda(T)]^{d(1+\delta_{2})}\right.$
$\displaystyle+\left.k_{4}T^{1+\gamma 2d}\sup_{(z,\tau)\in
B_{T}^{\delta_{1},\delta_{2}}}\left\\{C(z,\tau)\right\\}\int_{B_{T}^{\delta_{1},\delta_{2}}}\left(1-\frac{\tau}{T}\right)C^{m}(z,\tau)\psi_{T^{\gamma},K}(z)dzd\tau\right\\}.$

Hence, as $T\rightarrow\infty,$

$\displaystyle\frac{\mbox{Var}(S_{2,T})}{\sigma_{m,K}^{2}(T)}\leq
k_{5}\left\\{\frac{1}{\frac{\sigma_{m,K}^{2}(T)}{T^{1+\delta_{1}}T^{\gamma
d(1+\delta_{2})}}}+k_{6}\sup_{(z,\tau)\in
B_{T}^{\delta_{1},\delta_{2}}}\left\\{C(z,\tau)\right\\}\right.$
$\displaystyle\hskip 28.45274pt\left.\hskip
56.9055pt\times\frac{\int_{B_{T}^{\delta_{1},\delta_{2}}}\left(1-\frac{\tau}{T}\right)C^{m}(z,\tau)\psi_{T^{\gamma},K}(z)dzd\tau}{\int_{B_{T}^{\delta_{1},\delta_{2}}\cup\overline{B}_{T}^{\delta_{1},\delta_{2}}}\left(1-\frac{\tau}{T}\right)C^{m}(z,\tau)\psi_{T^{\gamma},K}(z)dzd\tau}\right\\}\rightarrow
0,$ (20)

where we have applied that from Condition 1, for all $T>0,$

$\frac{\int_{B_{T}^{\delta_{1},\delta_{2}}}\left(1-\frac{\tau}{T}\right)C^{m}(z,\tau)\psi_{T^{\gamma},,K}(z)dzd\tau}{\int_{B_{T}^{\delta_{1},\delta_{2}}\cup\overline{B}_{T}^{\delta_{1},\delta_{2}}}\left(1-\frac{\tau}{T}\right)C^{m}(z,\tau)\psi_{T^{\gamma},K}(z)dzd\tau}\leq
1,$

and that from Condition 4(i), $\sup_{(z,\tau)\in
B_{T}^{\delta_{1},\delta_{2}}}\left\\{C(z,\tau)\right\\}\to 0,$ $T\to\infty,$
as well as from Condition 4(ii) (see also Remark 1 and equation (13))

$\displaystyle\frac{\sigma_{m,K}^{2}(T)}{T^{1+\delta_{1}}T^{\gamma
d(1+\delta_{2})}}\to\infty$

as $T\to\infty.$ From equations (17), (18) and (20),

$\mbox{Var}\left(Y_{T}-Y_{m,T}\right)\to 0,\quad T\to\infty,$

as we wanted to prove.

##  4 Sojourn functionals for spatiotemporal fields

We consider the first Minkowski functional

$\displaystyle M_{1}(T)$ $\displaystyle=$ $\displaystyle\left|\left\\{0\leq
t\leq T,\ x\in T^{\gamma}K;\ Z(x,t)\geq u\right\\}\right|$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{T^{\gamma}K}G_{u}(Z(x,t))dxdt,\quad u\geq 0,\
\gamma\geq 0,$

where $G_{u}(z)=\mathbb{I}_{z\geq u},$ that is, $G_{u}(Z(x,t))$ is the
indicator function of the set

$\left\\{0\leq t\leq T,\ x\in T^{\gamma}K;\ Z(x,t)\geq u\right\\}.$

Then,

$E[M_{1}(T)]=(1-\Phi(u))T|K|T^{\gamma d},$

and from Theorem 1 with $m=1$ we arrive to the following result

###### Theorem 2

Under Conditions 1–4 with $m=1,$ the random variable

$\displaystyle\left\\{M_{1}(T)-(1-\Phi(u))|K|T^{1+\gamma d}\right\\}\times$
$\displaystyle\times\left(\phi(u)\left[2|K|^{2}T^{1+2\gamma
d}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{\mathcal{D}(T^{\gamma}K)}C(z,\tau)\psi_{T^{\gamma},K}(z)dzd\tau\right]^{1/2}\right)^{-1}$

converges to a standard normal distribution as $T\to\infty.$

Note that the case $\gamma=0$ was proved in Theorem 2 of Leonenko and
Ruiz–Medina [20]. In particular, for the ball
$K=\mathcal{B}(1)\subset\mathbb{R}^{d},$ the random variable

$\displaystyle\left\\{M_{1}(T)-(1-\Phi(u))\pi^{d/2}T^{1+\gamma
d}\left[\Gamma\left(\frac{d}{2}+1\right)\right]^{-1}\right\\}\left[\phi(u)\left[(8\pi^{d}\Gamma^{-2}\left(d/2\right)(1/d))\right.\right.$
$\displaystyle\left.\left.\times T^{1+\gamma
d}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2T^{\gamma}}z^{d-1}C(z,\tau)I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)dzd\tau\right]^{1/2}\right]^{-1}$

is asymptotically distributed as a standard normal random variable, as
$T\to\infty.$

### Examples

Let us analyze Condition 4 with $m=1$ for nonseparable covariance functions in
the Gneiting class (see Gneiting [11]). This family of covariance functions is
given by

$\widetilde{C}(\left\|x\right\|,\tau)=\frac{\sigma^{2}}{[\psi(\tau^{2})]^{d/2}}\varphi\left(\frac{\left\|x\right\|^{2}}{\psi(\tau^{2})}\right),\text{
}\sigma^{2}\geq 0,\text{ }(x,\tau)\in\mathbb{R}^{d}\times\mathbb{R},$ (21)

in terms of a completely monotone function $\varphi$ and a positive function
$\psi(u),$ $u\geq 0,$ with a completely monotone derivative.

