# Distance covariance for random fields

Muneya Matsui  Department of Business Administration, Nanzan University, 18
Yamazato-cho, Showa-ku, Nagoya 466-8673, Japan<EMAIL_ADDRESS>,  Thomas
Mikosch  Department of Mathematics, University of Copenhagen,
Universitetsparken 5, DK-2100 Copenhagen, Denmark<EMAIL_ADDRESS>,
Rasool Roozegar  Department of Statistics, Yazd University, P.O. Box
89195-741, Yazd, Iran<EMAIL_ADDRESS>and  Laleh Tafakori  Department of
Mathematical Sciences, School of Science, RMIT University, Melbourne, VIC,
Australia<EMAIL_ADDRESS>
###### Abstract.

We study an independence test based on distance correlation for random fields
$(X,Y)$. We consider the situations when $(X,Y)$ is observed on a lattice with
equidistant grid sizes and when $(X,Y)$ is observed at random locations. We
provide asymptotic theory for the sample distance correlation in both
situations and show bootstrap consistency. The latter fact allows one to build
a test for independence of $X$ and $Y$ based on the considered discretizations
of these fields. We illustrate the performance of the bootstrap test in a
simulation study involving fractional Brownian and infinite variance stable
fields. The independence test is applied to Japanese meteorological data,
which are observed over the entire area of Japan.
Keywords: Empirical characteristic function, distance covariance, random
field, independence test

###### 1991 Mathematics Subject Classification:

Primary 62E20; Secondary 62G20 62M99 60F05 60F25

##  1\. Introduction to model

###  1.1. Distance covariance in Euclidean space

It is well known that two $q$\- and $r$-dimensional random vectors ${\bf X}$
and ${\bf Y}$, respectively, are independent if and only if their joint
characteristic function factorizes, i.e.,

$\displaystyle\varphi_{{\bf X},{\bf Y}}({\bf s},{\bf t})$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\big{[}\exp(i\,{\bf s}^{\top}{\bf X}+i{\bf
t}^{\top}{\bf Y})\big{]}$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\big{[}\exp(i\,{\bf s}^{\top}{\bf
X})\big{]}\,{\mathbb{E}}\big{[}\exp(i\,{\bf t}^{\top}{\bf Y})\big{]}$
$\displaystyle=$ $\displaystyle\varphi_{\bf X}({\bf s})\,\varphi_{\bf Y}({\bf
t})\,,\qquad{\bf s}\in{\mathbb{R}}^{q}\,,{\bf t}\in{\mathbb{R}}^{r}\,.$

However, this identity is difficult to check if one has data at the disposal;
a replacement of the corresponding characteristic functions by empirical
versions does not lead to powerful statistical tools for detecting
independence between ${\bf X}$ and ${\bf Y}$. In the univariate case, Klebanov
and Zinger [11], Feuerverger [8], and later Székely et al. [20, 17, 18, 19] in
the general multivariate case recommended to use a weighted $L^{2}$-distance
between $\varphi_{{\bf X},{\bf Y}}$ and $\varphi_{\bf X}\,\varphi_{\bf Y}$:
for $\beta\in(0,2)$, the distance covariance between ${\bf X}$ and ${\bf Y}$
is given by

$\displaystyle T_{\beta}({\bf X},{\bf
Y})=c_{q}c_{r}\int_{{\mathbb{R}}^{q+r}}\big{|}\varphi_{{\bf X},{\bf Y}}({\bf
s},{\bf t})-\varphi_{\bf X}({\bf s})\varphi_{\bf Y}({\bf t})\big{|}^{2}|{\bf
s}|^{-(q+\beta)}|{\bf t}|^{-(r+\beta)}\,d{\bf s}\,d{\bf t}\,,$

where the constants $c_{d}$ for $d\geq 1$ are chosen such that

(1.1) $\displaystyle c_{d}\,\int_{{\mathbb{R}}^{d}}(1-\cos({\bf
s}^{\prime}{\bf x}))\,|{\bf x}|^{-(d+\beta)}d{\bf x}=|{\bf s}|^{\beta}\,.$

Here and in what follows, we suppress the dependence of the Euclidean norm
$|\cdot|$ on the dimension; it will always be clear from the context what the
dimension is.

The quantity $T_{\beta}({\bf X},{\bf Y})$ is finite under suitable moment
conditions on ${\bf X},{\bf Y}$. The corresponding distance correlation is
given by

(1.2) $\displaystyle R_{\beta}({\bf X},{\bf Y})=\dfrac{T_{\beta}({\bf X},{\bf
Y})}{\sqrt{T_{\beta}({\bf X},{\bf X})}\sqrt{T_{\beta}({\bf Y},{\bf Y})}}\,.$

Of course, ${\bf X}$ and ${\bf Y}$ are independent if and only if
$R_{\beta}({\bf X},{\bf Y})=T_{\beta}({\bf X},{\bf Y})=0$.

Thanks to the choice of the weight function $|{\bf s}|^{-(q+\beta)}|{\bf
t}|^{-(r+\beta)}$ and (1.1), $T_{\beta}({\bf X},{\bf Y})$ has an explicit
form: assuming that $({\bf X}_{i},{\bf Y}_{i})$, $i=1,2,\ldots,$ are iid
copies of $({\bf X},{\bf Y})$, we have

(1.3) $\displaystyle T_{\beta}({\bf X},{\bf Y})$ $\displaystyle=$
$\displaystyle{\mathbb{E}}[|{\bf X}_{1}-{\bf X}_{2}|^{\beta}|{\bf Y}_{1}-{\bf
Y}_{2}|^{\beta}]+{\mathbb{E}}[|{\bf X}_{1}-{\bf
X}_{2}|^{\beta}]{\mathbb{E}}[|{\bf Y}_{1}-{\bf Y}_{2}|^{\beta}]$
$\displaystyle-2\,{\mathbb{E}}[|{\bf X}_{1}-{\bf X}_{2}|^{\beta}|{\bf
Y}_{1}-{\bf Y}_{3}|^{\beta}]\,,$

and $R_{\beta}(c{\bf X},c{\bf Y})=R_{\beta}({\bf X},{\bf Y})$ for
$c\in{\mathbb{R}}$, i.e., $R_{\beta}$ is scale-invariant.

###  1.2. Distance covariance for random fields on a lattice in $[0,1]^{d}$

Székely et al. [18] showed that distance correlation fails for high-
dimensional vectors ${\bf X},{\bf Y}$ if their components are independent.
Therefore Matsui et al. [13] and Dehling et al. [5] required some dependence
structure on the components. They extended distance covariance and distance
correlation to stochastic processes. In particular, [5] studied
discretizations of stochastic processes $X,Y$ on $[0,1]$. Instead of using the
vectors $(X(t_{i}))_{i=1,\ldots,p}$ and $(Y(t_{i}))_{i=1,\ldots,p}$ for
partitions $t_{0}=0<t_{1}<\cdots<t_{p}=1$, $\Delta_{i}=(t_{i-1},t_{i}]$ such
that $p=p_{n}\to\infty$ and
$\delta_{n}=\max_{i}|\Delta_{i}|=\max_{i}(t_{i}-t_{i-1})\to 0$ as
$n\to\infty$, one introduces the weighted vectors

(1.4) $\displaystyle{\bf
X}^{(p)}=\big{(}\sqrt{t_{i}-t_{i-1}}\,X(t_{i}))_{i=1,\ldots,p}\quad\mbox{and}\quad{\bf
Y}^{(p)}=\big{(}\sqrt{t_{i}-t_{i-1}}\,Y(t_{i}))_{i=1,\ldots,p}.$

Here we introduce a direct analog of this approach to $[0,1]^{d}$ for $d>1$.
We assume that any random field $Z=(Z({\bf u}))_{{\bf u}\in[0,1]^{d}}$ of
interest is observed on a lattice with constant mesh size. We start by
partitioning $[0,1]$ into equidistant points $t_{i}=i/q$, $i=0,1,\ldots,q$,
for some positive integer $q$. From them we construct a lattice in $[0,1]^{d}$
via the points

$\displaystyle{\bf t}_{{\bf i}}=(t_{i_{1}},\ldots,t_{i_{d}}),\qquad{\bf
i}=(i_{1},\ldots,i_{d})\in\\{0,1,\ldots,q\\}^{d}\,.$

We have $p:=q^{d}$ lattice points in $(0,1]^{d}$. We discretize $Z$ on the
cells $\Delta_{{\bf i}}=({\bf t}_{{\bf i}-{\mathbf{1}}},{\bf t}_{\bf i}]$ with
volume

(1.5) $\displaystyle|\Delta|=|\Delta_{{\bf i}}|=p^{-1}\,.$

Consider the vector

$\displaystyle{\bf Z}_{p}=\big{(}|\Delta|^{1/2}Z({\bf t}_{{\bf
i}})\big{)}\,,\qquad{\bf i}\in\Pi_{d}=\\{1,\ldots,q\\}^{d}\,,$

and a step-function approximation to $Z$:

(1.6) $\displaystyle Z^{(p)}({\bf t})=\sum_{{\bf i}\in\Pi_{d}}Z({\bf t}_{\bf
i}){\mathbf{1}}({\bf t}\in\Delta_{\bf i}),\qquad{\bf t}\in[0,1]^{d}\,.$

For a measurable bounded square-integrable field $Z$ on $[0,1]^{d}$ we have
for a.e. sample path

(1.7) $\displaystyle|{\bf Z}_{p}|^{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{p}\,\sum_{{\bf i}\in\Pi_{d}}Z^{2}({\bf t}_{\bf
i})=\sum_{{\bf i}\in\Pi_{d}}Z^{2}({\bf t}_{{\bf i}})\,|\Delta_{{\bf
i}}|=\|Z^{(p)}\|^{2}_{2}$ $\displaystyle\to$
$\displaystyle\int_{[0,1]^{d}}Z^{2}({\bf t})\,d{\bf t}=\|Z\|_{2}^{2}\,,\qquad
p\to\infty,$

where $\|\cdot\|_{2}$ denotes the $L_{2}$-norm on $[0,1]^{d}$. Motivated by
(1.3) and this approximation, we define the distance covariance
$T_{\beta}(X,Y)$, $\beta\in(0,2)$, between two random fields $X,Y$ on some
bounded Borel set $B\subset{\mathbb{R}}^{d}$ of finite positive Lebesgue
measure by

(1.8) $\displaystyle T_{\beta}(X,Y)$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\big{[}\|X_{1}-X_{2}\|_{2}^{\beta}\,\|Y_{1}-Y_{2}\|_{2}^{\beta}\big{]}+{\mathbb{E}}\big{[}\|X_{1}-X_{2}\|_{2}^{\beta}\big{]}\,{\mathbb{E}}\big{[}\|Y_{1}-Y_{2}\|_{2}^{\beta}\big{]}$
$\displaystyle-2\,{\mathbb{E}}\big{[}\|X_{1}-X_{2}\|_{2}^{\beta}\|Y_{1}-Y_{3}\|_{2}^{\beta}\big{]}\,,$

where $(X_{i},Y_{i})$, $i=1,2,\ldots,$ are iid copies of $(X,Y)$, and the
distance correlation $R_{\beta}(X,Y)$ is defined correspondingly. (The
$L^{2}$-norm $\|\cdot\|_{2}$ is to be understood on $B$.) A further motivation
is the following fact:

######  Lemma 1.1.

Assume that $\beta\in(0,2)$, $X,Y$ are random fields on $B$ which are square-
integrable, bounded, stochastically continuous and
${\mathbb{E}}[\|X\|_{2}^{\beta}+\|Y\|_{2}^{\beta}+\|X\|_{2}^{\beta}\|Y\|_{2}^{\beta}]<\infty.$
Then $T_{\beta}(X,Y)=0$ if and only if $X,Y$ are independent.

For $\beta\leq 1$ the statement follows from Lyons [12], who also addressed
the more general problem of distance covariance in metric spaces, and for
$\beta\in(0,2)$ a straightforward modification of the proofs in Dehling et al.
[5] for random fields on general Borel sets $B$ yields the result.

Sample versions of $T_{\beta}$ and $R_{\beta}$ are given by

$\displaystyle\qquad T_{n,\beta}(X,Y)$ $\displaystyle=$
$\displaystyle\frac{1}{n^{2}}\sum_{k,l=1}^{n}\|X_{k}-X_{l}\|_{2}^{\beta}\|Y_{k}-Y_{l}\|_{2}^{\beta}+\frac{1}{n^{2}}\sum_{k,l=1}^{n}\|X_{k}-X_{l}\|_{2}^{\beta}\frac{1}{n^{2}}\sum_{k,l=1}^{n}\|Y_{k}-Y_{l}\|_{2}^{\beta}$
$\displaystyle\qquad-2\frac{1}{n^{3}}\sum_{k,l,m=1}^{n}\|X_{k}-X_{l}\|_{2}^{\beta}\|Y_{k}-Y_{m}\|_{2}^{\beta}\,,$
(1.10) $\displaystyle\qquad R_{n,\beta}(X,Y)$ $\displaystyle=$
$\displaystyle\dfrac{T_{n,\beta}(X,Y)}{\sqrt{T_{n,\beta}(X,X)T_{n,\beta}(Y,Y)}}\,.$

Since $T_{n,\beta}$ is a $V$-statistic, under suitable moment conditions
$T_{n,\beta}(X,Y)$ and $R_{n,\beta}(X,Y)$ are consistent estimators of
$T_{\beta}(X,Y)$ and $R_{\beta}(X,Y)$, respectively, and if $X,Y$ are
independent, one also has

$\displaystyle n\,R_{n,\beta}(X,Y)\stackrel{{\scriptstyle
d}}{{\rightarrow}}\sum_{i=1}^{\infty}\lambda_{i}\,(N_{i}^{2}-1)+{\rm const},$

for a square-summable real sequence $(\lambda_{i})$, iid $N(0,1)$ random
variables $(N_{i})$. These asymptotic results in combination with the fact
that $T_{\beta}(X,Y)=0$ if and only if $X,Y$ are independent encourage one to
build a statistical test about independence of $X,Y$ on the quantities
$T_{n,\beta}(X,Y)$.

Unfortunately, a sample of paths of $(X,Y)$ is rarely at our disposal, and so
one has to think about discretizations of $(X,Y)$. Motivated by the Riemann
sum approximation (1.7) possible choices are the lattice discretizations
$(X^{(p)},Y^{(p)})$ in (1.6) and, for an iid sample $(X_{i},Y_{i})$,
$i=1,\ldots,n$, the sample version $T_{n,\beta}(X^{(p)},Y^{(p)})$ as
replacement of the test statistic $T_{n,\beta}(X,Y)$. In agreement with [5] in
the case $d=1$ we will show that this idea can be made to work for $d>1$ if
$p=p_{n}\to\infty$.

###  1.3. Distance covariance for random fields at random locations

In this paper, we follow a second path of research.

  * •

We introduce distance covariance $T_{\beta}(X,Y)$ and $R_{\beta}(X,Y)$ for
random fields $X,Y$ on some Borel set $B\subset{\mathbb{R}}^{d}$ of positive
Lebesgue measure.

  * •

We define $T_{\beta}(X^{(p)},Y^{(p)})$ and $R_{\beta}(X^{(p)},Y^{(p)})$ for
non-lattice based discretizations $X^{(p)},Y^{(p)}$ of $X,Y$ on $B$. In
contrast to (1.4), we choose a random number $N_{p}$ of random locations
$({\bf U}_{i})$ where the processes $X,Y$ are observed. Typically, these
locations are uniformly distributed on $B$ and
$N_{p}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}\infty$ as $p$ increases
with the sample size $n$ to infinity.

The second idea has already been advocated in Matsui et al. [13]. There it was
assumed that $({\bf U}_{i})$ are the points of a Poisson process with constant
intensity $p$. For statistical purposes this means that one would have to
observe a sample of iid copies $(X_{i}^{(p)},Y_{i}^{(p)})$ of
$(X^{(p)},Y^{(p)})$ at random locations that change across $i$ and their
number would change as well. The asymptotic analysis of this setting is not
very elegant and does not lead beyond consistency of
$T_{n,\beta}(X^{(p)},Y^{(p)})$.

To be precise, we consider bounded stochastically continuous square-integrable
random fields $X,Y$ on a bounded Borel set $B\subset{\mathbb{R}}^{d}$ of
positive Lebesgue measure. We define the distance covariance $T_{\beta}(X,Y)$
between $X,Y$ as in (1.8) but the norm is now given by the (standardized)
$L^{2}$-norm on $B$:

(1.11) $\displaystyle\|f\|_{2}=\Big{(}|B|^{-1}\int_{B}f^{2}({\bf u})\,d{\bf
u}\Big{)}^{1/2}\,.$

For ease of notation, we assume without loss of generality that the Lebesgue
measure $|B|$ of $B$ is one. The distance correlation $R_{\beta}(X,Y)$ is
defined correspondingly.

Consider the point process

(1.12) $\displaystyle N^{(p)}(\cdot)=\sum_{i=1}^{N_{p}}\varepsilon_{{\bf
U}_{i}}(\cdot)=\\#\\{i\leq N_{p}:{\bf U}_{i}\in\cdot\\}\,,\qquad p>0\,,$

on the state space $B$ where $({\bf U}_{i})$ is iid uniform on $B$ independent
of the counting number $N_{p}\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}\infty$ as $p\to\infty$. If $N_{p}$ is Poisson
distributed and $p$ is fixed then $N^{(p)}$ constitutes a homogeneous Poisson
process on $B$ ; see p. 132 in Resnick [14]. In view of the order statistics
property of any homogeneous Poisson process $N$ on $B$ we also know that the
points of $N$, conditionally on $N(B)$, are iid uniform on $B$. However, the
representation (1.12) goes beyond the order statistics property since we
require the point sequence $({\bf U}_{i})$ to be the same for all $p>0$.

Recycling the notation $X^{(p)},Y^{(p)}$, the discretizations of $X,Y$ are now
given by

$\displaystyle X^{(p)}({\bf u})=\sum_{i=1}^{N_{p}}X({\bf
u}){\mathbf{1}}_{\\{{{\bf U}_{i}}\\}}({\bf u})\quad\text{and}\quad
Y^{(p)}({\bf u})=\sum_{i=1}^{N_{p}}Y({\bf u}){\mathbf{1}}_{{\\{{\bf
U}_{i}\\}}}({\bf u})\,,\qquad{\bf u}\in B\,,$

where $(N^{(p)})_{p>0}$ and $X,Y$ are independent. If $N_{p}=0$,
$X^{(p)}=Y^{(p)}=0$ by convention. Again recycling the symbol $\|\cdot\|_{2}$,
we write

(1.13) $\displaystyle\|X^{(p)}-Y^{(p)}\|_{2}^{2}:=\dfrac{1}{N}_{p}\sum_{i:{\bf
U}_{i}\in B}(X-Y)^{2}({\bf U}_{i})$ $\displaystyle=$
$\displaystyle\dfrac{1}{N}_{p}\,\int_{B}(X-Y)^{2}\,d\,N^{(p)}\,,$

with the convention that the right-hand side is zero if $N_{p}=0$. We notice
that for a.e. realization of $(N^{(p)})_{p>0}$, ${\bf U}_{1},\ldots,{\bf
U}_{N_{p}}$ are getting arbitrarily tight in $B$ since we assume
$N_{p}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}\infty$ as $p\to\infty$.
If we assume that $X,Y$ are path-wise square-integrable on $B$ and we
condition on $(N^{(p)})_{p>0}$ then for a.e. realization of $X,Y$ we have the
Riemann sum approximation $\|X^{(p)}-Y^{(p)}\|_{2}^{2}\to\|X-Y\|_{2}^{2}$.

In what follows, we define the sample versions
$T_{n,\beta}(X,Y),R_{n,\beta}(X,Y)$ in the natural way by applying the
$L^{2}$-norm (1.11) on $B$. We assume that the iid sequence $((X_{i},Y_{i}))$
is independent of $(N^{(p)})_{p>0}$. Then the random locations $({\bf U}_{i})$
are the same for any sample size $n$, and $((X_{i}^{(p)},Y_{i}^{(p)}))$ are
iid, conditional on $(N^{(p)})_{p>0}$. Moreover, we define
$T_{n,\beta}(X^{(p)},Y^{(p)}),R_{n,\beta}(X^{(p)},Y^{(p)})$ accordingly by
using the notation in (1.13).

The random discretizations of $X,Y$ have some advantages over the lattice
case:

  * •

they can be defined on quite general sets $B$,

  * •

the random fields can be observed on irregularly spaced locations,

  * •

the smoothness of the field does not play a significant role for the
asymptotic theory of the sample distance covariance.

###  1.4. Organization of the paper

We provide the necessary asymptotic theory for the two aforementioned methods:

  1. (i)

in Section 2 for random locations and increasing intensity $p$ on a bounded
Borel set $B\subset{\mathbb{R}}^{d}$ of positive Lebesgue measure,

  2. (ii)

in Section 3 for the lattice case on $B=[0,1]^{d}$ for increasing $p$ which is
the total (deterministic) number of grid points.

  3. (iii)

in Section 4 we combine the two methods by averaging the observations in each
cell of the regular lattice on $B=[0,1]^{d}$.

Since the (non-Gaussian) limit distributions of the considered discretized
sample distance covariances are not tractable, in Section 5 we consider some
modifications of these quantities. Roughly speaking, we find approximating
degenerate $U$-statistics to the sample distance covariances and apply
bootstrap techniques tailored for these $U$-statistics. We show the
consistency of the bootstrap. In the case of random locations, this part is
quite delicate and rather different from the theory developed in Dehling et
al. [5] in the lattice case for $d=1$. Indeed, we show bootstrap consistency
for these U-statistics conditional on the random locations and their number.
In Section 6 we illustrate how the convergence of the sample distance
correlation depends on the sample size and the number of discretization
points. We also show how the bootstrap performs for the sample distance
correlation on selected discretized random fields, in particular for
fractional Brownian and infinite variance stable Lévy sheets. In Section 7 we
apply the sample distance correlation for testing the independence of Japanese
meteorological data observed at stations all over Japan. We interpret their
number as random and their location as uniformly distributed.

In the remaining sections we provide the proofs of the main results.

##  2\. Random field at random locations

###  2.1. Technical conditions

We use the notation and assumptions of Section 1.3. In particular, $X,Y$ are
measurable bounded stochastically continuous random fields on a bounded Borel
set $B\subset{\mathbb{R}}^{d}$ of positive Lebesgue measure with Riemann
square-integrable sample paths on $B$. Also recall that $p=p_{n}\to\infty$ and
$N_{p}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}\infty$ as $n\to\infty$.

In what follows, $c$ denotes any positive constant whose value is not of
interest.

The following result gives some asymptotic results for the sample distance
covariance. They are the basis for proving Theorem 2.6.

######  Proposition 2.1.

Choose $\beta\in(0,2)$ and assume that $X,Y$ are independent. Consider the
following conditions:

  1. (1a)

$X,Y$ have finite second moment and $\int_{B}{\mathbb{E}}\big{[}X^{2}({\bf
u})+Y^{2}({\bf u})\big{]}\,d{\bf u}<\infty$.

  2. (1b)

$X,Y$ have infinite second moment but ${\mathbb{E}}\big{[}\sup_{{\bf u}\in
B}|X|^{\beta}({\bf u})+\sup_{{\bf u}\in B}|Y|^{\beta}({\bf u})\big{]}<\infty$.

  3. (2a)

$X$ has finite second moment, $\big{[}\beta\in(0,1]$ and
$\int_{B}{\mathbb{E}}[X^{2}({\bf u})]\,d{\bf u}<\infty\big{]}$ or
$\big{[}\beta\in(1,2)$ and $\int_{B}{\mathbb{E}}[|X|^{2\beta}({\bf u})]\,d{\bf
u}<\infty\big{]}$.

  4. (2b)

$X$ has infinite second moment, $\beta\in(0,1)$ and ${\mathbb{E}}[\sup_{{\bf
u}\in B}|X|^{2\beta}({\bf u})]<\infty$.

If (1a) or (1b) hold then

(2.1)
$\displaystyle{\mathbb{E}}\big{[}\big{|}T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)\big{|}\;\big{|}\;N^{(p)}\big{]}\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}0\,,\qquad n\to\infty\,.$

If (2a) or (2b) hold then

(2.2)
$\displaystyle{\mathbb{P}}\big{(}|T_{n,\beta}(X^{(p)},X^{(p)})-T_{n,\beta}(X,X)|>\varepsilon\;\big{|}\;N^{(p)}\big{)}\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}0\,,\qquad n\to\infty\,.$

The proof of this proposition is given in Section 8.

###  2.2. Some examples

We consider some examples of random fields and discuss the fulfillment of the
conditions in Proposition 2.1.

######  Example 2.2.

Let $X({\bf u})=B^{H}({\bf u})$, ${\bf u}\in[0,1]^{d}$, be a fractional
Brownian sheet with Hurst parameter $H=(H_{1},\ldots,H_{d})\in(0,1)^{d}$
introduced by Kamont [9]; see also Ayache and Xiao [4]. It is a centered
Gaussian random field with continuous sample paths and covariance function

(2.3) $\displaystyle{\rm cov}(B^{H}({\bf s}),B^{H}({\bf
t}))=\prod_{i=1}^{d}\frac{1}{2}(|s_{i}|^{2H_{i}}+|t_{i}|^{2H_{i}}-|s_{i}-t_{i}|^{2H_{i}}),\qquad{\bf
s},{\bf t}\in[0,1]^{d}\,.$

Moreover, for all $\gamma>0$, ${\mathbb{E}}[|B^{H}({\bf u})|^{\gamma}]$ is a
continuous function of ${\bf u}$. Therefore all conditions on $X$ are
satisfied.

######  Example 2.3.

Let $H\in(0,1)$ and $X^{H}$ be a centered Gaussian random field on $[0,1]^{d}$
with covariance function

$\displaystyle{\rm cov}(X^{H}({\bf s}),X^{H}({\bf
t}))=\dfrac{1}{2}\big{(}|{\bf s}|^{2H}+|{\bf t}|^{2H}-|{\bf s}-{\bf
t}|^{2H}\big{)}\,,\qquad{\bf s},{\bf t}\in[0,1]^{d}\,.$

This process is called Lévy fractional Brownian field; see Samorodnitsky and
Taqqu [15], p. 393. It has stationary increments and continuous sample paths
and moments of any order.

######  Example 2.4.

Let $X$ be a centered Lévy sheet on $[0,1]^{d}$ with Lévy-Khintchine triplet
$(\mu,\sigma^{2},\nu)$; see Khoshnevisan and Xiao [10], Example 2.1. This
means that, for disjoint intervals $A_{i}\in[0,1]^{d}$, $i=1,\ldots,k$, and
$k\geq 1$, the increments $X(A_{1}),\ldots,X(A_{k})$ are independent and for
each interval $A=({\bf s},{\bf t}]$, the characteristic function of the
increment

$X(A)=\sum_{k_{1}\in\\{0,1\\}}\cdots\sum_{k_{d}\in\\{0,1\\}}(-1)^{d-\sum_{l=1}^{d}k_{l}}X(s_{1}+k_{1}(t_{1}-s_{1}),\ldots,s_{d}+k_{d}(t_{d}-s_{d})),$

is given by

(2.4) $\displaystyle{\mathbb{E}}[{\rm e}\,^{iuX(A)}]=\exp(-|A|\,\Psi(u))\,,$

where

$\Psi(u)=i\mu u+u^{2}\sigma^{2}/2+\int_{\mathbb{R}}({\rm
e}\,^{iux}-1-ux{\mathbf{1}}_{\\{|x|\leq 1\\}})\nu(dx).$

If $X$ has finite $\gamma$th moment for some $\gamma>0$ then
${\mathbb{E}}[|X({\bf u})|^{\gamma}]$ is a continuous function of ${\bf u}$
and the function ${\mathbb{E}}[|\sup_{u\in[0,1]^{d}}X({\bf u})|^{\gamma}]$ is
finite as well. The field $X$ has càdlàg sample paths which are bounded and
square-integrable on $[0,1]^{d}$.

######  Example 2.5.

Consider a symmetric $\alpha$-stable Lévy sheet on $[0,1]^{d}$ for some
$\alpha\in(0,2)$; see for example Ehm [7], Samorodnitsky and Taqqu [15]. This
field has infinite variance and is a Lévy random sheet whose increments have a
symmetric $\alpha$-stable distribution, i.e. the characteristic function of
$X(A)$ is given by (2.4) with $\Psi(u)=c^{\alpha}|u|^{\alpha}$ for some $c>0$.
It has moments of order $\gamma\in(0,\alpha)$.

###  2.3. Main result

The following statement is the main result in the case of random locations.

######  Theorem 2.6.

Choose $\beta\in(0,2)$ and assume that $X,Y$ are independent,

  1. 1.

If (1a) or (1b) of Proposition 2.1 hold then for all $\varepsilon>0$ along
a.e. sample path of $(N^{(p)})_{p>0}$, as $n\to\infty$ and $p=p_{n}\to\infty$,

$\displaystyle{\mathbb{P}}\big{(}\big{|}T_{n,\beta}(X^{(p)},Y^{(p)})\big{|}>\varepsilon\mid
N^{(p)}\big{)}\to 0\,.$

  2. 2.

If $\big{[}$(1a) and (2a) both for $X,Y\big{]}$ or $\big{[}$(1b) and (2b) both
for $X,Y\big{]}$ of Proposition 2.1 hold then we also have for all
$\varepsilon>0$ along a.e. sample path of $(N^{(p)})_{p>0}$, as $n\to\infty$
and $p=p_{n}\to\infty$,

$\displaystyle{\mathbb{P}}\big{(}\big{|}R_{n,\beta}(X^{(p)},Y^{(p)})\big{|}>\varepsilon\mid
N^{(p)}\big{)}\to 0\,.$

###### Proof.

1. From Proposition 2.1, in particular from (2.1), we conclude that $T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)\to 0$ as $n\to\infty$ in probability conditional on $N^{(p)}$. The rest is analogous to the derivations in Dehling et al. [5]: $T_{n,\beta}(X,Y)$ is a $V$-statistic and satisfies the strong law of large numbers $T_{n,\beta}(X,Y)\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}T_{\beta}(X,Y)=0$ under the required moment conditions on $X,Y$.
2. Under the additional conditions (2a), (2b) both for $X,Y$ we also have $T_{n,\beta}(X^{(p)},X^{(p)})-T_{n,\beta}(X,X)\to 0$ and $T_{n,\beta}(Y^{(p)},Y^{(p)})-T_{n,\beta}(Y,Y)\to 0$ as $n\to\infty$ in probability conditional on $N^{(p)}$. Moreover, by the strong law of large numbers $T_{n,\beta}(X,X)\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}T_{\beta}(X,X)$ and $T_{n,\beta}(Y,Y)\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}T_{\beta}(Y,Y)$, hence $R_{n,\beta}(X,Y)\to R_{\beta}(X,Y)=0$ in probability conditional on $N^{(p)}$. ∎

In Section 5 we provide much stronger asymptotic results under stronger
conditions. In particular, we show that $n\,T_{n,\beta}(X^{(p)},Y^{(p)})$
conditional on $(N^{(p)})_{p>0}$ has the same weak limit as
$n\,T_{n,\beta}(X^{(p)},Y^{(p)})$, and we also show consistency of a suitable
bootstrap procedure.

##  3\. Random field at a lattice

In the present and next sections we consider sampling schemes of $(X,Y)$ on a
deterministic lattice on $B=[0,1]^{d}$. We will assume conditions on the
smoothness of $(X,Y)$ from which we can derive convergence rates of
$T_{n,\beta}(X^{(p)},Y^{(p)})$ to $T_{n,\beta}(X,Y)$. Here we closely follow
Dehling et al. [5] who dealt with the lattice case for $d=1$. The following
conditions and results are adaptations of those in [5].

###  3.1. Technical conditions

We use the notation and assumptions of Section 1.2. In what follows, we
introduce moment and smoothness conditions on the fields $X,Y$ and require
certain rates for $p=p_{n}\to\infty$ as $n\to\infty$. The conditions are
separated in two parts depending on whether $(X,Y)$ have finite second moments
or not.

If $X,Y$ have finite second moments we will work under the following
conditions.
Condition (A): finite second moments

  1. (A1)

Smoothness of increments. There exist $\gamma_{X},\gamma_{Y}>0$ and $c>0$ such
that

$\displaystyle{\rm var}\big{(}X({\bf t})-X({\bf s})\big{)}\leq c\,|{\bf
t}-{\bf s}|^{\gamma_{X}}\qquad\mbox{and}\qquad{\rm var}\big{(}Y({\bf
t})-Y({\bf s})\big{)}\leq c\,|{\bf t}-{\bf s}|^{\gamma_{Y}}\,.$

  2. (A2)

Additional moment conditions. If $\beta\in(1,2)$ we have

$\displaystyle\max_{{\bf t}\in[0,1]^{d}}{\mathbb{E}}[|X({\bf
t})|^{2(2\beta-1)}]+\max_{{\bf t}\in[0,1]^{d}}{\mathbb{E}}[|Y({\bf
t})|^{2(2\beta-1)}]<\infty\,.$

  3. (A3)

Growth condition on $p=p_{n}\to\infty$. We have

$\displaystyle
p^{-1}n^{2/\big{(}d^{-1}(\gamma_{X}\wedge\gamma_{Y})\,(\beta\wedge
1)\big{)}}\to 0\,,\qquad n\to\infty\,.$

If $X,Y$ possibly have infinite second moments we will work under the
following conditions:
Condition (B): infinite second moments Assume $\beta\in(0,2)$.

  1. (B1)

Finite $\beta$th moment.

$\displaystyle{\mathbb{E}}\big{[}\max_{{\bf t}\in(0,1]^{d}}|X({\bf
t})|^{\beta}\big{]}<\infty\;\;\mbox{and}\;\;{\mathbb{E}}\big{[}\max_{{\bf
t}\in(0,1]^{d}}|Y({\bf t})|^{\beta}\big{]}<\infty\,.$

  2. (B2)

Smoothness of increments. There exist $\gamma_{X},\gamma_{Y}>0$ and $c>0$ such
that

$\displaystyle\max_{{\bf i}\in\Pi_{d}}{\mathbb{E}}\big{[}\max_{{\bf
t}\in\Delta_{{\bf i}}}|X({\bf t})-X({\bf t}_{\bf i})|^{\beta}\big{]}\leq
c\,p^{-\gamma_{X}/d}\ \mbox{and}\ \max_{{\bf
i}\in\Pi_{d}}{\mathbb{E}}\big{[}\max_{t\in\Delta_{\bf i}}|Y({\bf t})-Y({\bf
t}_{\bf i})|^{\beta}\big{]}\leq c\,p^{-\gamma_{Y}/d}\,.$

  3. (B3)

Additional moment and smoothness conditions.