In the following two examples, we will consider two different scenarios
corresponding to some special cases of function $\varphi$ and $\psi$ in (21)
defining the correlation function $C_{Z}$ of a mean–square continuous
zero–mean Gaussian random field $\\{Z(x,t),\ x\in\mathbb{R}^{d},\
t\in\mathbb{R}\\}.$

Example 1. Let $E_{\nu}(-z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{k}}{\Gamma(\nu
k+1)},$ $z\geq 0,$ be the Mittag–Leffler function (see, e.g., Gorenflo et al.
[12]). Consider in equation (21),

$\varphi(z)=E_{\nu}(-z^{\widetilde{\gamma}}),\quad z\geq 0,\quad 0<\nu\leq
1,\quad 0<\widetilde{\gamma}<1,$ (22)

which is a complete monotone function, that is,

$(-1)^{r}\frac{d^{r}}{dz^{r}}\varphi(z)\geq 0,$

for all $r=0,1,2,\dots,$ and $0<\nu\leq 1$ (see Barnoff–Nielsen and Leonenko
[4]). From Theorem 4 in Simon [31], for $z\in\mathbb{R}_{+},$ and
$\nu\in(0,1)$

$\displaystyle\frac{1}{1+\Gamma(1-\nu)z}$ $\displaystyle\leq$ $\displaystyle
E_{\nu}(-z)\leq\frac{1}{1+[\Gamma(1+\nu)]^{-1}z}.$ (23)

The function $\psi(\tau)=(1+a\tau^{\alpha})^{\beta},$ $a>0,$ $0<\alpha\leq 1,$
$0<\beta\leq 1,$ $\tau\geq 0,$ has completely monotone derivatives (see
Gneiting [11]).

Example 2. Consider now, in equation (21),

$\displaystyle\varphi(z)$ $\displaystyle=$
$\displaystyle\frac{1}{(1+\widetilde{c}z^{\widetilde{\gamma}})^{\nu}},\ z>0,\
\widetilde{c}>0,\ 0<\widetilde{\gamma}\leq 1,\ \nu>0$
$\displaystyle\psi(\tau)$ $\displaystyle=$
$\displaystyle(1+a\tau^{\alpha})^{\beta},\ a>0,\,0<\alpha\leq 1,\ 0<\beta\leq
1,\ \tau\geq 0.$ (24)

It is known (see Gneiting [11]) that the function $\varphi(z)$ is completely
monotone while function $\psi(\tau)$ has completely monotone derivatives. We
restrict our attention to the ball $\mathcal{B}(1).$

In particular, in Example 1, from equations (9) and (23),

$\displaystyle\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2T^{\gamma}}z^{d-1}C(z,\tau)I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)dzd\tau$
$\displaystyle\geq\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2T^{\gamma}}\frac{z^{d-1}\sigma^{2}}{(a\tau^{2\alpha}+1)^{\beta
d/2}[1+\Gamma(1-\nu)\left(z^{2\widetilde{\gamma}}/((a\tau^{2\alpha}+1)^{\beta\widetilde{\gamma}})\right)]}$
$\displaystyle\hskip 142.26378pt\times
I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)dzd\tau$
$\displaystyle=T^{1+\gamma d-\alpha\beta
d-2\widetilde{\gamma}\gamma+2\alpha\beta\widetilde{\gamma}}\int_{0}^{1}\left(1-u\right)\int_{0}^{2}\frac{x^{d-1}\sigma^{2}}{(au^{2\alpha}+T^{-2\alpha})^{\beta
d/2}}$ $\displaystyle\hskip
85.35826pt\times\frac{I_{1-\left(\frac{x}{2}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)}{T^{2\alpha\beta\widetilde{\gamma}-2\widetilde{\gamma}\gamma}+\Gamma(1-\nu)\left[x^{2\widetilde{\gamma}}/(au^{2\alpha}+T^{-2\alpha})^{\beta\widetilde{\gamma}}\right]}dudx,$

where the last equality in equation (4) has been obtained by applying the
change of variables $\tau=Tu$ and $z=T^{\gamma}x.$ From (4), Condition 4(ii)
holds if $1+\gamma d-\alpha\beta
d-2\gamma\widetilde{\gamma}+2\alpha\beta\widetilde{\gamma}>\gamma
d\delta_{2}+\delta_{1}$ for some $\delta_{1},\delta_{2}\in(0,1).$ In
particular, a sufficient condition for Condition 4(ii) to hold is
$\gamma>\alpha\beta,$ and
$\widetilde{\gamma}<\frac{1}{2(\gamma-\alpha\beta)}.$ Under this condition,
one can consider, for instance, $\gamma d-\alpha\beta d>\gamma d\delta_{2},$
and $1-2\gamma\widetilde{\gamma}+2\alpha\beta\widetilde{\gamma}>\delta_{1},$
and hence, $0<\delta_{2}<1-\frac{\alpha\beta}{\gamma}<1,$
$0<\delta_{1}<1-2\widetilde{\gamma}(\gamma-2\alpha\beta)<1.$ In Example 2, in
a similar way, one can obtain