$\displaystyle{\mathbb{E}}\big{[}\max_{{\bf t}\in(0,1]^{d}}|X({\bf
t})|^{2\beta}\big{]}<\infty\;\;\mbox{and}\;\;{\mathbb{E}}\big{[}\max_{{\bf
t}\in(0,1]^{d}}|Y({\bf t})|^{2\beta}\big{]}<\infty\,.$

If $\beta\in(0,1)$ there also exist
$\gamma_{X}^{\prime},\gamma_{Y}^{\prime}>0$ and $c>0$ such that

$\displaystyle\max_{{\bf i}\in\Pi_{d}}{\mathbb{E}}\big{[}\max_{{\bf
t}\in\Delta_{\bf i}}|X({\bf t})-X({\bf t}_{\bf i})|^{2\beta}\big{]}\leq
c\,p^{-\gamma_{X}^{\prime}/d}\mbox{ and }\ \max_{{\bf
i}\in\Pi_{d}}{\mathbb{E}}\big{[}\max_{{\bf t}\in\Delta_{\bf i}}|Y({\bf
t})-Y({\bf t}_{\bf i})|^{2\beta}\big{]}\leq c\,p^{-\gamma_{Y}^{\prime}/d}\,.$

  4. (B4)

Growth condition on $p=p_{n}\to\infty$. We have
$\beta/2+(\gamma_{X}\wedge\gamma_{Y})/d>1$ and

$\displaystyle p^{-1}n^{(\beta/(\beta\wedge
1))\big{/}(\beta/2+(\gamma_{X}\wedge\gamma_{Y})/d-1)}\to 0\,,\quad
n\to\infty\,.$

In the lattice case the following result is the analog of Proposition 2.1.

######  Proposition 3.1.

Assume the following conditions.

  1. (1)

$X,Y$ are independent stochastically continuous bounded processes on
$[0,1]^{d}$ defined on the same probability space.

  2. (2)

If $X,Y$ have finite expectations, then these are assumed to be equal to $0$.

  3. (3)

$\beta\in(0,2)$.

  4. (4)

$p=p_{n}\to\infty$ as $n\to\infty$.

Then the following statements hold.

  1. 1.

If also (A1) holds then there is a constant $c$ such that for any $n\geq 1$,

(3.1)
$\displaystyle{\mathbb{E}}\big{[}|T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)|\big{]}\leq
c\,p^{-d^{-1}(\gamma_{X}\wedge\gamma_{Y})\,(\beta\wedge 1)/2}\,.$

  2. 2.

If also (B1),(B2) hold then there is a constant $c$ such that for any $n\geq
1$,

(3.2)
$\displaystyle{\mathbb{E}}\big{[}|T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)|\big{]}\leq
c\,\big{(}p^{1-(\beta/2+(\gamma_{X}\wedge\gamma_{Y})/d)}\big{)}^{(\beta\wedge
1)/\beta}\,.$

  3. 3.

If also (A1), (A2) hold in the finite variance case and
$\big{[}\mbox{$\beta\in(0,1)$, {\rm(B3)}}\ and\
1<\beta+(\gamma_{X}^{\prime}\wedge\gamma_{Y}^{\prime})/d\big{]}$ in the
infinite variance case, then

$\displaystyle
T_{n,\beta}(X^{(p)},X^{(p)})-T_{n,\beta}(X,X)\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}0\,,\qquad
n\to\infty\,,$

and the analogous result holds for $Y$.

The proof of this result is completely analogous to the case $d=1$ given in
[5] and therefore omitted.

A comparison of Propositions 2.1 and 3.1 shows that in the former result one
does not need conditions on the smoothness of the sample paths such as (A1),
(B2) or (B3). In the latter result, the smoothness parameters
$\gamma_{X},\gamma_{Y}$ appear explicitly in the approximation rates in (3.1)
and (3.2). For example, consider two iid Brownian sheets $X,Y$. According to
Example 3.2 below, $\gamma_{X}=\gamma_{Y}=1$ since $H_{i}=0.5$. Then the
right-hand side of (3.1) turn into $c\,p^{-(\beta\wedge 1)/(2d)}$. In
Proposition 2.1 the parameter $p$ has a distinct meaning. Assume that
$(N_{p})_{p>0}$ are Poisson variables, e.g. $N_{p}=M(p\,B)$ for a unit rate
homogeneous Poisson process $M$ on ${\mathbb{R}}^{d}$. When restricted to $B$,
$M(p\,\cdot)$ has intensity $p|B|$ and we have
$N_{p}/p\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}|B|$ and therefore
$p|B|$ is a rough approximation of $N_{p}$. We observe that (3.1) is the
analog of (2.1). In the latter case, we do not have a convergence rate. This
is perhaps not surprising since knowledge of $X,Y$ on the grid points of a
lattice in combination with the smoothness of the sample paths of $X,Y$ often
give us additional information about the sample path inside a cell
$\Delta_{\bf i}$. In Section 4 we will moderate between the two sampling
schemes: we will average the observations at random locations in each cell of
the lattice. This property allows one to use smoothness properties of the
random field also for observations at random locations.

Thus, if the observations are given on a lattice it is preferable to use
distance covariance on it. However, the random location case has the advantage
that smoothness does not matter and it works on quite general bounded sets
$B$.

######  Example 3.2.

Let $X({\bf u})=B^{H}({\bf u})$, ${\bf u}\in[0,1]^{d}$, be a fractional
Brownian sheet with Hurst parameter $H=(H_{1},\ldots,H_{d})\in(0,1)^{d}$; see
Example 2.2. From (2.3) we have

$\displaystyle{\rm var}(B^{H}({\bf t})-B^{H}({\bf s}))$
$\displaystyle=\Big{(}\prod_{i=1}^{d}|t_{i}|^{2H_{i}}-\prod_{i=1}^{d}0.5(|t_{i}|^{2H_{i}}+|s_{i}|^{2H_{i}}-|t_{i}-s_{i}|^{2H_{i}})\Big{)}$
$\displaystyle\quad+\Big{(}\prod_{i=1}^{d}|s_{i}|^{2H_{i}}-\prod_{i=1}^{d}0.5(|t_{i}|^{2H_{i}}+|s_{i}|^{2H_{i}}-|t_{i}-s_{i}|^{2H_{i}})\Big{)}$
$\displaystyle=\sum_{i=1}^{d}\Big{(}\prod_{j=1}^{i-1}|t_{j}|^{2H_{j}}\,\dfrac{1}{2}\big{(}|t_{i}|^{2H_{i}}-|s_{i}|^{2H_{i}}+|t_{i}-s_{i}|^{2H_{i}}\big{)}$
$\displaystyle\quad+\prod_{j=1}^{i-1}|s_{j}|^{2H_{j}}\,\dfrac{1}{2}\big{(}|s_{i}|^{2H_{i}}-|t_{i}|^{2H_{i}}+|t_{i}-s_{i}|^{2H_{i}}\big{)}\Big{)}\prod_{k=i+1}^{d}\frac{1}{2}(|t_{k}|^{2H_{k}}+|s_{k}|^{2H_{k}}-|t_{k}-s_{k}|^{2H_{k}}).$

Thus we have for some constant $c>0$,

${\rm var}(B^{H}({\bf t})-B^{H}({\bf s}))\leq
c\sum_{i=1}^{d}|t_{i}-s_{i}|^{(2H_{i})\wedge 1}\leq c|{\bf t}-{\bf
s}|^{(2\min_{i}H_{i})\wedge 1}.$

Hence (A1) is satisfied with $\gamma_{X}=(2\min_{i=1,\ldots,d}H_{i})\wedge 1$.
Moreover, for any $\gamma>0$, ${\mathbb{E}}[|B^{H}({\bf u})|^{\gamma}]$ is
continuous. Hence (A2) holds.

######  Example 3.3.

Let $X^{H}({\bf u})$, ${\bf u}\in[0,1]^{d}$, for some $H\in(0,1)$ be a Lévy
fractional Brownian field; see Example 2.3. It has all moments and stationary
increments. Hence

$\displaystyle{\rm var}(X^{H}({\bf t})-X^{H}({\bf s}))=|{\bf t}-{\bf
s}|^{2H}\,.$

Hence (A1) is satisfied with $\gamma_{X}=2H$. Moreover, for any $\gamma>0$,
${\mathbb{E}}[|X^{H}({\bf u})|^{\gamma}]$ is continuous. Hence (A2) holds.

######  Example 3.4.

Let $X$ be a Lévy sheet on $[0,1]^{d}$; see Example 2.4. Recall that $X({\bf
t})=X([\bf 0,{\bf t}])$. Therefore

(3.3) $\displaystyle X({\bf t})-X({\bf s})=X([{\bf 0},{\bf t}]\setminus([{\bf
0},{\bf s}]\cap[{\bf 0},{\bf t}]))-X([\bf 0,{\bf s}]\setminus([\bf 0,{\bf
s}]\cap[\bf 0,{\bf t}])),$

where the two random variables on the right-hand side are defined on disjoint
sets, hence they are independent. We observe that the set $\big{(}[\bf 0,{\bf
s}]\cup[\bf 0,{\bf t}]\big{)}\backslash\big{(}[\bf 0,{\bf s}]\cap[\bf 0,{\bf
t}]\big{)}$ for ${\bf s},{\bf t}\in[0,1]^{d}$ is the union of a finite number
(only depending on $d$) of disjoint sets $({\bf a},{\bf b}]$ such that for
some $1\leq j\leq d$ and a constant $c>0$, $|({\bf a},{\bf b}]\big{|}\leq
c\,|t_{j}-s_{j}|$. Thus, if $X$ has finite second moment, we have for some
constants $c_{1},c_{2}$,

$\displaystyle{\rm var}(X({\bf t})-X({\bf s}))$ $\displaystyle\leq$
$\displaystyle c_{1}\max_{i=1,\ldots,d}|t_{i}-s_{i}|\leq c_{2}|{\bf t}-{\bf
s}|\,.$

Hence (A1) holds with $\gamma_{X}=1$. If the moment condition
${\mathbb{E}}[|X({\bf u})|^{2(2\beta-1)}]<\infty$ holds for some ${\bf u}$, it
is finite for all ${\bf u}$ and also continuous. Hence (A2) holds for $X$.

######  Example 3.5.

Consider a symmetric $\alpha$-stable Lévy sheet on $[0,1]^{d}$ for some
$\alpha\in(0,2)$; see Example 2.5. It has moments of order
$\beta\in(0,\alpha)$. Since $X$ is symmetric and has independent increments an
application of Levy’s maximal inequality yields

$\displaystyle{\mathbb{P}}\Big{(}\max_{{\bf t}\in\Delta_{{\bf i}}}|X({\bf
t})-X({\bf t}_{{\bf i}})|>x\Big{)}\leq 2\,{\mathbb{P}}\big{(}|X({\bf t}_{{\bf
i}})-X({\bf t}_{{\bf i}-\bf 1})|>x\big{)}\,,\qquad x>0\,.$

By definition of a symmetric $\alpha$-stable sheet

$\displaystyle{\mathbb{E}}\big{[}|X({\bf t}_{{\bf i}})-X({\bf t}_{{\bf i}-\bf
1})|^{\beta}\big{]}$ $\displaystyle=$ $\displaystyle c\,\big{|}[{\bf 0},{\bf
t}_{{\bf i}}]\backslash[{\bf 0},{\bf t}_{{\bf i}-\bf
1}]\big{|}^{\beta/\alpha}\leq c\,p^{-\beta/(\alpha\,d)}\,.$

Thus (B2) holds with $\gamma_{X}=\beta/\alpha$. The same argument yields (B3)
with $\gamma_{X}^{\prime}=2\beta/\alpha$ for $\beta\in(0,0.5\alpha)$.

###  3.2. Main result in the lattice case

Now we proceed with the main theorem. In the case $d=1$ it corresponds to
Theorem 3.1 in Dehling et al. [5].

######  Theorem 3.6.

Assume the following conditions:

  1. (1)

$X,Y$ are independent stochastically continuous bounded processes on
$[0,1]^{d}$.

  2. (2)

If $X,Y$ have finite expectations, then they are centered.

  3. (3)

$\beta\in(0,2)$.

  4. (4)

$p=p_{n}\to\infty$ as $n\to\infty$.

Then the following statements hold.

  1. 1.

If either (A1) or $\big{[}\mbox{{\rm(B1),(B2)} and
$1<\beta/2+(\gamma_{X}\wedge\gamma_{Y})/d$}\big{]}$ are satisfied then

$T_{n,\beta}(X^{(p)},Y^{(p)})\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}0$

holds.

  2. 2.

If either (A1),(A3) or (B1),(B2),(B4) hold then

(3.4) $\displaystyle n\,T_{n,\beta}(X^{(p)},Y^{(p)})\stackrel{{\scriptstyle
d}}{{\rightarrow}}\sum_{i=1}^{\infty}\lambda_{i}(N_{i}^{2}-1)+c,$

for an iid sequence of standard normal random variables $(N_{i})$, a constant
$c$, and a square summable sequence $(\lambda_{i})$.

  3. 3.

If either (A1),(A2) or $\big{[}\mbox{$\beta\in(0,1)$, {\rm(B1)-(B3)}}\ and\
1<\beta+(\gamma_{X}^{\prime}\wedge\gamma_{Y}^{\prime})/d\big{]}$ hold then

$\displaystyle
R_{n,\beta}(X^{(p)},Y^{(p)})\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}0\,.$

  4. 4.

If either (A1)-(A3) or $\big{[}\mbox{$\beta\in(0,1)$, {\rm(B1)-(B4)}}\ and\
1<\beta+(\gamma_{X}^{\prime}\wedge\gamma_{Y}^{\prime})/d\big{]}$ hold then
$n\,R_{n,\beta}(X^{(p)},Y^{(p)})$ converges to a scaled version of the limit
in (3.4).

###### Proof.

1. Under the assumptions, Proposition 3.1, yields that $T_{n,\beta}(X,Y)-T_{n,\beta}(X^{(p)},Y^{(p)})$
$\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}0$ while
$T_{n,\beta}(X,Y)\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}T_{\beta}(X,Y)=0$ by the strong law of large numbers for
$V$-statistics. The statement follows.
2. Under the additional conditions, where $1<\beta/2+(\gamma_{X}\wedge\gamma_{Y})/d$ is implied by $\rm(B4)$, we have from Proposition 3.1, that $n|T_{n,\beta}(X,Y)-T_{n,\beta}(X^{(p)},Y^{(p)})|\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}{0}$. By degeneracy of the $V$-statistics $nT_{n,\beta}(X,Y)$ converges in distribution to a sum of independent weighted $\chi^{2}$ random variables, so does $nT_{n,\beta}(X^{(p)},Y^{(p)})$.
3\. and 4. They follow by combining (1) and (2) with Proposition 3.1, which
allows one to switch from $T_{n,\beta}$ to $R_{n,\beta}$. We omit further
details. ∎

##  4\. Random field at random locations grouped in lattice cells

In this section we consider a combination of the two previous sampling schemes
for $(X,Y)$. For the sake of argument we restrict ourselves to fields on
$B=[0,1]^{d}$. We assume that the fields are observed at the locations $({\bf
U}_{i})_{i=1,\ldots,p}$ which are iid uniformly distributed on $B$ and
$p=p_{n}\to\infty$ is a deterministic integer sequence. This time we average
the randomly scattered ${\bf U}_{i}$ in each cell of a regular lattice grid.

Similarly to Section 1.2 we partition $B$ into $\tilde{p}=[p/\log p]$ disjoint
cells $\Delta_{i},i=1,\ldots,\tilde{p}$ with side length $\tilde{p}^{-1/d}$
and volume $\tilde{p}^{-1}$. The main idea of this approach is to average the
observations $(X({\bf U}_{j}),Y({\bf U}_{j}))$ in a given cell. Since $p$
increases slightly faster than the number of cells $\tilde{p}$ the probability
that there is no observation in a given cell decreases at a certain rate.

For any random field $Z$ on $B$ we consider the discretization (again abusing
notation),

(4.1) $\displaystyle Z^{(p)}({\bf
u})=\mbox{$\sum^{\prime}$}{\mathbf{1}}_{\Delta_{j}}({\bf u})\overline{Z}_{j},$

where $\sum^{\prime}$ denotes summation over those $j=1,\ldots,\tilde{p}$ such
that $\\#\Delta_{j}=\\#\\{i\leq p:{\bf U}_{i}\in\Delta_{j}\\}\neq 0$ and

$\displaystyle\overline{Z}_{j}=\sum_{k:{\bf U}_{k}\in\Delta_{j}}\dfrac{Z({\bf
U}_{k})}{\\#\Delta_{j}}\,.$

We also recycle the notation

(4.2) $\displaystyle\|Z^{(p)}\|_{2}^{2}$ $\displaystyle:=$
$\displaystyle\int_{[0,1]^{d}}\mbox{$\sum^{\prime}$}{\mathbf{1}}_{\Delta_{j}}({\bf
u})\,\overline{Z}_{j}^{2}\,d{\bf
u}={\tilde{p}}^{-1}\mbox{$\sum^{\prime}$}\overline{Z}_{j}^{2}\,.$

###  4.1. Technical conditions

The results in this section parallel those in the lattice case 3. The
conditions are similar to (A) and (B); we also use the notation from (A) and
(B) in the sequel.

If $X,Y$ have finite second moments we will need the following condition.

  1. (A3’)

Growth condition on $p=p_{n}\to\infty$.

$\displaystyle(p^{-1}\vee\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d})n^{2/(\beta\wedge
1)}\to 0\,,\qquad n\to\infty\,.$

If $X,Y$ possibly have infinite second moments we will need the following
condition.
Assume $\beta\in(0,2)$ and

  1. (B4’)

Growth condition on $p=p_{n}\to\infty$. We have

$\displaystyle(p^{-1}\vee\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d})p^{1-\beta/2}n^{\beta/(1\wedge\beta)}\to
0\,,\quad n\to\infty\,.$

If we remove $p^{-1}$ in (A3’) and (B4’) and replace $\tilde{p}$ by $p$ then
we recover (A3) and (B4).

The following result is an analog of Proposition 2.1 for the discretizations
$X^{(p)},Y^{(p)}$ of $X,Y$ defined via (4.1). In contrast to the latter case,
we have explicit rates of convergence for
${\mathbb{E}}\big{[}|T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)|\big{]}\to
0$. These rates are achieved due to the averaging of values in the cells
$\Delta_{j}$. Then one can also exploit the smoothness of the field over these
small cells.

######  Proposition 4.1.

Assume the following conditions.

  1. (1)

$X,Y$ are independent stochastically continuous bounded processes on
$[0,1]^{d}$ defined on the same probability space.

  2. (2)

If $X,Y$ have finite expectations, then these are assumed to be equal to $0$.

  3. (3)

$\beta\in(0,2)$.

Then the following statements hold.

  1. 1.

If also (A1) holds then there is a constant $c$ such that for all $n\geq 1$,

$\displaystyle{\mathbb{E}}\big{[}|T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)|\big{]}\leq
c\,\big{(}\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d}+p^{-1}\big{)}^{(\beta\wedge
1)/2}\,.$

  2. 2.

If also (B1),(B2) hold then there is a constant $c$ such that for all $n\geq
1$,

$\displaystyle{\mathbb{E}}\big{[}|T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)|\big{]}\leq
c\,\big{(}p^{1-\beta/2}(\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d}+p^{-1})\big{)}^{(\beta\wedge
1)/\beta}\,.$

  3. 3.

Under the additional conditions (A1), (A2) in the finite variance case and
under $\big{[}\beta\in(0,1)$, (B1)-(B3) and $1<\beta+\gamma_{X}^{\prime}/d$
$\big{]}$ in the infinite variance case,

$\displaystyle
T_{n,\beta}(X^{(p)},X^{(p)})-T_{n,\beta}(X,X)\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}0\,,\qquad
n\to\infty\,.$

The proof of Proposition 4.1 is rather technical and given in Section 9.

###  4.2. Main result

The following is the main result of this section.

######  Theorem 4.2.

Assume the following conditions:

  1. 1.

$X,Y$ are independent stochastically continuous bounded processes on
$[0,1]^{d}$.

  2. 2.

If $X,Y$ have finite expectations, then they are centered.

  3. 3.

$\beta\in(0,2)$ and $p_{n}\to\infty$ as $n\to\infty$.

Then the following statements hold.

  1. (1)

If either (A1) or $\big{[}\mbox{{\rm(B1),(B2)} and
$1<\beta/2+(\gamma_{X}\wedge\gamma_{Y})/d$}\big{]}$ are satisfied then

$T_{n,\beta}(X^{(p)},Y^{(p)})\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}0$

hold.

  2. (2)

If either (A1),(A3’) or (B1),(B2),(B4’) hold then

$\displaystyle n\,T_{n,\beta}(X^{(p)},Y^{(p)})\stackrel{{\scriptstyle
d}}{{\rightarrow}}\sum_{i=1}^{\infty}\lambda_{i}(N_{i}^{2}-1)+c$

for an iid sequence of standard normal random variables $(N_{i})$, a constant
$c$, and a square summable sequence $(\lambda_{i})$.

  3. (3)

If either (A1),(A2) or $\big{[}\mbox{$\beta\in(0,1)$, {\rm(B1)-(B3)}}\ and\
1<\min(\beta+(\gamma_{X}^{\prime}\wedge\gamma_{Y}^{\prime})/d,\beta/2+(\gamma_{X}\wedge\gamma_{Y})/d)\big{]}$
hold then

$\displaystyle
R_{n,\beta}(X^{(p)},Y^{(p)})\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}0\,.$

  4. (4)

If either (A1),(A2),(A3’) or $\big{[}\mbox{$\beta\in(0,1)$,
{\rm(B1)-(B3),(B4')}}\ and\
1<\beta+(d^{-1}(\gamma_{X}^{\prime}\wedge\gamma_{Y}^{\prime}))\wedge
1)\big{]}$ hold then $n\,R_{n,\beta}(X^{(p)},Y^{(p)})$ converges to a scaled
version of the limit in ((2)).

One can follow the lines of the proof of Theorem 3.6.

##  5\. The bootstrap for $T_{n,\beta}$

In this section we introduce a bootstrap procedure for $T_{n,\beta}(X,Y)$ and
$T_{n,\beta}(X^{(p)},Y^{(p)})$.

###  5.1. $T_{n,\beta}(X,Y)$ as a degenerate $V$-statistic

We recall some facts from the Appendix in Dehling et al. [5] (see also Lyons
[12]) and adapt them to the situation of a random field on $B$. We assume that
$Z_{i}=(X_{i},Y_{i})$, $i=1,2,\ldots,$ is an iid sequence with generic element
$Z=(X,Y)$ whose components are bounded Riemann square-integrable random fields
on $B$, and
${\mathbb{E}}[\|X\|_{2}^{\beta}+\|Y\|_{2}^{\beta}+\|X\|_{2}^{\beta}\|Y\|_{2}^{\beta}]<\infty$
for some $\beta\in(0,2)$. Under these assumptions, $T_{n,\beta}(X,Y)$ has
representation as a $V$-statistic of order 4 with symmetric degenerate kernel
of order 1.

We start with the kernel

$\displaystyle f((x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3}),(x_{4},y_{4}))\
(=:f(z_{1},z_{2},z_{3},z_{4}))$
$\displaystyle\quad=\|x_{1}-x_{2}\|_{2}^{\beta}\|y_{1}-y_{2}\|_{2}^{\beta}+\|x_{1}-x_{2}\|_{2}^{\beta}\|y_{3}-y_{4}\|_{2}^{\beta}-2\|x_{1}-x_{2}\|_{2}^{\beta}\|y_{1}-y_{3}\|_{2}^{\beta}.$

From this representation,

$T_{n,\beta}(X,Y)=\frac{1}{n^{4}}\sum_{1\leq i,j,k,l\leq
n}f(Z_{i},Z_{j},Z_{k},Z_{l}).$

Then one can define the corresponding symmetric kernel via the usual
symmetrization as

(5.1)
$h(z_{1},z_{2},z_{3},z_{4})=\frac{1}{24}\sum_{(l_{1},l_{2},l_{3},l_{4})\mbox{
permutation of }(1,2,3,4)}f(z_{l_{1}},z_{l_{2}},z_{l_{3}},z_{l_{4}}).$

The kernel $h$ is at least $1$-degenerate: under the null hypothesis of
independence between $X$ and $Y$,

$\displaystyle{\mathbb{E}}[f(z_{1},Z_{2},Z_{3},Z_{4})]+{\mathbb{E}}[f(Z_{2},z_{1},Z_{3},Z_{4})]+{\mathbb{E}}[f(Z_{2},Z_{3},z_{1},Z_{4})]$
$\displaystyle\qquad+{\mathbb{E}}[f(Z_{2},Z_{3},Z_{4},z_{1})]=0\,.$

Still under the null hypothesis of independence between $X$ and $Y$,

(5.2) $\displaystyle{\mathbb{E}}[h(z_{1},z_{2},(X_{3},Y_{3}),(X_{4},Y_{4}))]$
$\displaystyle=$
$\displaystyle\frac{1}{6}\big{(}\|x_{1}-x_{2}\|_{2}^{\beta}+{\mathbb{E}}[\|X_{1}-X_{2}\|_{2}^{\beta}]-{\mathbb{E}}[\|x_{1}-X\|_{2}^{\beta}]-{\mathbb{E}}[\|x_{2}-X\|_{2}^{\beta}]\big{)}$
$\displaystyle\ \
\times\bigl{(}\|y_{1}-y_{2}\|_{2}^{\beta}+{\mathbb{E}}[\|Y_{1}-Y_{2}\|_{2}^{\beta}]-{\mathbb{E}}[\|y_{1}-Y\|_{2}^{\beta}]-{\mathbb{E}}[\|y_{2}-Y\|_{2}^{\beta}]\big{)}\,,$

and the right-hand side is not constant. Hence, the kernel $h$ is precisely
$1$-degenerate. One can follow the lines of the proof in the Appendix in [5]
to show the following result (the only necessary change is the replacement of
$[0,1]$ by $B$).

######  Lemma 5.1.

If $X,Y$ are independent and
${\mathbb{E}}[\|X\|_{2}^{\beta}+\|Y\|_{2}^{\beta}]<\infty$ for some
$\beta\in(0,2)$ then $T_{n,\beta}(X,Y)$ has representation as a $V$-statistic
of order $4$ with symmetric $1$-degenerate kernel $h$. The corresponding
$U$-statistic $\widetilde{T}_{n,\beta}(X,Y)$ is obtained from
$T_{n,\beta}(X,Y)$ by restricting the summation to indices
$(i_{1},i_{2},i_{3},i_{4})$ with mutually distinct components. Then

$\displaystyle\qquad
n\,\big{(}T_{n,\beta}(X,Y)-\widetilde{T}_{n,\beta}(X,Y)\big{)}\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}{\mathbb{E}}[\|X_{1}-X_{2}\|_{2}^{\beta}]{\mathbb{E}}[\|Y_{1}-Y_{2}\|_{2}^{\beta}]\,,\qquad
n\to\infty\,.$

An immediate consequence of this result is that, up to an additive constant,
$(n\,T_{n,\beta}(X,Y))$ and $(n\,\widetilde{T}_{n,\beta}(X,Y))$ have the same
limit distribution which is indicated in (1.2). In what follows, we will focus
on $\widetilde{T}_{n,\beta}(X,Y)$.

###  5.2. Bootstrapping $T_{n}(X,Y)$

For the sake illustration of the method we restrict ourselves to $\beta=1$ and
suppress $\beta$ in the notation. We also assume that

(5.3) $\displaystyle\int_{B}{\mathbb{E}}[X^{2}({\bf u})+Y^{2}({\bf u})]d{\bf
u}<\infty\,.$

A generic element $Z=(X,Y)$ has trajectory $z=(x,y)$ on
$B\subset{\mathbb{R}}^{d}$.

Under the assumptions of Lemma 5.1, $\widetilde{T}_{n}(X,Y)$ has
representation as $U$-statistic of order 4 with a 1-degenerate symmetric
kernel $h(x_{1},x_{2},x_{3},x_{4})$. Applying the Hoeffding decomposition to
$\widetilde{T}_{n}(X,Y)$, the limit distributions (modulo a change of
location/scale) of $nT_{n}(X,Y)$ and the following normalized version of
$\widetilde{T}_{n}(X,Y)$ coincide:

$\displaystyle U_{n}(Z)=\dfrac{1}{n}\sum_{1\leq i\neq j\leq
n}h_{2}(Z_{i},Z_{j};F_{Z})$

where $F_{Z}=F_{X}\times F_{Y}$ and $h_{2}$ is defined by

$\displaystyle h_{2}(z_{1},z_{2};F_{Z})=$
$\displaystyle{\mathbb{E}}[h(z_{1},z_{2},Z_{3},Z_{4})]-{\mathbb{E}}[h(z_{1},Z_{2},Z_{3},Z_{4})]$
$\displaystyle-{\mathbb{E}}[h(Z_{1},z_{2},Z_{3},Z_{4})]+{\mathbb{E}}[h(Z_{1},Z_{2},Z_{3},Z_{4})]\,.$

We write $F_{n}$ for the empirical distribution of the sample
$Z_{1},\ldots,Z_{n}$ and $(Z_{n1}^{\ast},\ldots,Z_{nn}^{\ast})$ for a
bootstrap sample, i.e., given $Z_{1},\ldots,Z_{n}$ it is iid with distribution
$F_{n}$. Arcones and Giné [3] proved that the correct bootstrap version of
$U_{n}(Z)$ is

$\displaystyle U_{n}(Z^{\ast})=\dfrac{1}{n}\sum_{1\leq i\neq j\leq
n}h_{2}(Z_{ni}^{\ast},Z_{nj}^{\ast};F_{n})\,.$

The fact that the limiting distributions of $U_{n}(Z)$ and $U_{n}(Z^{\ast})$
coincide follows from Dehling and Mikosch [6]; see also Corollary 5.2 in [5]
in the case $B=[0,1]$.

######  Proposition 5.2.

Under the aforementioned conditions, and if also
${\mathbb{E}}[|h(Z_{i_{1}},\ldots,Z_{i_{4}})|^{2}]<\infty$ for all indices
$1\leq i_{1}\leq\cdots\leq i_{4}\leq 4$, we have

$\displaystyle
d_{2}\big{(}{\mathcal{L}}\big{(}U_{n}(Z)\big{)}\;,{\mathcal{L}}\big{(}U_{n}(Z^{\ast})\big{)}\big{)}\to
0\,,\qquad n\to\infty\,,$

for almost all realizations of $(Z_{i})$. Here $d_{2}$ denotes the Wasserstein
distance of order $2$.

The additional second moment assumption on $h$ is satisfied for our kernel.
Note that it suffices to consider the non-symmetric kernel $f$, and to show
that ${\mathbb{E}}[(f(Z_{i_{1}},Z_{i_{2}},Z_{i_{3}},Z_{i_{4}}))^{2}]<\infty$
for all indices $1\leq i_{1},\ldots,i_{4}\leq 4$. For our specific kernel,
this condition reads as

$\displaystyle{\mathbb{E}}\big{[}\big{(}\|X_{i_{1}}-X_{i_{2}}\|\big{[}\|Y_{i_{1}}-Y_{i_{2}}\|+\|Y_{i_{3}}-Y_{i_{4}}\|-2\|Y_{i_{1}}-Y_{i_{3}}\|\big{]}\big{)}^{2}\big{]}<\infty,$

and this holds under the moment conditions in this paper.

The Wasserstein distance $d_{2}$ metrizes weak convergence and moment
convergence up to the second order. Therefore Proposition 5.2 proves bootstrap
consistency for $U_{n}(Z^{\ast})$. In particular, $U_{n}(Z^{\ast})$ and
$n\,T_{n}(X,Y)$ have the same limit distribution up to some change of
scale/location. From now, we will work with $U_{n}(Z)$ as a surrogate of the
normalized sample distance covariance $n\,T_{n}(X,Y)$ and with its bootstrap
version $U_{n}(Z^{\ast})$.

###  5.3. Bootstrapping $U_{n}(Z^{(p)})$

Our goal is to show that we are allowed to replace $Z=(X,Y)$ in $U_{n}(Z)$ by
the corresponding discretizations $Z^{(p)}=(X^{(p)},Y^{(p)})$ as well as the
corresponding result for $U_{n}(Z^{\ast})$. We restrict ourselves to the case
of random locations studied in Section 2. The corresponding bootstrap
consistency results in the lattice case of Section 3 follow by a
straightforward adaptation of the results in Dehling et al. [5] who considered
the case $d=1$ and $B=[0,1]$.

We start by showing that $U_{n}(Z)$ and $U_{n}(Z^{(p)})$ are close
conditionally on $N^{(p)}$.

######  Lemma 5.3.

Assume that $X,Y$ are independent, (5.3) holds, and $N_{p}\to\infty$ a.s.as
$p=p_{n}\to\infty$. Then

(5.4) $\displaystyle{\mathbb{E}}[(U_{n}(Z)-U_{n}(Z^{(p)}))^{2}\mid N^{(p)}]\to
0\,,\qquad n\to\infty\,,$

for a.e. realization of $(N^{(p)})$.