$\displaystyle\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2T^{\gamma}}z^{d-1}C(z,\tau)I_{1-\left(\frac{z}{2T^{\gamma}}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)dzd\tau$
$\displaystyle=T^{1+\gamma d-\alpha\beta
d-2\widetilde{\gamma}\gamma\nu+2\alpha\beta\widetilde{\gamma}\nu}\int_{0}^{1}\left(1-u\right)\int_{0}^{2}\frac{x^{d-1}\sigma^{2}}{(au^{2\alpha}+T^{-2\alpha})^{\beta
d/2}}$ $\displaystyle\hskip
85.35826pt\times\frac{I_{1-\left(\frac{x}{2}\right)^{2}}\left(\frac{d+1}{2},\frac{1}{2}\right)}{T^{2\alpha\beta\widetilde{\gamma}\nu-2\widetilde{\gamma}\gamma\nu}+\widetilde{c}\left[x^{2\widetilde{\gamma}\nu}/(au^{2\alpha}+T^{-2\alpha})^{\beta\widetilde{\gamma}\nu}\right]}dudx.$

Condition 4(ii) is then satisfied if $1+\gamma d-\alpha\beta
d-2\gamma\widetilde{\gamma}\nu+2\alpha\beta\widetilde{\gamma}\nu>\gamma
d\delta_{2}+\delta_{1},$ for some $\delta_{1},\delta_{2}\in(0,1).$ In
particular, a sufficient condition for Condition 4(ii) to hold is
$\gamma>\alpha\beta,$ and
$\widetilde{\gamma}\nu<\frac{1}{2(\gamma-\alpha\beta)}.$ Under this condition,
one can consider, for instance, $\gamma d-\alpha\beta d>\gamma d\delta_{2},$
and
$1-2\gamma\widetilde{\gamma}\nu+2\alpha\beta\widetilde{\gamma}\nu>\delta_{1},$leading
to $0<\delta_{2}<1-\frac{\alpha\beta}{\gamma}<1,$
$0<\delta_{1}<1-2\nu\widetilde{\gamma}(\gamma-2\alpha\beta)<1.$

##  5 Reduction theorem for time varying subordination

For each fixed $T>0,$ let $G_{T}\in L_{2}(\mathbb{R},\phi(z)dz)$ such that

$G_{T}(z)=\sum_{q=0}^{\infty}\frac{\mathcal{J}_{q}(T)}{q!}H_{q}(z),\quad\mathcal{J}_{q}(T)=\int_{\mathbb{R}}G_{T}(z)H_{q}(z)\phi(z)dz.$

For example, one can consider the indicator function with moving threshold,

$G_{T}(Z(x,t))=\mathbb{I}_{Z(x,t)\geq u(T)},$

or, equivalently,

$G_{T}(z)=\mathbb{I}_{z\geq u(T)}\in L_{2}(\mathbb{R},\phi(z)dz),$

for each fixed $T>0.$ Here, $u:\mathbb{R}_{+}\rightarrow\mathbb{R}$ be such
that $u(T)\rightarrow\infty,$ as $T\rightarrow\infty.$ In this special case,
$G_{T}(z)$ admits the following expansion in the Hermite polynomial basis:

$\displaystyle\mathcal{J}_{0}(T)$ $\displaystyle=$
$\displaystyle\int_{u(T)}^{\infty}\phi(\xi)d\xi=1-\Phi(u(T))$
$\displaystyle\mathcal{J}_{q}(T)$ $\displaystyle=$
$\displaystyle\int_{u(T)}^{\infty}H_{q}(x)\phi(x)dx=H_{q-1}(u(T))\phi(u(T)),\
q\geq 1.$ (27)

In the general case, $G_{T}\in L_{2}(\mathbb{R},\phi(z)dz),$ the asymptotic
behavior of the following functionals is analyzed:

$\displaystyle A(T)$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Lambda(T)K}G_{T}(Z(x,t))dxdt=\sum_{n=0}^{\infty}\frac{\mathcal{J}_{n}(T)}{n!}\xi_{n}(T),$
(28)

where, as before, $\Lambda(T)=T^{\gamma},\gamma\geq 0,$ and

$\xi_{n}(T)=\int_{0}^{T}\int_{T^{\gamma}K}H_{n}(Z(x,t))dxdt,\quad\forall
n\in\mathbb{N},\ T>0.$

Their mean and variances can be computed as follows:

$\displaystyle E[A(T)]$ $\displaystyle=$ $\displaystyle T^{1+\gamma
d}|K|\mathcal{J}_{0}(T).$ $\displaystyle\mbox{Var}(A(T))$ $\displaystyle=$
$\displaystyle\mathbb{E}[A(T)-\mathbb{E}[A(T)]]^{2}=\sum_{q=m}^{\infty}\frac{\mathcal{J}_{q}^{2}(T)}{(q!)^{2}}\sigma^{2}_{q,K\Lambda(T)}(T)$
(29) $\displaystyle\sigma_{q,\Lambda(T)K}^{2}(T)$ $\displaystyle=$
$\displaystyle
2q!T|K|^{2}T^{2d\gamma}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{\mathcal{D}(T^{\gamma}K)}C^{q}(z,\tau)$
$\displaystyle\hskip 170.71652pt\times\psi_{T^{\gamma}K}(z)dzd\tau.$

We assume that function $G_{T}$ has rank $m\geq 1,$ for every $T>0,$ i.e.,
Condition 3 is satisfied. The derivation of Theorem 3 below is obtained under
the following additional condition.

Condition 5. For some $m\geq 1,$

$\displaystyle\overline{\lim}_{T\to\infty}\frac{\mbox{Var}(A(T))}{\frac{\mathcal{J}_{m}^{2}(T)}{(m!)^{2}}\sigma^{2}_{m,K\Lambda(T)}(T)}\leq
1.$ (31)

###### Theorem 3

Under equation (31) in Condition 5,

$\displaystyle Y(T)$ $\displaystyle=$
$\displaystyle\frac{\left[A(T)-\mathbb{E}[A(T)]\right]}{|\mathcal{J}_{m}(T)|\sigma_{m,T^{\gamma}K}(T)/m!}$
(33) and $\displaystyle
Y_{m}(T)=\frac{\mbox{sgn}(\mathcal{J}_{m}(T))\int_{0}^{T}\int_{T^{\gamma}K}H_{m}(Z(x,t))dxdt}{\sigma_{m,T^{\gamma}K}(T)}$

have the same limiting distributions as $T\to\infty$ (if one of it exists).