###### Proof.

We consider

(5.5) $\displaystyle U_{n}(Z)-U_{n}(Z^{(p)}\mid
N^{(p)}):=\frac{1}{n}\sum_{1\leq i\neq j\leq
n}\big{(}h_{2}(Z_{i},Z_{j})-h_{2}(Z_{i}^{(p)},Z_{j}^{(p)}\mid
N^{(p)})\big{)}\,,$

where

$\displaystyle h_{2}(z_{1},z_{2};F_{Z}^{(p)}\mid N^{(p)})$
$\displaystyle:={\mathbb{E}}[h(z_{1},z_{2},Z_{3}^{(p)},Z_{4}^{(p)})\mid
N^{(p)}]-{\mathbb{E}}[h(z_{1},Z_{2}^{(p)},Z_{3}^{(p)},Z_{4}^{(p)})\mid
N^{(p)}]$
$\displaystyle\quad-{\mathbb{E}}[h(Z_{1}^{(p)},z_{2},Z_{3}^{(p)},Z_{4}^{(p)})\mid
N^{(p)}]+{\mathbb{E}}[h(Z_{1}^{(p)},Z_{2}^{(p)},Z_{3}^{(p)},Z_{4}^{(p)})\mid
N^{(p)}]\,.$

By construction, given $N^{(p)}$, $U_{n}(Z)-U_{n}(Z^{(p)}\mid N^{(p)})$ only
depends on the sample $Z_{1},\ldots,Z_{n}$ and is a $U$-statistic of order $2$
with symmetric $1$-degenerate kernel. By Lemma A in Serfling [16, p. 183], we
have

(5.6) $\displaystyle{\mathbb{E}}\big{[}\big{(}U_{n}(Z)-U_{n}(Z^{(p)}\mid
N^{(p)})\big{)}^{2}\mid N^{(p)}\big{]}$ $\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\big{[}\big{(}h_{2}(Z_{1},Z_{2})-h_{2}(Z_{1}^{(p)},Z_{2}^{(p)}\mid
N^{(p)})\big{)}^{2}\mid N^{(p)}\big{]}$ $\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\big{[}\big{(}h(Z_{1},Z_{2},Z_{3},Z_{4})-h(Z_{1}^{(p)},Z_{2}^{(p)},Z_{3}^{(p)},Z_{4}^{(p)})\big{)}^{2}\mid
N^{(p)}\big{]}$ $\displaystyle\leq$ $\displaystyle
c{\mathbb{E}}\big{[}\big{(}f(Z_{1},Z_{2},Z_{3},Z_{4})-f(Z_{1}^{(p)},Z_{2}^{(p)},Z_{3}^{(p)},Z_{4}^{(p)})\big{)}^{2}\mid
N^{(p)}\big{]}$ $\displaystyle\leq$ $\displaystyle c\
{\mathbb{E}}\big{[}\big{(}\|X_{1}-X_{2}\|_{2}\|Y_{1}-Y_{2}\|_{2}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\|Y_{1}^{(p)}-Y_{2}^{(p)}\|_{2}\big{)}^{2}$
$\displaystyle\qquad+(\|X_{1}-X_{2}\|_{2}\|Y_{3}-Y_{4}\|_{2}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\|Y_{3}^{(p)}-Y_{4}^{(p)}\|_{2}\big{)}^{2}$
$\displaystyle\qquad+(\|X_{1}-X_{2}\|_{2}\|Y_{1}-Y_{3}\|_{2}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\|Y_{1}^{(p)}-Y_{3}^{(p)}\|_{2}\big{)}^{2}\mid
N^{(p)}\big{]}\,.$

Now we can proceed as in the proof of Proposition 2.1 to show that the right-
hand side converges to zero as $n\to\infty$ along a.e. sample path of
$(N^{(p)})_{p>0}$. We illustrate this for the first term on the right-hand
side. We have by independence of $(X_{i})$ and $(Y_{i})$,

(5.7)
$\displaystyle{\mathbb{E}}\big{[}\big{(}\|X_{1}-X_{2}\|_{2}\|Y_{1}-Y_{2}\|_{2}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\|Y_{1}^{(p)}-Y_{2}^{(p)}\|_{2}\big{)}^{2}\mid
N^{(p)}\big{]}$ $\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\big{[}\big{(}\|X_{1}-X_{2}\|_{2}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\big{)}^{2}\mid
N^{(p)}\big{]}\,{\mathbb{E}}[\|Y_{1}-Y_{2}\|_{2}^{2}]$
$\displaystyle+c\,{\mathbb{E}}\big{[}\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}^{2}\mid
N^{(p)}\big{]}\,{\mathbb{E}}\big{[}\big{(}\|Y_{1}-Y_{2}\|_{2}-\|Y_{1}^{(p)}-Y_{2}^{(p)}\|_{2}\big{)}^{2}\mid
N^{(p)}\big{]}\,.$

The expectation ${\mathbb{E}}[\|Y_{1}-Y_{2}\|_{2}^{2}]$ is finite by (5.3).
For the same reason and by Riemann-sum approximation along a.e. sample path of
$(N^{(p)})_{p>0}$,

$\displaystyle{\mathbb{E}}\big{[}\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}^{2}\mid
N^{(p)}\big{]}$ $\displaystyle=$
$\displaystyle\dfrac{1}{N_{p}}\sum_{i=1}^{N_{p}}{\mathbb{E}}\big{[}(X_{1}-X_{2})^{2}({\bf
U}_{i})\mid{\bf U}_{i}\big{]}$ $\displaystyle\to$
$\displaystyle\int_{B}{\mathbb{E}}\big{[}(X_{1}-X_{2})^{2}({\bf
u})\big{]}\,d{\bf u}\,.$

We have by dominated convergence

$\displaystyle{\mathbb{E}}\big{[}\big{(}\|X_{1}-X_{2}\|_{2}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\big{)}^{2}\mid(N^{(q)})_{q>0}\big{]}$
$\displaystyle\leq{\mathbb{E}}\big{[}\big{|}\int_{B}(X_{1}-X_{2})^{2}({\bf
u})d{\bf
u}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}^{2}\big{|}\mid(N^{(q)})_{q>0}\big{]}$
$\displaystyle\to 0\,,\qquad p\to\infty\,,$

and the corresponding result holds if we replace $(X_{1},X_{2})$ by
$(Y_{1},Y_{2})$. Hence the right-hand side in (5.7) converges to zero along
a.e. sample path of $(N^{(p)})$. ∎

Our next goal is to show that $U_{n}(Z^{\ast})$ and $U_{n}(Z^{(p)\ast}\mid
N^{(p)})$ are asymptotically close where the latter quantity is defined as in
(5.5) if we replace the distribution of $Z$ by $F_{n}$. We write ${\rm
var}^{\ast}$ for the variance with respect to bootstrap probability measure.

We will need some stronger assumptions to achieve this goal.

######  Lemma 5.4.

We assume that $(N_{p})$ are Poisson variables such that
${\mathbb{E}}[N_{p}]=p=p_{n}$ and $N_{p}\to\infty$ a.s. as $n\to\infty$,
$\sum_{n}p_{n}^{-1/2}<\infty$ and

$\displaystyle\int_{B}{\mathbb{E}}[X^{4}({\bf u})+Y^{4}({\bf u})]\,d{\bf
u}<\infty\,.$

Then for a.e. sample paths of $(N^{(p)})$ and $(Z_{i})$,

$\displaystyle{\mathbb{P}}\big{(}U_{n}(Z^{\ast})-U_{n}(Z^{\ast(p)}\mid
N^{(p)})\to 0\,,n\to\infty\mid(Z_{i}),(N^{(p)})\big{)}=1\,.$

###### Proof.

We observe that given $N^{(p)}$,

$\displaystyle U_{n}(Z^{\ast})-U_{n}(Z^{\ast(p)}\mid
N^{(p)})=\frac{1}{n}\sum_{1\leq i\neq j\leq
n}\big{(}h_{2}(Z_{i}^{\ast},Z_{j}^{\ast})-h_{2}(Z_{i}^{\ast(p)},Z_{j}^{\ast(p)}\mid
N^{(p)})\big{)}$

is a $U$-statistic of order $2$ with a $1$-degenerate kernel. Again applying
the variance formula in Lemma A of [16], p. 183, we obtain

$\displaystyle{\rm var}^{\ast}(U_{n}(Z^{\ast})-U_{n}(Z^{\ast(p)}\mid
N^{(p)})\mid N^{(p)})$ $\displaystyle\leq c\,{\rm
var}^{\ast}(h_{2}(Z_{1}^{\ast},Z_{2}^{\ast})-h_{2}(Z_{1}^{\ast(p)},Z_{2}^{\ast(p)}\mid
N^{(p)})\mid N^{(p)})$
$\displaystyle=c\,\frac{1}{n^{2}}\sum_{k,\ell=1}^{n}\big{(}h_{2}(Z_{k},Z_{\ell})-h_{2}(Z_{k}^{(p)},Z_{\ell}^{(p)}\mid
N^{(p)})\big{)}^{2}.$

Taking expectations on both sides, we have

$\displaystyle{\mathbb{E}}\big{[}{\rm
var}^{\ast}(U_{n}(Z^{\ast})-U_{n}(Z^{\ast(p)}\mid N^{(p)})\mid
N^{(p)})\big{]}$ $\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\big{[}\big{(}h_{2}(Z_{1},Z_{2})-h_{2}(Z_{1}^{(p)},Z_{2}^{(p)}\mid
N^{(p)})\big{)}^{2}\big{]}\,.$

Now appeal to (5.6) and take expectations on both sides. Keeping in mind
(5.7), we have to bound expressions of the following type:

(5.8) $\displaystyle
c\,{\mathbb{E}}\Big{[}{\mathbb{E}}\big{[}\big{(}\|X_{1}-X_{2}\|_{2}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\big{)}^{2}\mid
N^{(p)}\big{]}\,{\mathbb{E}}[\|Y_{1}-Y_{2}\|_{2}^{2}]$
$\displaystyle+c\,{\mathbb{E}}\big{[}\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}^{2}\,\big{(}\|Y_{1}-Y_{2}\|_{2}-\|Y_{1}^{(p)}-Y_{2}^{(p)}\|_{2}\big{)}^{2}\mid
N^{(p)}\big{]}\Big{]}$ $\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\big{[}\big{(}\|X_{1}-X_{2}\|_{2}-\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\big{)}^{2}\big{]}$
$\displaystyle+c\,\big{(}{\mathbb{E}}\big{[}\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}^{4}\big{]}\big{)}^{1/2}\,\big{(}{\mathbb{E}}\big{[}\big{(}\|Y_{1}-Y_{2}\|_{2}-\|Y_{1}^{(p)}-Y_{2}^{(p)}\|_{2}\big{)}^{4}\big{]}\big{)}^{1/2}\,.$

We write $A=X_{1}-X_{2}$. Then we have for the first expression on the right-
hand side

(5.9) $\displaystyle
c\,{\mathbb{E}}\big{[}\big{(}\|A\|_{2}-\|A^{(p)}\|_{2}\big{)}^{2}\big{]}$
$\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\Big{[}{\mathbb{E}}\Big{[}N_{p}^{-1}\Big{|}\sum_{i=1}^{N_{p}}\Big{(}A^{2}({\bf
U}_{i})-\int_{B}A^{2}({\bf u})d{\bf
u}\Big{)}\Big{|}\;\Big{|}A,N_{p}\Big{]}\Big{]}$ $\displaystyle\leq$
$\displaystyle
c\,{\mathbb{E}}\Big{[}\Big{(}{\mathbb{E}}\Big{[}\Big{(}N_{p}^{-1}\sum_{i=1}^{N_{p}}\Big{(}A^{2}({\bf
U}_{i})-\int_{B}A^{2}({\bf u})d{\bf
u}\Big{)}\Big{)}^{2}\;\Big{|}A,N_{p}\Big{]}\Big{)}^{1/2}\Big{]}$
$\displaystyle=$ $\displaystyle
c\,{\mathbb{E}}\big{[}N_{p}^{-1/2}{\mathbf{1}}(N_{p}>0)\big{]}\,{\mathbb{E}}\Big{[}\big{(}{\rm
var}\big{(}A^{2}({\bf U}_{1})\mid A\big{)}\big{)}^{1/2}\Big{]}$
$\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\big{[}N_{p}^{-1/2}{\mathbf{1}}(N_{p}>0)]\,\Big{(}\int_{B}{\mathbb{E}}[(X_{1}-X_{2})^{4}({\bf
u})]d{\bf u}\Big{)}^{1/2}\,.$

The second factor is finite by the moment assumptions on $X$. Using the
Poisson structure of $N_{p}$, we can apply Lemma 4.1 to the first factor and
conclude that it has the asymptotic order $O(p^{-1/2})$. Now we consider the
first expectation in (5.8):

$\displaystyle{\mathbb{E}}\big{[}\|A^{(p)}\|_{2}^{4}\big{]}={\mathbb{E}}\Big{[}\Big{(}N_{p}^{-1}\sum_{i=1}^{N_{p}}A^{2}({\bf
U}_{i})\Big{)}^{2}\Big{]}$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\Big{[}N_{p}^{-1}\sum_{i=1}^{N_{p}}A^{4}({\bf
U}_{i})\Big{]}$ $\displaystyle=$
$\displaystyle{\mathbb{E}}[{\mathbf{1}}(N_{p}>0)]\,\int_{B}{\mathbb{E}}\big{[}(X_{1}-X_{2})^{4}({\bf
u})\big{]}d{\bf u}\,.$

The right-hand side is finite in view of the moment conditions on $X$. Next we
turn to the second expression in (5.8). Write $C=Y_{1}-Y_{2}$. It remains to
bound

$\displaystyle\big{(}{\mathbb{E}}\big{[}\big{(}\|C\|_{2}-\|C^{(p)}\|_{2}\big{)}^{4}\big{]}\big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle
c\,\Big{(}{\mathbb{E}}\Big{[}{\mathbb{E}}\Big{[}\Big{(}N_{p}^{-1}\sum_{i=1}^{N_{p}}\Big{(}C^{2}({\bf
U}_{i})-\int_{B}C^{2}({\bf u})d{\bf
u}\Big{)}\Big{)}^{2}\;\Big{|}C,N_{p}\Big{]}\Big{]}\Big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle
c\,\big{(}{\mathbb{E}}\big{[}N_{p}^{-1}{\mathbf{1}}(N_{p}>0)]\big{)}^{1/2}\,\Big{(}\int_{B}{\mathbb{E}}[C^{4}({\bf
u})]\,d{\bf u}\Big{)}^{1/2}$ $\displaystyle=$ $\displaystyle
O(p^{-1/2})\,,\qquad n\to\infty\,.$

Here we again used Lemma 4.1. Summarizing the bounds above, we conclude that
for $\varepsilon>0$,

$\displaystyle\sum_{n}{\mathbb{P}}(|U_{n}(Z^{\ast})-U_{n}(Z^{\ast(p))}\mid
N^{(p)})|>\varepsilon)$
$\displaystyle\leq\varepsilon^{-2}\sum_{n}{\mathbb{E}}\Big{[}{\rm
var}^{\ast}\big{(}U_{n}(Z^{\ast})-U_{n}(Z^{\ast(p)})\mid N^{(p)})\mid
N^{(p)}\big{)}\Big{]}$ $\displaystyle\leq c\,\sum_{n}p_{n}^{-1/2}<\infty\,.$

Therefore the Borel-Cantelli lemma implies that

$\displaystyle U_{n}(Z^{\ast})-U_{n}(Z^{\ast(p)}\mid N^{(p)})\to 0\,,\qquad
n\to\infty\,.$

for a.e. sample path of $(Z_{i})$ and $(N^{(p)})_{p>0}$. ∎

######  Corollary 5.5.

Under the conditions of Lemma 5.4, $(n\,T_{n}(X,Y))$ and $(U_{n}(Z^{\ast(p)})$
(conditional on $(N^{(p)})$) have the same limit distribution in (3.4) (up to
changes of scale/location).

This result means that the modified bootstrap for
$(n\,T_{n}(X^{(p)},Y^{(p)}))$ is consistent.

###### Proof.

Following the discussion in Sections 5.1 and 5.3, $(nT_{n}(X,Y))$,
$(U_{n}(Z))$ and $(U_{n}(Z^{\ast}))$ (given the data) have the same limit
distribution (up to possible changes of scale/location). In this section we
proved that $(U_{n}(Z))$ and $(U_{n}(Z^{(p)}))$ (given $(N^{(p)})$) and
$(U_{n}(Z^{\ast}))$ and $(U_{n}(Z^{\ast(p)}))$ (given the data and
$(N^{(p)})$) have the same limit distribution. This finishes the proof. ∎

It remains to show the bootstrap consistency for
$(n\,R_{n}(X^{(p)},Y^{(p)}))$. We consider the following modified bootstrap
version of the latter sequence:

$\displaystyle\dfrac{n\,U_{n}(Z^{\ast(p)})}{\sqrt{T_{n}(X^{(p)},X^{(p)})\,T_{n}(Y^{(p)},Y^{(p)})}}.$

Since under our moment conditions, $T_{n}(X,X)\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}T(X,X)$ and $T_{n}(Y,Y)\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}T(Y,Y)$ by the strong law of large numbers for
$V$-statistics it remains to show that

(5.10) $\displaystyle
T_{n}(X^{(p)},X^{(p)})-T_{n}(X,X)\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}0\,,\qquad n\to\infty\,,$

and the corresponding result for $(T_{n}(Y^{(p)},Y^{(p)}))$. This is the
content of the following lemma.

######  Lemma 5.6.

Under the conditions of Lemma 5.4 and summability of $(p_{n}^{-1/4})$, (5.10)
and the corresponding result for $(T_{n}(Y^{(p)},Y^{(p)}))$ hold.

###### Proof.

We have the decomposition

$\displaystyle T_{n}(X,X)-T_{n}(X^{(p)},X^{(p)})$ $\displaystyle=$
$\displaystyle\dfrac{1}{n^{2}}\sum_{i,j=1}^{n}\big{(}\|X_{i}-X_{j}\|_{2}^{2}-\|X_{i}^{(p)}-X_{j}^{(p)}\|_{2}^{2})$
$\displaystyle+\Big{[}\Big{(}\dfrac{1}{n^{2}}\sum_{i,j=1}^{n}\|X_{i}-X_{j}\|_{2}\Big{)}^{2}-\Big{(}\dfrac{1}{n^{2}}\sum_{i,j=1}^{n}\|X_{i}^{(p)}-X_{j}^{(p)}\|_{2}\Big{)}^{2}\Big{]}$
$\displaystyle-2\dfrac{1}{n^{3}}\sum_{i,j,k=1}^{n}\Big{[}(\|X_{i}-X_{j}\|_{2}-\|X_{i}^{(p)}-X_{j}^{(p)}\|_{2})\,\|X_{i}-X_{k}\|_{2}$
$\displaystyle+\|X_{i}^{(p)}-X_{j}^{(p)}\|_{2}\,(\|X_{i}-X_{k}\|_{2}-\|X_{i}^{(p)}-X_{k}^{(p)}\|_{2})\Big{]}$
$\displaystyle=:$ $\displaystyle I_{1}+I_{2}-2(I_{31}+I_{32})\,.$

Writing $A=X_{1}-X_{2}$, we see that

$\displaystyle{\mathbb{E}}[|I_{1}|]\leq{\mathbb{E}}\big{[}|\|A\|_{2}^{2}-\|A^{(p)}\|_{2}^{2}|\big{]}={\mathbb{E}}\Big{[}{\mathbb{E}}\Big{[}N_{p}^{-1}\Big{|}\sum_{i=1}^{N_{p}}\Big{(}A^{2}({\bf
U}_{i})-\int_{B}A^{2}({\bf u})d{\bf
u}\Big{)}\Big{|}\;\Big{|}A,N_{p}\Big{]}\Big{]}\,.$

Now we can proceed as for (5.9) and conclude that the right-hand side is
$O(p^{-1/2})$. Since we assume summability of $(p_{n}^{-1/4})$ the Borel-
Cantelli lemma yields $I_{1}\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}0$.

For $I_{2}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}0$ it suffices to
prove that

$\displaystyle
I_{4}:=\dfrac{1}{n^{2}}\sum_{i,j=1}^{n}\big{(}\|X_{i}-X_{j}\|_{2}-\|X_{i}^{(p)}-X_{j}^{(p)}\|_{2}\big{)}\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}0\,,\qquad n\to\infty\,$

since, by the strong law of large numbers for $U$-statistics,
$n^{-2}\sum_{i,j=1}^{n}\|X_{i}-X_{j}\|_{2}\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}{\mathbb{E}}[\|X_{1}-X_{2}\|_{2}]$. We have

$\displaystyle{\mathbb{E}}[|I_{4}|]$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\big{[}\big{|}\|A\|_{2}-\|A^{(p)}\|_{2}\big{|}\big{]}\leq\big{(}{\mathbb{E}}\big{[}\big{|}\|A\|_{2}-\|A^{(p)}\|_{2}\big{|}^{2}\big{]}\big{)}^{1/2}.$

Following the proof in (5.9), the right-hand side is bounded by
$c\,({\mathbb{E}}[N_{p}^{-1/2}{\mathbf{1}}(N_{p}>0)])^{1/2}=O(p^{-1/4})$. The
summability of $(p_{n}^{-1/4})$ and the Borel-Cantelli lemma prove that
$I_{4}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}0$.

We focus on showing $I_{32}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}0$;
the case $I_{31}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}0$ is
analogous. We have

$\displaystyle{\mathbb{E}}[|I_{32}|]$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\big{[}\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}\,\big{|}\|X_{1}-X_{3}\|_{2}-\|X_{1}^{(p)}-X_{3}^{(p)}\|_{2}\big{|}\big{]}$
$\displaystyle\leq$
$\displaystyle\big{(}{\mathbb{E}}\big{[}\|A^{(p)}\|_{2}^{2}\big{]}\big{)}^{1/2}\,\big{(}{\mathbb{E}}\big{[}\big{(}\|A\|_{2}-\|A^{(p)}\|_{2}\big{)}^{2}\big{]}\big{)}^{1/2}\,.$

The first expected value is bounded uniformly for $p$. Another application of
(5.9) shows that the right-hand side is of the order $O(p_{n}^{-1/4})$.
Another application of the Borel-Cantelli lemma proves
$I_{32}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}0$. This finishes the
proof. ∎

##  6\. A Monte Carlo study

###  6.1. Finite sample study of the convergence

In this section we conduct a Monte Carlo study of the finite sample behavior
of the sample distance correlation for $\beta=1$; in what follows we suppress
$\beta$ in the notation. We consider a fractional Brownian sheet (fB for
short) with Hurst parameter $H=(H_{1},H_{2})\in(0,1)^{2}$, in particular,
$H_{i}\in\\{1/4,1/2,3/4\\}$. As a heavy-tailed alternative we choose symmetric
$1.8$-stable Lévy sheets. The observations are given either on a lattice or at
random locations in $[0,1]^{2}$. In the lattice case we take $q$ equidistant
grid points on each side of $[0,1]^{2}$, resulting in a lattice of $p=q\times
q$ points. We choose sample sizes $n\in\\{100,200,300\\}$ and repeat the Monte
Carlo simulations 500 times for each process.

######  Figure 6.1.

Boxplots for $R_{n}(X^{(p)},Y^{(p)})$ based on 500 simulations of independent
sheets $X,Y$. The sheets are simulated on a $p=q\times q$ equidistant lattice
on $[0,1]^{2}$ for $q=30$. Top and middle row: fB sheets for different choices
of Hurst coefficients $H_{i}\in\\{1/4,1/2,3/4\\}$, $i=1,2$ and increasing
sample size $n$. Bottom row: Effects of increasing $n$ (for fixed $q=100$) and
increasing $q$ (for fixed $n=300$). The left (right) graphs are based on
Brownian (1.8-stable) sheets.

######  Figure 6.2.

Boxplots for $R_{n}(X^{(p)},Y^{(p)})$ based on 500 simulations of dependent fB
and $1.8$-stable sheets $X,Y$ in the lattice case. The parameters $H_{1}$ and
$H_{2}$ and the values $n,p$ are the same as for the corresponding graph at
the same location in Figure 6.1. The construction of $X,Y$ is based on (6.1)
and (6.2) for fB and 1.8-stable sheets for $\rho=0.5$, respectively.

|  |
---|---|---
|  |

######  Figure 6.3.

Boxplots for $R_{n}(X^{(p)},Y^{(p)})$ based on 500 simulations of independent
fB and 1.8-stable sheets $X,Y$ at $N_{p}\sim{\rm Poisson}(p)$, $p=1000$,
uniformly distributed locations in $[0,1]^{2}$. The parameters $H_{1}$ and
$H_{2}$ and the values $n$ are the same as for the corresponding graph at the
same location in Figure 6.1. Top and middle row: fB sheets for different
choices of Hurst parameters $H_{i}\in\\{1/4,1/2,3/4\\}$, $i=1,2$, and
increasing sample size $n$. Bottom row: Effects of increasing $p$ for
independent Brownian sheets and $n=100,300$ (left two). In the two right
graphs we consider the case of independent $1.8$-stable sheets $X,Y$ with
increasing $n$ ($p$) and fixed $p=300$ ($n=300$).

|  |
---|---|---
|  |

######  Figure 6.4.

Boxplots for $R_{n}(X^{(p)},Y^{(p)})$ from 500 replications of dependent fB
and $1.8$-stable sheets $X,Y$, respectively. We consider independent copies of
$(X,Y)$ at a random Poisson $N_{p}$ number (with parameter $p$) of iid
uniformly distributed locations $({\bf U}_{i})$ on $(0,1)^{2}$. The parameters
$H_{1}$ and $H_{2}$, the values $n,p$ and the dependence parameter $\rho=0.5$
are the same as for the corresponding graph at the same location in Figure
6.3.

For the simulation of fB sheets we calculate their covariance function

$\displaystyle{\rm cov}(B^{H}({\bf s}),B^{H}({\bf
t}))=\prod_{i=1}^{d}\frac{1}{2}(|s_{i}|^{2H_{i}}+|t_{i}|^{2H_{i}}-|s_{i}-t_{i}|^{2H_{i}}),\qquad{\bf
s},{\bf t}\in[0,1]^{d}\,,$

at the lattice points and use the multivariate Gaussian random generator in R
(mvfast) while for Brownian (this is the case $H_{1}=H_{2}=0.5$) and stable
sheets we use the independent and stationary increment property: for each cell
in $[0,1]^{2}$ we draw properly scaled independent Gaussian or stable random
variables and sum up, starting from the origin until we reach the boundary of
$[0,1]^{2}$. (We generated Brownian sheets with both methods, leading to very
similar results.)

Figure 6.1 shows boxplots for $R_{n}(X^{(p)},Y^{(p)})$ based on simulations
from independent fields $X,Y$ on the described lattice. In the top and middle
rows, $q=30$, hence $p=900$, and $X,Y$ are fB sheets for different choices of
$H_{1}$ and $H_{2}$. We see the influence of the smoothness of the sample
paths: the larger $H_{i}$, the smoother the sample paths and the closer
$R_{n}(X^{(p)},Y^{(p)})$ to zero. In the bottom row we simulate from Brownian
and $1.8$-stable sheets and examine the effect of increasing
$n\in\\{100,200,300\\}$ for fixed $q=100$ and of increasing
$q\in\\{100,200,300\\}$ for fixed $n=300$. An increase of $n$ apparently
improves the performance of $R_{n}(X^{(p)},Y^{(p)})$: the larger $n$ the
closer it is to zero. On the other hand, if we fix $n$ one hardly sees a
change for different values of $p=q\times q$. It is surprising for us that the
sample distance correlation for independent $1.8$-stable sheets outperforms
the corresponding sample distance correlation for independent fB sheets: one
detects independence between $X$ and $Y$ already for medium sample sizes.

In Figure 6.2 we confront the results of Figure 6.1 with the sample distance
correlation for some dependent $X,Y$, again on the lattice. We consider iid fB
sheets $X,X^{\prime}$ and define $Y\stackrel{{\scriptstyle d}}{{=}}X$ by

(6.1) $\displaystyle
Y=\rho\,X+(1-\rho^{2})^{1/2}\,X^{\prime}\,,\qquad\rho\in(0,1)\,.$

This choice of $X,Y$ leads to the correlation $\rho$ between $X$ and $Y$. In
Figure 6.2 we choose $\rho=0.5$. We present boxplots for the same choices of
$H_{1}$ and $H_{2}$ as in the top and middle rows of Figure 6.1. The bottom
row graphs are based on dependent Brownian/$1.8$-stable sheets for the same
$n$ and $p$ as at the bottom of Figure 6.1. We consider iid $1.8$-stable
symmetric Lévy sheets $X,X^{\prime}$ and define $Y$ by

(6.2) $\displaystyle
Y=\rho\,X+(1-\rho^{1.8})^{1/1.8}X^{\prime}\,,\qquad\rho\in(0,1)\,.$

Then in particular, $X\stackrel{{\scriptstyle d}}{{=}}Y$. In Figure 6.2 we
choose $\rho=0.5$. The graphs in Figure 6.2 are in stark contrast to those in
Figure 6.1; they clearly point at the dependence of $X,Y$. Again, the value of
$p$ seems irrelevant.

We also examine boxplots in the random location setting. In the lattice case
in the top and middle graphs of Figures 6.1 and 6.2 we had $p=900$ points in
$[0,1]^{2}$. In the random location case we choose uniform locations ${\bf
U}_{i}$ on $[0,1]^{2}$ whose number $N_{p}$ is Poisson with parameter
$p=1000$, i.e., on average there are 1000 points. In Figures 6.3 and 6.4 we
keep the same parameters and sample sizes as in the previous two figures. In
Figure 6.3 we illustrate the case of independent $X,Y$. Comparing Figures 6.3
and 6.1, we observe similar finite sample behavior: the sample size $n$ is
more relevant for convergence to zero than the number $N_{p}$ and the random
locations. The bottom graphs in Figure 6.3 look less convincing (the median of
the boxplot is higher than in the lattice case) but this is due to the fact
that we do not have the information from $100\times 100=10000$ locations but
only from $N_{p}\approx 1000$. (The choice of a Poisson variable $N_{p}$ with
parameter $p=10000$ leads to a complexity which we could not handle on our
laptops.) Figure 6.4 corresponds to the setting of dependent $X,Y$ in Figure
6.2 with random locations. The graphs in both figures show quite convincingly
the difference between dependence and independence of $X$ and $Y$. There is
one significant difference to Figures 6.1 and 6.2: the distribution of
$R_{n}(X^{(p)},Y^{(p)})$ in Figures 6.3 and 6.4 is more spread than in the
lattice case. This is due to the additional uncertainty of the random
locations.

###  6.2. Size and power of the distance correlation test

We illustrate the performance of the bootstrap procedure for the test for
independence based on distance correlation in the cases of fixed locations on
a lattice and of randomly scattered locations. We focus on independent pairs
$X,Y$, Brownian or $1.8$-stable sheets. Given a sample
$(X_{1}^{(p)},Y_{1}^{(p)}),\ldots,(X_{n}^{(p)},Y_{n}^{(p)})$, we draw $500$
bootstrap samples. From each bootstrap sample we calculate the sample distance
correlation and from the corresponding bootstrap distribution the
$(1-\xi)$-quantile $q^{\ast}_{1-\xi}$. Finally, we verify whether

(6.3) $\displaystyle R_{n}(X^{(p)},Y^{(p)})\geq q^{\ast}_{1-\xi}\,.$

Then we repeat this procedure $M\in\\{500,1000\\}$ times and count the
successes of (6.3).

In Table 6.5 we choose $q=100$ in the lattice case and observe the empirical
rejection rates for independent Brownian and $1.8$-stable sheets $X,Y$: each
cell of the table corresponds to a given sample size $n$, test level $\xi$ and
iteration number $M$. In Table 6.6 we consider the simulation results in the
lattice case for dependent standard Brownian sheets with correlation
$\rho\in(0,1)$ and $M=500$, where the correlation $\rho$ is accomplished
through (6.1). We also consider dependent $1.8$-stable Lévy sheets $X,Y$ with
dependence parameter $\rho$ introduced in (6.2). In agreement with the theory,
the rejection rates increase as $n$ and $\rho$ increase.

| Brownian sheets | 1.8-stable sheets
---|---|---
$M$ | $500$ | $1000$ | $500$ | $1000$
$n\setminus\xi$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$
100 | 12.8 | 6.2 | 11.6 | 6.2 | 11.4 | 3.6 | 9.2 | 2.6
200 | 9.4 | 4.8 | 9.6 | 4.8 | 8.2 | 3.4 | 8.4 | 4.8
300 | 8.8 | 4.4 | 11.6 | 6.0 | 9.2 | 3.8 | 7.0 | 3.2
400 | 12.8 | 7.0 | 12.2 | 7.0 | 8.6 | 4.2 | 9.6 | 4.2

######  Table 6.5.

Bootstrap size of $n\,R_{n}(X^{(p)},Y^{(p)})$: lattice case. Empirical
rejection rates of bootstrap test based on $nR_{n}(X^{(p)},Y^{(p)})$ for
independent Brownian and $1.8$-stable sheets $X,Y$ with $M=500,1000$
iterations. We choose $q=100$.

Table 6.7 shows the empirical rejection rates for independent Brownian and
$1.8$-stable sheets $X,Y$ in the random observation setting. We generate the
Poisson number $N_{p}$ and iid uniform locations $({\bf U}_{j})_{j\leq
N_{p}}$. To generate the discretized Brownian sheets we calculate the
correlation at the given random lattice and use the multivariate Gaussian
random generator in R (mvfast) and choose $p=1000$. For generating a stable
sheet at random locations, we first proceed as in the lattice case, generating
independent stable random variables for each cell of the random lattice, where
the lattice is constructed by cutting $[0,1]^{2}$ at all marginal points of
$({\bf U}_{j})$. Then we calculate the sheet at ${\bf U}_{j}$ by summing the
cells, starting from the origin. The computational complexity for doing this
is high and therefore we restrict ourselves to the smaller Poisson parameter
$p=500$. As discussed in Section 4 the bootstrap of
$Z^{(p)}=(X^{(p)},Y^{(p)})$ is conducted conditionally on $N_{p}$. In Tables
6.7 and 6.8 we present results for $N_{500}=492$ and $N_{1000}=982$. We have
examined the bootstrap procedure for several realizations of $N_{p}$ with
$p=500,1000$, but we did not find differences in the performance.