Proof. The proof is straightforward from Condition 5. From expression of
$\mbox{Var}(A(T))$ in equation (29),

$\mbox{Var}(A(T))\geq\frac{\mathcal{J}^{2}_{m}(T)\sigma^{2}_{m,\Lambda(T)K}(T)}{(m!)^{2}}.$
(34)

Considering $R(T)=Y(T)-Y_{m}(T),$ or, equivalently, $Y(T)=R(T)+Y_{m}(T),$
since $\mbox{Var}(Y_{m}(T))=1,$ from equation (34), under equation (31),

$1=\overline{\lim}_{T\to\infty}\frac{\mbox{Var}(A(T))}{\frac{\mathcal{J}_{m}^{2}(T)}{(m!)^{2}}\sigma^{2}_{m,K\Lambda(T)}(T)}=1+\overline{\lim}_{T\to\infty}\mbox{Var}(R(T)).$

Thus, $\lim_{T\to\infty}\mbox{Var}(R(T))=0.$

##  6 First Minkowski functional with moving level

We consider the following special case:

$\displaystyle M(T)$ $\displaystyle=$ $\displaystyle\left|\left\\{0\leq t\leq
T,\ x\in T^{\gamma}K;\ Z(x,t)\geq u(T)\right\\}\right|$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{T^{\gamma}K}\mathbb{I}_{Z(x,t)\geq u(T)}dxdt$
$\displaystyle=$ $\displaystyle\left(1-\Phi(u(T))\right)|K|T^{\gamma
d}+\sum_{q=1}^{\infty}\frac{\mathcal{J}_{q}(T)}{q!}\int_{0}^{T}\int_{T^{\gamma}K}H_{q}(Z(x,t))dxdt,$

where

$\mathcal{J}_{q}(T)=\phi(u(T))H_{q-1}(u(T)),$

and $u(T)$ is continuous function such that $u(T)\to\infty,$ as $T\to\infty.$

In the next result, the following condition is assumed:

Condition 6. Assume that Condition 4 is satisfied, and $u(T)$ is such that
$u^{2}(T)=o(log(T)),$ and $u^{2}(T)\sup_{(z,\tau)\in
B_{T}^{\beta_{1},\beta_{2}}}C(z,\tau)\to 0,$ as $T\to\infty,$ where

$B_{T}^{\beta_{1},\beta_{2}}=\\{(z,\tau);\ \tau\geq T^{\beta_{1}}\ \mbox{or}\
z\geq T^{\gamma\beta_{2}}\\},$ (36)

for some $\beta_{1}\in(0,\delta_{1}),$ $\beta_{2}\in(0,\delta_{2}).$

###### Remark 2

Note that, for $\Lambda(T)=T^{\gamma},$ $\gamma\geq 0,$
$u^{2}(T)=o(\log(T[\Lambda(T)]^{d}))=o((\gamma d+1)\log(T)),$ $T\to\infty$ if
and only if $u^{2}(T)=o(log(T)).$ In particular, for any $\varepsilon_{i}>0,$
$i=1,2,$
$u^{2}(T)=o(\log(T^{\varepsilon_{1}}[\Lambda(T)]^{d\varepsilon_{2}}))=o((\gamma
d\varepsilon_{2}+\varepsilon_{1})\log(T))=o(\log(T)).$

###### Theorem 4

Under Conditions 1–3, and Condition 6, as $T\rightarrow\infty,$ the random
variables

$\frac{M(T)-T^{1+\gamma
d}|K|(1-\Phi(u(T)))}{\phi(u(T))\left[2T|K|^{2}T^{2\gamma
d}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{\mathcal{D}(T^{\gamma}K)}C(z,\tau)\psi_{\rho_{T^{\gamma}K}}(z,\tau)dzd\tau\right]^{1/2}}$

and

$\frac{\int_{0}^{T}\int_{T^{\gamma}K}Z(x,t)dxdt}{\left[2T|K|^{2}T^{2d\gamma}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{\mathcal{D}(T^{\gamma}K)}C(z,\tau)\psi_{\rho_{T^{\gamma}K}}(z,\tau)dzd\tau\right]^{1/2}}$

have the same asymptotic distribution, that is, a standard normal
distribution, where $M(T)$ has been introduced in (6).

Proof. It is known that for bivariate normal density (see, e.g., equation
10.8.3 in Cramer and Leadbetter [8])

$\displaystyle\phi(x,y,\rho)=\frac{1}{2\pi\sqrt{1-\rho^{2}}}\exp\left(-\frac{1}{2(1-\rho^{2})}(x^{2}+y^{2}-2\rho
xy)\right),$ (37)

the following identity holds:

$\displaystyle\int_{u(T)}^{\infty}\int_{u(T)}^{\infty}\phi(x,y,\rho)dxdy$
$\displaystyle=\left(\int_{u(T)}^{\infty}\phi(y)dy\right)^{2}+\frac{1}{2\pi}\int_{0}^{\rho}\exp\left(-\frac{u^{2}(T)}{1+v}\right)\frac{dv}{\sqrt{1-v^{2}}}.$
(38)