In Table 6.8 we illustrate the power of the test in the random observation
case. The dependence parameter $\rho$ and the sample size $n$ are the same as
in Table 6.6. Again, $p=1000$ ($p=500$) are the Poisson parameters of $N_{p}$
for Brownian (stable) sheets. When comparing Tables 6.8 and 6.6, the empirical
powers are quite similar, despite the different models for
$(X^{(p)},Y^{(p)})$.

| Dependent Brownian sheets | Dependent 1.8-stable Lévy sheets
---|---|---
$n$ | $100$ | $200$ | $300$ | $100$ | $200$ | $300$
$\rho\setminus\xi$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$
0.1 | $15.2$ | $29.2$ | $28.7$ | $38.2$ | $36.8$ | $51.0$ | $14.2$ | $25.6$ | $27,2$ | $42.2$ | $40.0$ | $53.6$
0.2 | $50.0$ | $65.0$ | $80.6$ | $87.6$ | $93.2$ | $96.4$ | $46.4$ | $63.2$ | $76.0$ | $86.4$ | $87.6$ | $92.8$
0.3 | $85.8$ | $93.6$ | $98.6$ | $99.0$ | $100$ | $100$ | $79.4$ | $86.8$ | $94.0$ | $94.8$ | $95.6$ | $96.8$
0.4 | $98.8$ | $99.6$ | $100$ | $100$ | $100$ | $100$ | $90.2$ | $93.6$ | $95.8$ | $96.8$ | $97.0$ | $98.0$
0.5 | $100$ | $100$ | $100$ | $100$ | $100$ | $100$ | $94.0$ | $95.4$ | $96.8$ | $97.4$ | $98.0$ | $98.6$

######  Table 6.6.

Bootstrap power of $nR_{n}(X^{(p)},Y^{(p)})$ with dependent stable sheets:
lattice case. Empirical rejection rates of bootstrap test based on
$nR_{n}(X^{(p)},Y^{(p)})$ for dependent Brownian sheets and $1.8$-stable
sheets. We choose $q=100$. In each cell of the table the rejection rate is
calculated from $M=500$ iterations.

| Brownian sheets | 1.8-stable sheets
---|---|---
$n$ | $\xi=0.1$ | $\xi=0.05$ | $\xi=0.1$ | $\xi=0.05$
100 | 12.2 | 5.4 | 10.6 | 4.0
200 | 11.4 | 6.0 | 8.6 | 4.0
300 | 10.6 | 3.6 | 8.5 | 4.4
400 | 8.8 | 5.6 | 9.0 | 3.8

######  Table 6.7.

Bootstrap size of $nR_{n}(X^{(p)},Y^{(p)})$: random location case. Empirical
rejection rates of bootstrap test based on $nR_{n}(X^{(p)},Y^{(p)})$ for
independent Brownian and $1.8$-stable sheets $X,Y$ with $M=500$ iterations.
There is a Poisson number $N_{p}$ of iid uniform locations on $(0,1)^{2}$, the
Poisson parameter is $p=1000$ ($p=500$) for Brownian (stable) sheets.

| Dependent Brownian sheets | Dependent 1.8-stable sheets
---|---|---
$n$ | $100$ | $200$ | $300$ | $100$ | $200$ | $300$
$\rho\setminus\xi$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$ | $0.05$ | $0.1$
0.1 | $17.0$ | $27.0$ | $26.2$ | $37.2$ | $38.4$ | $50.2$ | $14.0$ | $26.8$ | $26.2$ | $42.0$ | $39.8$ | $52.4$
0.2 | $52.2$ | $67.2$ | $79.6$ | $88.4$ | $95.4$ | $98.8$ | $47.0$ | $63.0$ | $77.4$ | $85.4$ | $88.4$ | $94.4$
0.3 | $88.2$ | $93.0$ | $99.0$ | $99.4$ | $100$ | $100$ | $81.6$ | $88.4$ | $92.6$ | $94.6$ | $96.2$ | $97.4$
0.4 | $99.4$ | $99.6$ | $100$ | $100$ | $100$ | $100$ | $92.8$ | $94.4$ | $95.6$ | $96.8$ | $97.8$ | $98.0$
0.5 | $100$ | $100$ | $100$ | $100$ | $100$ | $100$ | $94.4$ | $95.4$ | $96.8$ | $97.4$ | $98.2$ | $98.2$

######  Table 6.8.

Bootstrap power of $nR_{n}(X^{(p)},Y^{(p)})$: random location case. Empirical
rejection rates of bootstrap test based on $nR_{n}(X^{(p)},Y^{(p)})$ for
dependent Brownian and $1.8$-stable sheets. The number $N_{p}$ is Poisson
distributed with $p=1000$ ($p=500$) for Brownian (stable) sheets. In each cell
of the table the rejection rate corresponds to $M=500$ iterations.

##  7\.  An application to Japanese meteorological data

We apply our results to Japanese meteorological data. We choose the 3 most
fundamental factors: temperature (temp), precipitation (prec) and wind speed
(wind) which have been observed for a long time and over a wide range of
Japan. The data are available in various formats at the web-page of the
Japanese Meteorological Agency $\sc
https://www.data.jma.go.jp/gmd/risk/obsdl$. Since daily data include many
zeros and can be sparse, especially for precipitation, monthly average data
are taken. From January 1980 to January 2021 we have $493$ monthly data at
$783$ observation stations. There exist more such points in Japan. They are,
however, subject to problems such as change of position, many missing data, or
only precipitation is observed. We removed these points, but still have plenty
of points left; see the red dots in the map of Japan in Figure 7.1. Missing
values are observed at less than $30$ stations, corresponding to less than
$10$ out of $493$ months in total. A missing value at a station is replaced by
the average value at the station.

######  Figure 7.1.

Distribution of $783$ meteorological observation points scattered all over
Japan.

######  Figure 7.2.

Japanese $8$ regions
---

In our analysis we focus on the distance correlations for the pairs (prec &
temp), (prec & wind), (temp & wind) denoted by $R_{pt}$, $R_{pw}$, $R_{wt}$.
In a preliminary analysis we conduct pair-wise independence tests between
temp, prec and wind at each station. Here we use the classical sample distance
correlation/correlation coefficients (Peasen, Spearman, and Kendall) between
two components of a random vector. At each station we use the time series of
$493$ monthly observations and conduct bootstrap tests for pair-wise
independence based on $1000$ resamples. We follow the procedure described in
Section 6 with significance levels $\xi\in\\{0.05,0.01\\}$. For other
correlation coefficient tests, we use the “corr.test” R package. The results
are reported in Table 7.3. Each cell contains the number of rejections of the
hypothesis of pair-wise independence between (prec & temp), (prec & wind),
(temp & wind). The hypothesis of independence is overwhelmingly rejected at
the majority of stations while the distance correlation based test tends to
detect dependence more often than other correlation tests.

rejection number | prec & temp | prec & wind | temp & wind
---|---|---|---
methods $\setminus\,\xi$ | 0.05 | 0.01 | 0.05 | 0.01 | 0.05 | 0.01
Peason Corr. | 734 | 720 | 599 | 524 | 689 | 654
Spearman Corr. | 731 | 710 | 599 | 539 | 691 | 662
Kendall Corr. | 730 | 710 | 599 | 538 | 689 | 661
Dist. Corr. | 782 | 773 | 698 | 635 | 769 | 748

######  Table 7.3.

Rejection numbers of pair-wise independence for the 783 stations.

We interpret the monthly meteorological observations as realizations of a
random field on the area of Japan. We regard the monthly data as iid random
field observations and conduct independence tests on them. Figure 7.1 shows
quite convincingly that the stations are scattered randomly all over Japan, so
the assumption of uniformly distributed locations is plausible. We follow the
approach of random field at random locations from Sections 1.3 and 2. We
define distance as in (1.13) and rely on Theorem 2.6 for the theoretical
background on the independence test. The tests are based on the bootstrap
procedure introduced in the simulation study (Section 6); the theory for the
bootstrap was developed in Section 5.

The first line in Table 7.4 yields the pair-wise sample distance correlations
for the whole of Japan. They clearly hint at dependence between the 3 factors,
and this is also confirmed by bootstrap tests of the sample distance
correlation at different significance levels; see the right part of the table.
Next we study dependence locally in the distinct regions of Japan and at
different seasons of the year.

Most parts of Japan are classified as humid subtropical climate area. However,
the country stretches out from the far north to the far south, and the weather
is strongly influenced by the seasonal monsoon wind. The changes caused by the
4 seasons are significant; cf. [1]. Therefore we also test for independence at
regional level in each season. The seasonal samples consist of $4\times
123=492$ months, discarding the last month. Seasons are defined as common in
the Northern hemisphere: March–May, June–August, September–November,
December–February.

Table 7.4 provides the pair-wise (distance) correlations for annual and
quarterly meteorological data for the whole of Japan (Ja, 783 stations) as
well as $8$ main regions (indicated in Figure 7.2): Hokkaido (Ho, 159), Tohoku
(To, 137), Kanto (Ka, 73), Chubu (Chb, 140), Kinki (Ki,65), Chugoku (Chg, 71),
Shikoku (Shi, 40) and Kyusyu (Kyu, 98). After the names of the regions and
seasons the pair-wise distance correlations $R_{\cdot}$ and the results of the
independence tests based on the bootstrap for $R_{\cdot}$ are presented for
$1000$ resamples, following the previous section.

| (Dist.) Corr. | $R_{pt}$ | $R_{pw}$ | $R_{tw}$ | $R_{pt}$ | $R_{pw}$ | $R_{tw}$
---|---|---|---|---|---|---|---
Ja | Year | .567 | .477 | .693 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .220 | .272 | .354 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Summer | .329 | .287 | .208 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Autumn | .493 | .391 | .543 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .211 | .327 | .192 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
Ho | Year | .404 | .272 | .655 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .226 | .217 | .351 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Summer | .292 | .205 | .201 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Autumn | .329 | .325 | .644 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .259 | .309 | .140 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
To | Year | .398 | .338 | .668 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .121 | .146 | .297 | $\bigcirc$ | $\times$ | $\bigcirc$
| Summer | .165 | .222 | .233 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Autumn | .407 | .324 | .524 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .228 | .240 | .111 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
Ka | Year | .471 | .308 | .539 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .125 | .154 | .344 | $\bigcirc$ | $\bigtriangleup$ | $\bigcirc$
| Summer | .190 | .142 | .158 | $\bigcirc$ | $\times$ | $\bigcirc$
| Autumn | .387 | .321 | .457 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .086 | .176 | .202 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
Chb | Year | .394 | .366 | .679 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .159 | .184 | .371 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Summer | .153 | .201 | .120 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Autumn | .379 | .256 | .400 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .131 | .308 | .182 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
Ki | Year | .429 | .211 | .426 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .144 | .156 | .301 | $\bigcirc$ | $\bigtriangleup$ | $\bigcirc$
| Summer | .204 | .188 | .133 | $\bigcirc$ | $\bigtriangleup$ | $\bigcirc$
| Autumn | .402 | .173 | .190 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .135 | .233 | .208 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
Chg | Year | .311 | .225 | .514 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .112 | .127 | .221 | $\bigcirc$ | $\bigtriangleup$ | $\bigcirc$
| Summer | .165 | .151 | .161 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Autumn | .322 | .179 | .303 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .082 | .194 | .182 | $\bigtriangleup$ | $\bigcirc$ | $\bigcirc$
Shi | Year | .438 | .262 | .542 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .146 | .193 | .341 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Summer | .196 | .270 | .157 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Autumn | .414 | .195 | .233 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .194 | .141 | .261 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
Kyu | Year | .460 | .211 | .354 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Spring | .177 | .184 | .274 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Summer | .338 | .181 | .134 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Autumn | .406 | .173 | .172 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
| Winter | .193 | .161 | .189 | $\bigcirc$ | $\bigcirc$ | $\bigcirc$

######  Table 7.4.

Correlation and distance correlation of Japanese meteorological data. Pair-
wise (distance) correlations for annual and quarterly meteorological data of
the whole of Japan and $8$ main regions. Each row presents the corresponding
region and seasons, followed by $3$ cells with the sample distance
correlations and $3$ cells with the results of the independence tests. The
symbols $\bigcirc,\bigtriangleup,\times$ stand for rejection of the
independence hypothesis at both significance levels $\xi=0.05,\,0.01$, only at
$\xi=0.05$, no rejection, respectively.

We can extract several implications from the test results for the distance
correlation and its magnitude which are also in agreement with the research of
meteorologists; cf. [2]. The pair-wise distance correlations of the $3$
factors in the whole of Japan clearly imply dependence of all weather patterns
through the $4$ seasons. Especially, the values in autumn are rather high. The
situation is somewhat different if one looks at each region. The $3$ factors
still keep stable pair-wise dependence in autumn and winter in most regions
(an exception is the winter of Chugoku area), though the values of $R_{pt}$ in
winter are rather small in the southern region.

The autumn values are relatively high in most regions, in agreement with the
values for the whole country. Possible causes include the typhoon (tropical
cyclones) attacks in August and October, yielding a lot of precipitation and
strong wind speed together with changes in temperature. Rain fronts in autumn
can be another reason. While (prec & temp) and (temp & wind) indicate
dependence in all areas through the $4$ seasons, the relation between (prec &
wind) shows independence in several regions, especially in spring and summer.
One reason is that the baiu front, causing the rainy season, is stationary and
moves very slowly around in May–June. This fact may violate the dependence
relations between the $3$ factors.

##  8\. Proof of Proposition 2.1

Write for any Riemann square-integrable random field $Z$ on $B$ independent of
$(N^{(p)})_{p>0}$, and any $\beta\in(0,2]$,

$\displaystyle
D_{p}^{(\beta)}(Z)=\|Z^{(p)}\|_{2}^{\beta}-\|Z\|_{2}^{\beta}=\Big{(}N_{p}^{-1}\int_{B}Z^{2}\,dN^{(p)}\Big{)}^{\beta/2}-\Big{(}\int_{B}Z^{2}({\bf
u})\,d{\bf u}\Big{)}^{\beta/2}\,.$

Then we have

(8.1) $\displaystyle|D_{p}^{(\beta)}(Z)|\leq|D_{p}^{(2)}(Z)|^{\beta/2}\,.$

We start by proving (2.1). We have

(8.2) $\displaystyle
T_{n,\beta}(X^{(p)},Y^{(p)})-T_{n,\beta}(X,Y)=I_{1}+I_{2}-2\,I_{3},$

where

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\dfrac{1}{n^{2}}\sum_{k,l=1}^{n}\Big{(}\|X_{k}^{(p)}-X_{l}^{(p)}\|_{2}^{\beta}\,\|Y_{k}^{(p)}-Y_{l}^{(p)}\|_{2}^{\beta}-\|X_{k}-X_{l}\|_{2}^{\beta}\,\|Y_{k}-Y_{l}\|_{2}^{\beta}\Big{)}\,,$
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{n^{2}}\,\sum_{k,l=1}^{n}\|X_{k}^{(p)}-X_{l}^{(p)}\|_{2}^{\beta}\,\dfrac{1}{n^{2}}\,\sum_{k,l=1}^{n}\|Y_{k}^{(p)}-Y_{l}^{(p)}\|_{2}^{\beta}-\dfrac{1}{n^{2}}\,\sum_{k,l=1}^{n}\|X_{k}-X_{l}\|_{2}^{\beta}\dfrac{1}{n^{2}}\,\sum_{k,l=1}^{n}\|Y_{k}-Y_{l}\|_{2}^{\beta},$
$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\dfrac{1}{n^{3}}\sum_{k,l,m=1}^{n}\Big{(}\|X_{k}^{(p)}-X_{l}^{(p)}\|_{2}^{\beta}\,\|Y_{k}^{(p)}-Y_{m}^{(p)}\|_{2}^{\beta}-\|X_{k}-X_{l}\|_{2}^{\beta}\|Y_{k}-Y_{m}\|_{2}^{\beta}\Big{)}.$

We have

$\displaystyle|I_{1}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{n^{2}}\sum_{k,l=1}^{n}|D_{p}^{(\beta)}(X_{k}-X_{l})|\,\|Y_{k}^{(p)}-Y_{l}^{(p)}\|_{2}^{\beta}+\frac{1}{n^{2}}\sum_{k,l=1}^{n}|D_{p}^{(\beta)}(Y_{k}-Y_{l})|\,\|X_{k}-X_{l}\|_{2}^{\beta}$
$\displaystyle=:$ $\displaystyle I_{11}+I_{12}.$

Since the conditions on $X$ and $Y$ are symmetric it suffices to consider
$I_{11}$. It will be convenient to set $Z=X_{1}-X_{2}$. Since
$((X_{i},Y_{i}))$ are iid and independent of $N^{(p)}$ in view of (8.1) we get
the bound

$\displaystyle{\mathbb{E}}[I_{11}\mid N^{(p)}]$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\big{[}|D_{p}^{(\beta)}(Z)|\mid
N^{(p)}\big{]}{\mathbb{E}}\big{[}\|Y_{1}^{(p)}-Y_{2}^{(p)}\|^{\beta}_{2}\mid
N^{(p)}\big{]}$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\big{[}|D_{p}^{(2)}(Z)|^{\beta/2}\mid
N^{(p)}\big{]}\,{\mathbb{E}}\big{[}\|Y_{1}^{(p)}-Y_{2}^{(p)}\|^{\beta}_{2}\mid
N^{(p)}\big{]}\,.$

Assume that $X,Y$ have finite variance. Then, by the strong law of large
numbers,

$\displaystyle{\mathbb{E}}\big{[}|D_{p}^{(2)}(Z)|^{\beta/2}\mid
N^{(p)}\big{]}$ $\displaystyle\leq$
$\displaystyle\Big{(}\dfrac{1}{N_{p}}\sum_{i=1}^{N_{p}}{\mathbb{E}}[|Z|^{2}({\bf
U}_{i})\mid{\bf U}_{i}]+\int_{B}{\mathbb{E}}[|Z|^{2}({\bf u})]\,d{\bf
u}\Big{)}^{\beta/2}$ $\displaystyle\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}$
$\displaystyle\Big{(}2\,\int_{B}{\mathbb{E}}[|Z|^{2}({\bf u})]\,d{\bf
u}\Big{)}^{\beta/2}\,,\qquad p\to\infty\,.$