Here, $\rho=\widetilde{C}(\|x-y\|,|t-s|),$ and we know that

$E[M(T)]=(1-\Phi(u(T)))|K|T^{1+\gamma d}.$

From equation (38),

$\displaystyle E[M^{2}(T)]=\int_{[0,T]\times[0,T]}\int_{T^{\gamma}K\times
T^{\gamma}K}\mathbb{E}\left[\mathbb{I}_{Z(x,t)>u(T)}\mathbb{I}_{Z(y,s)>u(T)}\right]dxdydtds$
$\displaystyle=\int_{[0,T]\times[0,T]}\int_{T^{\gamma}K\times
T^{\gamma}K}\int_{u(T)}^{\infty}\int_{u(T)}^{\infty}\phi(u,w,\rho)dudwdxdydtds$
$\displaystyle=T^{2+2d\gamma}|K|^{2}\left[\int_{u(T)}^{\infty}\phi(u)du\right]^{2}$
$\displaystyle+\frac{1}{2\pi}\int_{0}^{T}\int_{0}^{T}\int_{T^{\gamma}K\times
T^{\gamma}K}\int_{0}^{\widetilde{C}(\|x-y\|,|t-s|)}\exp\left(-\frac{u^{2}(T)}{1+v}\right)\frac{dv}{\sqrt{1-v^{2}}}dxdydtds$
$\displaystyle=T^{2+2d\gamma}|K|^{2}[1-\Phi(u(T))]^{2}$
$\displaystyle+\frac{1}{2\pi}\int_{0}^{T}\int_{0}^{T}\int_{T^{\gamma}K\times
T^{\gamma}K}\int_{0}^{\widetilde{C}(\|x-y\|,|t-s|)}\exp\left(-\frac{u^{2}(T)}{1+v}\right)\frac{dv}{\sqrt{1-v^{2}}}dxdydtds.$

Thus,

$\displaystyle\mbox{Var}(M(T))$ $\displaystyle=$ $\displaystyle
2T|K|^{2}T^{2\gamma
d}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{\mathcal{D}(T^{\gamma}K)}z^{d-1}\widetilde{\psi}_{\rho_{T^{\gamma}K}}(z)$
$\displaystyle\times$
$\displaystyle\frac{1}{2\pi}\int_{0}^{C(z,\tau)}\exp\left(-\frac{u^{2}(T)}{1+v}\right)\frac{dv}{\sqrt{1-v^{2}}}dzd\tau,$

where
$\widetilde{\psi}_{\rho_{T^{\gamma}K}}(z)=(z^{-(d-1)})\psi_{\rho_{T^{\gamma}K}}(z),$
with $\psi_{\rho_{T^{\gamma}K}}(z)$ being, as before, the probability density
of the random variable $Z=\rho_{K}=\|P_{1}-P_{2}\|,$ where $P_{1}$ and $P_{2}$
are two independent random points with uniform distribution in $T^{\gamma}K$
(see also Remark 1). The proof is based on verifying that Condition 5 holds
under the assumptions made, and hence, applied Theorem 3 for $m=1.$
Specifically, we prove that

$\overline{\lim}_{T\to\infty}\frac{\mbox{Var}(M(T))}{\mathcal{J}_{1}^{2}(T)\sigma^{2}_{1,KT^{\gamma}}(T)}\leq
1.$

Note that
$\mathcal{J}_{1}(T)=\phi(u(T))=\frac{1}{\sqrt{2\pi}}\exp(-(u^{2}(T))/2)>0,$
and

$\displaystyle\frac{\exp\left(-\frac{u^{2}(T)}{1+v}\right)}{\phi^{2}(u(T))}$
$\displaystyle=$ $\displaystyle
2\pi\exp\left(u^{2}(T)\left(\frac{v}{1+v}\right)\right).$

Hence, from equations (6) and (6),

$\displaystyle\frac{\mbox{Var}(M(T))}{[\phi(u(T))]^{2}}=2T|K|^{2}T^{\gamma
2d}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{\mathcal{D}(T^{\gamma}K)}z^{d-1}\widetilde{\psi}_{\rho_{T^{\gamma}K}}(z)$
$\displaystyle\hskip
56.9055pt\times\int_{0}^{C(z,\tau)}\exp\left(u^{2}(T)\left(\frac{v}{1+v}\right)\right)\frac{dv}{\sqrt{1-v^{2}}}dzd\tau$
$\displaystyle=2T|K|^{2}T^{2\gamma
d}\left[\int_{\overline{B}_{T}^{\beta_{1},\beta_{2}}}+\int_{B_{T}^{\beta_{1},\beta_{2}}}\right]\left(1-\frac{\tau}{T}\right)z^{d-1}\widetilde{\psi}_{\rho_{T^{\gamma}K}}(z)$
$\displaystyle\hskip
56.9055pt\times\int_{0}^{C(z,\tau)}\exp\left(u^{2}(T)\left(\frac{v}{1+v}\right)\right)\frac{dv}{\sqrt{1-v^{2}}}dzd\tau$
$\displaystyle=S_{1}(T)+S_{2}(T),$ (42)

where the set $B_{T}^{\beta_{1},\beta_{2}}$ has been introduced in equation
(36), and the set $\overline{B}_{T}^{\beta_{1},\beta_{2}}$ is defined as

$\overline{B}_{T}^{\beta_{1},\beta_{2}}=\left\\{(z,\tau);\ 0\leq\tau\leq
T^{\beta_{1}},\ 0\leq z\leq T^{\gamma\beta_{2}}\right\\},$

with $\beta_{1}\in(0,\delta_{1}),$ $\beta_{2}\in(0,\delta_{2}).$

From Condition 6, applying Remark 2,
$u^{2}(T)=o\left(\log((T)^{\varepsilon_{1}}([\Lambda(T)]^{d})^{\varepsilon_{2}})\right),$
for any $\varepsilon_{1},\varepsilon_{2}>0.$ Keeping in mind that
$\frac{v}{1+v}\leq 1,$ we obtain

$\displaystyle S_{1}(T)\leq k_{8}\ T^{1+2\gamma
d}\int_{\overline{B}_{T}^{\beta_{1},\beta_{2}}}\left(1-\frac{\tau}{T}\right)z^{d-1}\widetilde{\psi}_{\rho_{T^{\gamma}K}}(z)$
$\displaystyle\hskip
56.9055pt\times\exp\left(u^{2}(T)\right)\int_{0}^{C(z,\tau)}\frac{dv}{\sqrt{1-v^{2}}}dzd\tau$
$\displaystyle=k_{8}\ T^{1+2\gamma
d}\exp\left(u^{2}(T)\right)\int_{\overline{B}_{T}^{\beta_{1},\beta_{2}}}\left(1-\frac{\tau}{T}\right)z^{d-1}\widetilde{\psi}_{\rho_{T^{\gamma}K}}(z)$
$\displaystyle\hskip 56.9055pt\times\mbox{arc sin}C(z,\tau)dzd\tau$
$\displaystyle\leq k_{8}\
T^{1+\varepsilon_{1}}T^{2d\gamma+d\gamma\varepsilon_{2}}\int_{\overline{B}_{T}^{\beta_{1},\beta_{2}}}\left(1-\frac{\tau}{T}\right)z^{d-1}\widetilde{\psi}_{\rho_{T^{\gamma}K}}(z)dzd\tau$
$\displaystyle\leq
k_{9}T^{1+\beta_{1}+\varepsilon_{1}}[\Lambda(T)]^{d+d\beta_{2}+d\varepsilon_{2}},$
(43)

for some positive constants $k_{8},$ and $k_{9}.$ In particular, for
$\varepsilon_{i}=\delta_{i}-\beta_{i},$ $i=1,2,$ under Condition 4(ii), as
$T\to\infty,$

$\displaystyle\frac{S_{1}(T)}{\sigma_{1,K}^{2}(T)}\leq\frac{k_{9}}{\frac{1}{T^{1+\delta_{1}}T^{\gamma
d(1+\delta_{2})}}\sigma_{1,K}^{2}(T)}\to 0.$ (44)

Let us know consider $S_{2}(T),$ under Condition 6,

$\displaystyle\sup_{(z,\tau)\in
B_{T}^{\beta_{1},\beta_{2}}}C(z,\tau)\rightarrow 0,\quad T\rightarrow\infty$
$\displaystyle B_{T}^{\beta_{1},\beta_{2}}=\\{(z,\tau);\ \tau\geq
T^{\beta_{1}}\ \mbox{or}\ z\geq T^{\gamma\beta_{2}}\\},$

and, since $\frac{1}{1+v}\leq 1$ and $\frac{1}{\sqrt{1-v^{2}}}\to 1,$ as
$T\to\infty,$ we obtain in $S_{2}(T)$,

$\displaystyle S_{2}(T)=2T^{1+2\gamma
d}|K|^{2}\int_{B_{T}^{\beta_{1},\beta_{2}}}\left(1-\frac{\tau}{T}\right)\psi_{\rho_{T^{\gamma}K}}(z)$
$\displaystyle\hskip
99.58464pt\times\int_{0}^{C(z,\tau)}\exp\left(u^{2}(T)\frac{v}{1+v}\right)\frac{dv}{\sqrt{1-v^{2}}}dzd\tau$
$\displaystyle\leq 2T^{1+2\gamma
d}|K|^{2}\int_{B_{T}^{\beta_{1},\beta_{2}}}\left(1-\frac{\tau}{T}\right)\psi_{\rho_{T^{\gamma}K}}(z)$
$\displaystyle\hskip
99.58464pt\times\int_{0}^{C(z,\tau)}\exp\left(u^{2}(T)v\right)\frac{dv}{\sqrt{1-v^{2}}}dzd\tau$
$\displaystyle\leq k_{10}2T^{1+2\gamma
d}|K|^{2}\exp\left(u^{2}(T)\sup_{(z,\tau)\in
B_{T}^{\beta_{1},\beta_{2}}}C(z,\tau)\right)$ $\displaystyle\hskip
99.58464pt\times\int_{B_{T}^{\beta_{1},\beta_{2}}}\left(1-\frac{\tau}{T}\right)\psi_{\rho_{T^{\gamma}K}}(z)C(z,\tau)dzd\tau,$

for some positive constant $k_{10}.$ Hence, for $T$ sufficiently large,

$\displaystyle\frac{S_{2}(T)}{\sigma_{1,K}^{2}(T)}\leq
k_{11}\exp\left(u^{2}(T)\sup_{(z,\tau)\in
B_{T}^{\beta_{1},\beta_{2}}}C(z,\tau)\right)$ $\displaystyle\hskip
42.67912pt\times\frac{\int_{B_{T}^{\beta_{1},\beta_{2}}}\left(1-\frac{\tau}{T}\right)\psi_{\rho_{T^{\gamma}K}}(z)C(z,\tau)dzd\tau}{\int_{\overline{B}_{T}^{\beta_{1},\beta_{2}}\cup
B_{T}^{\beta_{1},\beta_{2}}}\left(1-\frac{\tau}{T}\right)\psi_{\rho_{T^{\gamma}K}}(z)C(z,\tau)dzd\tau}\leq
1,$ (45)

since under Condition 6,

$\exp\left(u^{2}(T)\sup_{(z,\tau)\in
B_{T}^{\beta_{1},\beta_{2}}}C(z,\tau)\right)\to 1,\quad T\to\infty,$

and $k_{11}\leq 1,$ for sufficiently large $T.$ Finally, from (44)–(45),

$\displaystyle\vspace*{-3cm}\overline{lim}_{T\to\infty}\frac{\mbox{Var}(M(T))}{\phi^{2}(u(T))\sigma_{1,K}^{2}(T)}$
$\displaystyle=\overline{lim}_{T\to\infty}\left(\frac{S_{1}(T)}{\sigma_{1,K}^{2}(T)}+\frac{S_{2}(T)}{\sigma_{1,K}^{2}(T)}\right)\leq
1.$ (46)

Thus, the desired result follows from Theorem 3.

##  7 Sojourn functionals for a class of spherical random fields

This section derives a central limit result for sojourn functionals
subordinated to spatiotemporal Gaussian random fields homogeneous and
isotropic in space, and stationary in time, restricted to the unit sphere.

Let $\mathbb{S}_{d-1}=\\{x\in\mathbb{R}^{d};\ \|x\|=1\\}$ be the unit sphere
embedded into $\mathbb{R}^{d},$ for some $d\geq 2,$ and denote by
$d\nu_{d-1}(x)$ the normalized Riemannian measure on $\mathbb{S}_{d-1}.$
Denote by $\theta=\arccos\left(\left\langle x,x^{\prime}\right\rangle\right)$
the angle between two points $x,x^{\prime}\in\mathbb{S}_{d-1}.$ Hence, the
Euclidean distance is given by
$\|x-x^{\prime}\|=2\sin\left(\frac{\theta}{2}\right).$

Let $\left\\{Z(x,t),\ x\in\mathbb{R}^{d},\ t\in\mathbb{R}\right\\}$ be a
zero–mean Gaussian mean–square continuous spatiotemporal random field,
homogeneous and isotropic in space, and stationary in time, with covariance
function

$\displaystyle\widetilde{C}(\|x\|,\tau)=2^{\frac{d-2}{2}+1}\Gamma\left(\frac{d}{2}\right)\int_{0}^{\infty}\cos\left(\mu\tau\right)\int_{0}^{\infty}\frac{J_{\frac{d-2}{2}}\left(\lambda\|x\|\right)}{\left(\lambda\|x\|\right)^{\frac{d-2}{2}}}G(d\lambda,d\mu).$
(47)

Thus, its restriction to unit sphere $\mathbb{S}_{d-1}(1),$
$\left\\{T_{R}(x,t),\ x\in\mathbb{S}_{d-1}(1),\ t\in\mathbb{R}\right\\},$ has
covariance function

$\displaystyle
C_{R}(\cos(\theta),\tau)=2^{d/2}\Gamma\left(\frac{d}{2}\right)\int_{0}^{\infty}\int_{0}^{\infty}\cos(\mu\tau)$
$\displaystyle\hskip
56.9055pt\times\int_{0}^{\infty}\frac{J_{\frac{d-2}{2}}(2\lambda\sin\left(\theta/2\right))}{\left[2\lambda\sin\left(\theta/2\right)\right]^{\frac{d-2}{2}}}G(d\lambda,d\mu).$
(48)

Here, the spectral measure $F$ of $Z$ satisfies

$\displaystyle\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}F(d\widetilde{\omega},d\widetilde{\mu})$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}G(d\lambda,d\mu)<\infty,$
$\displaystyle G(\lambda,\mu)$ $\displaystyle=$
$\displaystyle\int_{\|\widetilde{\omega}\|<\lambda}\int_{|\widetilde{\mu}|<\mu}F(d\widetilde{\omega},d\widetilde{\mu}).$

_Condition K._ Assume that, as before, $\left\\{T_{R}(x,t),\
x\in\mathbb{S}_{d-1},\ t\in\mathbb{R}\right\\}$ be a zero–mean Gaussian
spherical isotropic spatiotemporal random field defined as the restriction to
$\mathbb{S}_{d-1}$ of a spatiotemporal, stationary in time, and spatially
homogeneous and isotropic random field, with covariance function (48).

Let now consider the first Minkowski functional subordinated to $T_{R}(x,t),$
given by

$\displaystyle
N_{T}=\int_{0}^{T}\int_{\mathbb{S}_{d-1}}\mathbb{I}_{T_{R}(x,t)\geq
u}d\nu_{d-1}(x)dt$ (49) $\displaystyle=$ $\displaystyle\left|\left\\{0\leq
t\leq T;\ T_{R}(x,t)\geq u,\ x\in\mathbb{S}^{d-1}\right\\}\right|$
$\displaystyle=$ $\displaystyle E[N_{T}]+\sum_{n\geq
1}\frac{\mathcal{J}_{n}}{n!}\xi_{n}(T),$

where $E[N_{T}]=(1-\Phi(u))\frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)},$
and $\mathcal{J}_{n}(u)=\phi(u)H_{n-1}(u),$ $n\geq 1,$ and

$\eta_{n}(T)=\int_{0}^{T}\int_{\mathbb{S}_{d-1}}H_{n}(T_{R}(x,t))d\nu_{d-1}(x)dt.$

Thus, $E[\eta_{n}(T)]=0,$ $E[\eta_{n}(T)\eta_{l}(T)]=0,$ $n\neq l,$ and

$\displaystyle\sigma_{n}^{2}(T)=E[\eta_{n}^{2}(T)]$ $\displaystyle=$
$\displaystyle
2n!\int_{0}^{T}\int_{0}^{T}\int_{\mathbb{S}_{d-1}\times\mathbb{S}_{d-1}}\widetilde{C}^{n}\left(\|x-x^{\prime}\|,|t-t^{\prime}|\right)d\nu(x)d\nu(x^{\prime})dtdt^{\prime}$
$\displaystyle=$ $\displaystyle
2n!T|\mathbb{S}_{d-1}|^{2}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)E\left(\widetilde{C}^{n}(\|W_{1}-W_{2}\|,\tau)\right)d\tau$
$\displaystyle=$ $\displaystyle
2n!T|\mathbb{S}_{d-1}|^{2}\frac{1}{\sqrt{\pi}}\Gamma\left(\frac{d}{2}\right)\Gamma^{-1}\left(\frac{d-1}{2}\right)$
$\displaystyle\hskip
28.45274pt\times\int_{0}^{T}\int_{0}^{2}\left(1-\frac{\tau}{T}\right)z^{d-2}\left(1-\frac{z^{2}}{4}\right)^{\frac{d-3}{2}}C^{n}(z,\tau)dzd\tau$
$\displaystyle=$ $\displaystyle
2n!T4\pi^{d-1/2}\left[\Gamma\left(\frac{d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)\right]^{-1}$
$\displaystyle\hskip
28.45274pt\times\int_{0}^{T}\int_{0}^{2}\left(1-\frac{\tau}{T}\right)z^{d-2}\left(1-\frac{z^{2}}{4}\right)^{\frac{d-3}{2}}C^{n}(z,\tau)dzd\tau,$

where $W_{1}$ and $W_{2}$ are two independent uniformly distributed random
vectors on $\mathbb{S}_{d-1}$ with probability density of its Euclidean
distance given by, for $0\leq z\leq 2$ (see Lemma 1.4.4 in Ivanov and Leonenko
[14])

$\frac{d}{dz}P\left[\|W_{1}-W_{2}\|\leq
z\right]=\frac{1}{\sqrt{\pi}}\Gamma\left(\frac{d}{2}\right)\Gamma^{-1}\left(\frac{d-1}{2}\right)z^{d-2}\left(1-\frac{z^{2}}{4}\right)^{\frac{d-3}{2}}.$
(51)

_Condition L_.

  * (i)

Assume that $\sup_{z\in[0,2]}C(z,\tau)\to 0$ as $\tau\to\infty.$

  * (ii)

There exists $\delta\in(0,1),$ such that

$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{T^{\delta}}\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2}z^{d-2}C(z,\tau)\left(1-\frac{z^{2}}{4}\right)^{\frac{d-3}{2}}dzd\tau=\infty.$

###### Remark 3

Note that, for the restriction to the sphere of a spatiotemporal Gaussian
random field with covariance function as in Example 1, Condition L holds if
$d\geq 3.$

Under _Conditions K and L_ , applying the corresponding reduction theorem, the
following central limit results follow.

###### Theorem 5

Assume that _Conditions K_ and _L_ hold. Then, the random variable

$\frac{N_{T}-T(1-\Phi(u))|\mathbb{S}_{d-1}|}{\phi(u)\left[\frac{8\pi^{d-\frac{1}{2}}}{\Gamma\left(\frac{d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)}T\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2}C(z,\tau)z^{d-2}\left(1-\frac{z^{2}}{4}\right)^{\frac{d-3}{2}}dzd\tau\right]^{1/2}}$

has asymptotically standard normal distribution as $T\rightarrow\infty.$

For moving levels when $u(T)\to\infty$ as $T\to\infty,$ we have the following
central limit result.

###### Theorem 6

Assume that _Conditions_ K and L hold, and there exists a $\beta\in(0,\delta)$
such that as $T\rightarrow\infty,$

$u^{2}(T)\sup_{u\in(0,2)}C(u,T^{\beta})\rightarrow 0,\quad
u^{2}(T)=o(\log(T)).$

Then, as $T\rightarrow\infty,$ the random variable

$\frac{N_{T}^{\star}-T(1-\Phi(u(T))|\mathbb{S}_{d-1}|}{\phi(u(T))\left[\frac{8\pi^{d-\frac{1}{2}}}{\Gamma\left(\frac{d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)}T\int_{0}^{T}\left(1-\frac{\tau}{T}\right)\int_{0}^{2}C(z,\tau)z^{d-2}\left(1-\frac{z^{2}}{4}\right)^{\frac{d-3}{2}}dzd\tau\right]^{1/2}},$

has asymptotically a standard normal distribution, where

$N_{T}^{\star}=\left|\left\\{0\leq t\leq T;\ T_{R}(x,t)\geq u(T),\
x\in\mathbb{S}_{d-1}\right\\}\right|.$

The proofs of Theorems 5 and 6 can be respectively obtained from equations
(LABEL:eqchaoscomp) and (51), in a similar way to the proofs of Theorems 2 and
4.

###### Remark 4

Special cases of stationary covariance functions on spheres cross time have
recently been analyzed in White and Porcu [35]. In our case, we paid special
attention to the family of nonseparable covariance functions introduced in
equation (11) in [35], providing competitive models in real data analysis
(e.g., surface air temperature modelling). Note that this covariance family
admits an integral representation of the form (47)– (48), according Schoenberg
[30] (see also equations (6)–(9) in Porcu et al. [29]).

## Acknowledgements

This work has been supported in part by projects MCIN/
AEI/PID2022-142900NB-I00, and CEX2020-001105-M MCIN/
AEI/10.13039/501100011033). Nikolai Leonenko (NL) would like to thank for
support and hospitality during the programme ”Fractional Differential
Equations” and the programme ”Uncertainly Quantification and Modelling of
Materials” in Isaac Newton Institute for Mathematical Sciences, Cambridge.
Also NL was partially supported under the ARC Discovery Grant DP220101680
(Australia), LMS grant 42997 (UK) and grant FAPESP 22/09201-8 (Brazil).

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