The right-hand side is finite by assumption (1a). If $X,Y$ have infinite
variance and $\beta\in(0,2)$ then

$\displaystyle{\mathbb{E}}[|D_{p}^{(2)}(Z)|^{\beta/2}\mid N_{p}]\leq
2\,{\mathbb{E}}\big{[}\sup_{{\bf u}\in B}|Z|^{\beta}({\bf u})\big{]}.$

The right-hand side is finite by assumption (1b). Conditional on
$(N^{(p)})_{p>0}$ we have the convergence

$\displaystyle|D_{p}^{(2)}(Z)|^{\beta/2}\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}0\,,\qquad p\to\infty\,,$

due the convergence of the Riemann sums $\|Z^{(p)}\|_{2}^{2}\to\|Z\|_{2}^{2}$
for a.e. realization of $Z$. Then an application of dominated convergence
yields

$\displaystyle{\mathbb{E}}\big{[}|D_{p}^{(2)}(Z)|^{\beta/2}\mid
N^{(p)}\big{]}\stackrel{{\scriptstyle\rm a.s.}}{{\rightarrow}}0\,.$

A similar argument shows that
$({\mathbb{E}}\big{[}\|Y_{1}^{(p)}-Y_{2}^{(p)}\|^{\beta/2}\mid
N^{(p)}\big{]})$ is bounded a.s. We conclude that
${\mathbb{E}}\big{[}|I_{1}|\mid N^{(p)}\big{]}\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}0\,,\,p\to\infty\,.$ We can deal with $I_{2},\,I_{3}$ in
the same way by observing that

$\displaystyle|I_{2}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{n^{2}}\sum_{k,l=1}^{n}|D_{p}^{(\beta)}(X_{k}-X_{l})|\,\dfrac{1}{n^{2}}\sum_{k,l=1}^{n}\|Y_{k}^{(p)}-Y_{l}^{(p)}\|_{2}^{\beta}$
$\displaystyle+\dfrac{1}{n^{2}}\sum_{k,l=1}^{n}\|X_{k}-X_{l}\|_{2}^{\beta}\frac{1}{n^{2}}\sum_{k,l=1}^{n}|D_{p}^{(\beta)}(Y_{k}-Y_{l})|,$

and

$\displaystyle|I_{3}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{n^{3}}\sum_{k,l,m=1}^{n}|D_{p}^{(\beta)}(X_{k}-X_{l})|\,\|Y_{k}^{(p)}-Y_{m}^{(p)}\|_{2}^{\beta}+\frac{1}{n^{3}}\sum_{k,l,m=1}^{n}|D_{p}^{(\beta)}(Y_{k}-Y_{m})|\,\|X_{k}-X_{l}\|_{2}^{\beta}.$

We omit further details.

Next we deal with the case $X=Y$. Again using the decomposition (8.2) and
writing $Z=X_{1}-X_{2}$, we observe that

$\displaystyle{\mathbb{E}}[|I_{1}|\,\mid\,N^{(p)}]$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}[|D_{p}^{(\beta)}(Z)|\,(\|Z^{(p)}\|_{2}^{\beta}+\|Z\|_{2}^{\beta})\,\mid\,N^{(p)}]$
$\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\big{[}\|Z\|_{2}^{2\beta}+\|Z^{(p)}\|_{2}^{2\beta}\,\mid\,N^{(p)}\big{]}\,.$

Since we already know that $D_{p}^{(\beta)}(Z)\stackrel{{\scriptstyle\rm
a.s.}}{{\rightarrow}}0$ conditionally on $(N^{(p)})_{p>0}$ we intend to use
dominated convergence to show that the left-hand side converges to zero a.s.
Therefore we will bound the right-hand terms. First assume that $X$ has finite
variance. We observe that

$\displaystyle{\mathbb{E}}\big{[}\|Z\|_{2}^{2\beta}\mid
N^{(p)}\big{]}={\mathbb{E}}\Big{[}\Big{(}\int_{B}Z^{2}({\bf u})\,d{\bf
u}\Big{)}^{\beta}\Big{]}\leq\left\\{\begin{array}[]{ll}\Big{(}\int_{B}{\mathbb{E}}[Z^{2}({\bf
u})\,d{\bf u}]\Big{)}^{\beta}&\text{for}\quad\beta\leq 1\,,\\\
\int_{B}{\mathbb{E}}[|Z|^{2\beta}]({\bf u})\,d{\bf
u}&\text{for}\quad\beta>1\,.\end{array}\right.$

The right-hand side is finite by assumption (2a). If $X$ has infinite variance
and $\beta\in(0,1)$ we have
${\mathbb{E}}\big{[}\|Z\|_{2}^{2\beta}\big{]}\leq{\mathbb{E}}\big{[}\sup_{{\bf
u}\in B}|Z|^{2\beta}({\bf u})\big{]}\,,$ and the right-hand side is finite by
assumption (2b). The same bounds apply to

$\displaystyle{\mathbb{E}}\big{[}\|Z\|_{2}^{2\beta}\mid N^{(p)}\big{]}$
$\displaystyle={\mathbb{E}}\big{[}\big{(}N_{p}^{-1}\sum_{i=1}^{N_{p}}Z^{2}({\bf
U}_{i})\big{)}^{\beta}\mid N^{(p)}\big{]}\,.$

Thus for any $\varepsilon>0$,
${\mathbb{P}}(|I_{1}|>\varepsilon\,\mid\,N^{(p)})\to 0$ as $n\to\infty$ along
a.e. sample path of $(N^{(p)})$.

We also have

$\displaystyle|I_{2}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{n^{2}}\sum_{k,l=1}^{n}|D_{p}^{(\beta)}(X_{k}-X_{l})|\frac{1}{n^{2}}\sum_{k,l=1}^{n}\Big{(}\|X_{k}^{(p)}-X_{l}^{(p)}\|_{2}^{\beta}+\|X_{k}-X_{l}\|_{2}^{\beta}\Big{)}=:I_{21}\,I_{22}.$

We have already proved

$\displaystyle{\mathbb{E}}[I_{21}^{(\beta)}\mid N^{(p)}]$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}[|D_{p}^{(\beta)}(Z)|\mid N^{(p)}]\to 0\,.$

Moreover,

$\displaystyle{\mathbb{E}}[I_{22}\mid
N^{(p)}]\leq{\mathbb{E}}[\|Z^{(p)}\|_{2}^{\beta}\mid
N^{(p)}]+{\mathbb{E}}[\|Z\|_{2}^{\beta}]\,.$

We have already shown that the right-hand side is bounded for a.e. sample path
of $(N^{(p)})_{p>0}$. Therefore we have for any $\varepsilon>0$ and a.e.
sample path of $(N^{(p)})_{p>0}$,

$\displaystyle{\mathbb{P}}\big{(}|I_{2}|>\varepsilon\mid N^{(p)}\big{)}\to
0\,,\qquad n\to\infty\,.$

We have

$\displaystyle|I_{3}|\leq\frac{1}{n^{3}}\sum_{k,l,m=1}^{n}|D_{p}^{(\beta)}(X_{k}-X_{l})|\,\big{(}\|X_{k}^{(p)}-X_{m}^{(p)}\|_{2}^{\beta}+\|X_{k}-X_{l}\|_{2}^{\beta}\big{)},$

hence by Cauchy-Schwarz,

$\displaystyle{\mathbb{E}}\big{[}|I_{3}|\mid N^{(p)}\big{]}$
$\displaystyle\leq$
$\displaystyle{\mathbb{E}}\big{[}|D_{p}^{(\beta)}(X_{1}-X_{2})|\big{(}\|X_{1}^{(p)}-X_{3}^{(p)}\|_{2}^{\beta}+\|X_{1}-X_{3}\|_{2}^{\beta}\big{)}\mid
N^{(p)}\big{]}$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\big{[}|D_{p}^{(2)}(X_{1}-X_{2})|^{\beta/2}\big{(}\|X_{1}^{(p)}-X_{3}^{(p)}\|_{2}^{\beta}+\|X_{1}-X_{3}\|_{2}^{\beta}\big{)}\mid
N^{(p)}\big{]}$ $\displaystyle\leq$
$\displaystyle\big{(}{\mathbb{E}}\big{[}|D_{p}^{(2)}(Z)|^{\beta}\mid
N^{(p)}\big{]}\big{)}^{1/2}\Big{(}\big{(}{\mathbb{E}}\big{[}\|Z^{(p)}\|_{2}^{2\beta}\mid
N^{(p)}\big{]}\big{)}^{1/2}+\big{(}{\mathbb{E}}\big{[}\|Z\|_{2}^{2\beta}\big{]}\big{)}^{1/2}\Big{)}\,.$

The right-hand side converges to zero by arguments similar to those above. We
finally conclude that

${\mathbb{P}}\big{(}|I_{3}|>\varepsilon\mid N^{(p)}\big{)}\to 0\,,\qquad
n\to\infty\,.$

This finishes the proof of (2.2).

##  9\. Proof of Proposition 4.1

######  Lemma 9.1.

$(1)$ Assume $\beta\in(0,2)$ and, if $X$ has finite second or $\beta$th
moment, we assume (A1) or (B1), respectively. Then there exists a constant
$c>0$ such that

$\displaystyle\sup_{p}{\mathbb{E}}[\|X^{(p)}\|^{\beta}_{2}\mid{\bf
U}^{(p)}]\leq c\qquad{\rm a.s.}$

where ${\bf U}^{(p)}=(U_{1},\ldots,U_{p})$.
$(2)$ If $\beta\in(1,2)$ and $\max_{{\bf t}\in B}{\mathbb{E}}[|X({\bf
t})|^{2(2\beta-1)}]<\infty$ then we have
${\mathbb{E}}[\|X\|^{2(2\beta-1)}_{2}]<\infty,$ and there exists a constant
$c>0$ such that

$\displaystyle\sup_{p}{\mathbb{E}}[\|X^{(p)}\|^{2(2\beta-1)}_{2}\mid{\bf
U}^{(p)}]\leq c\qquad{\rm a.s.}$

###### Proof.

$\rm(1)$ Assume (A1). Then $\sup_{{\bf t}\in B}{\mathbb{E}}[X^{2}({\bf
t})]=c_{0}<\infty$. In view of (4.2) and by Jensen’s inequality we have

$\displaystyle{\mathbb{E}}[\|X^{(p)}\|_{2}^{\beta}\mid{\bf U}^{(p)}]$
$\displaystyle=$
$\displaystyle{\mathbb{E}}\Big{[}\big{(}\tilde{p}^{-1}\mbox{$\sum^{\prime}$}\overline{X}_{j}^{2}\big{)}^{\beta/2}\Big{|}{\bf
U}^{(p)}\Big{]}$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\Big{[}\Big{(}\tilde{p}^{-1}\mbox{$\sum^{\prime}$}\sum_{\ell:{\bf
U}_{\ell}\in\Delta_{j}}X^{2}({\bf
U}_{\ell})/\\#\Delta_{j}\Big{)}^{\beta/2}\Big{|}{\bf U}^{(p)}\Big{]}$
$\displaystyle\leq$
$\displaystyle\Big{(}\tilde{p}^{-1}\mbox{$\sum^{\prime}$}\sum_{\ell:{\bf
U}_{\ell}\in\Delta_{j}}{\mathbb{E}}[X^{2}({\bf U}_{\ell})\mid{\bf
U}_{\ell}]/{\\#\Delta_{j}}\Big{)}^{\beta/2}\leq c_{0}^{\beta/2}\qquad{\rm
a.s.}$

Assume (B1). Then by similar arguments

$\displaystyle{\mathbb{E}}[\|X^{(p)}\|_{2}^{\beta}\mid{\bf U}^{(p)}]$
$\displaystyle\leq$ $\displaystyle c\,{\mathbb{E}}\big{[}\big{(}\sup_{{\bf
t}\in B}X^{2}({\bf t})\big{)}^{\beta/2}\big{]}\leq
c\,{\mathbb{E}}\big{[}\sup_{{\bf t}\in B}|X({\bf
t})|^{\beta}\big{]}<\infty\qquad{\rm a.s.}$

$(2)$ The first statement is immediate. Using Jensen’s inequality, we also
have

$\displaystyle{\mathbb{E}}[\|X^{(p)}\|^{2(2\beta-1)}_{2}\mid{\bf U}^{(p)}]$
$\displaystyle\leq$
$\displaystyle\tilde{p}^{-1}\mbox{$\sum^{\prime}$}\sum_{k:{\bf
U}_{k}\in\Delta_{j}}{\mathbb{E}}[|X({\bf U}_{k})|^{2(2\beta-1)}\mid{\bf
U}_{k}]/\\#\Delta_{j}$ $\displaystyle\leq$ $\displaystyle c\,\max_{{\bf t}\in
B}{\mathbb{E}}[|X({\bf t})|^{2(2\beta-1)}]<\infty\qquad{\rm a.s.}$

∎

Now we proceed with the proof of the proposition. In what follows, it will be
convenient to write $Z=X_{1}-X_{2}$. We use the decomposition (8.2) adjusted
to the present situation.
1. We assume (A1).
First we study $I_{1}$. By a symmetry argument it suffices to consider
$I_{11}$.
Assume $\beta\in(0,1]$. Using the independence of $X,Y$ and [Lemma 9.1 (1)],
we have

$\displaystyle{\mathbb{E}}[I_{11}]$
$\displaystyle={\mathbb{E}}[{\mathbb{E}}[I_{11}\mid{\bf U}^{(p)}]]$
$\displaystyle\leq{\mathbb{E}}\big{[}{\mathbb{E}}\big{[}|D_{p}^{(\beta)}(Z)|\,\big{|}\,{\bf
U}^{(p)}\big{]}\,{\mathbb{E}}[\|Y_{1}^{(p)}-Y_{2}^{(p)}\|_{2}^{\beta}\,\mid\,{\bf
U}^{(p)}]\big{]}$ (9.1) $\displaystyle\leq
c\,{\mathbb{E}}[|D_{p}^{(\beta)}(Z)|]\leq
c\,{\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{\beta}]\,,$

where we used $\beta\in(0,1]$ in the last step. We have

$\displaystyle{\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{\beta}]$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\Big{[}\Big{(}\sum_{j=1}^{\tilde{p}}\int_{\Delta_{j}}(\overline{Z}_{j}-Z({\bf
t}))^{2}\,d{\bf t}\Big{)}^{\beta/2}\Big{]}$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\Big{[}\Big{(}\mbox{$\sum^{\prime}$}\int_{\Delta_{j}}(\overline{Z}_{j}-Z({\bf
t}))^{2}\,d{\bf
t}\Big{)}^{\beta/2}\Big{]}+{\mathbb{E}}\Big{[}\Big{(}\sum_{j\leq\tilde{p}:\\#\Delta_{j}=0}\int_{\Delta_{j}}Z^{2}({\bf
t})\,d{\bf t}\Big{)}^{\beta/2}\Big{]}\,.$ $\displaystyle=:$ $\displaystyle
J_{1}+J_{2}\,.$

Then an application of Jensen’s inequality yields

$\displaystyle J_{2}$ $\displaystyle\leq$
$\displaystyle\Big{(}{\mathbb{E}}\Big{[}\sum_{j\leq\tilde{p}:\\#\Delta_{j}=0}\int_{\Delta_{j}}Z^{2}({\bf
t})\,d{\bf t}\Big{]}\Big{)}^{\beta/2}$ $\displaystyle=$
$\displaystyle\Big{(}{\mathbb{E}}\Big{[}\sum_{j\leq\tilde{p}:\\#\Delta_{j}=0}{\mathbb{E}}\big{[}\int_{\Delta_{j}}Z^{2}({\bf
t})\,d{\bf t}\big{|}{\bf U}^{(p)}\big{]}\Big{]}\Big{)}^{\beta/2}$
$\displaystyle\leq$ $\displaystyle c\,\Big{(}\max_{{\bf t}\in B}{\rm
var}(X({\bf
t}))\Big{)}^{\beta/2}\,\Big{(}{\mathbb{E}}\Big{[}\tilde{p}^{-1}\sum_{j\leq\tilde{p}}{\mathbf{1}}(\\#\Delta_{j}=0)\Big{]}\Big{)}^{\beta/2}$
$\displaystyle\leq$ $\displaystyle
c\,\Big{(}{\mathbb{P}}\Big{(}\sum_{k=1}^{p}{\mathbf{1}}_{\Delta_{1}}({\bf
U}_{k})=0\Big{)}\Big{)}^{\beta/2}$ $\displaystyle=$ $\displaystyle
c\,\big{(}(1-\tilde{p}^{-1})^{p}\big{)}^{\beta/2}\leq c\,p^{-\beta/2}\,.$

Now we turn to $J_{1}$. Then applications of Jensen’s inequality and (A1)
yield

$\displaystyle J_{1}$ $\displaystyle\leq$
$\displaystyle\Big{(}{\mathbb{E}}\Big{[}\mbox{$\sum^{\prime}$}\int_{\Delta_{j}}(\overline{Z}_{j}-Z({\bf
t}))^{2}\,d{\bf t}\Big{]}\Big{)}^{\beta/2}$ $\displaystyle\leq$
$\displaystyle\Big{(}{\mathbb{E}}\Big{[}\mbox{$\sum^{\prime}$}(\\#\Delta_{j})^{-1}\sum_{k:{\bf
U}_{k}\in\Delta_{j}}\int_{\Delta_{j}}(Z({\bf U}_{k})-Z({\bf t}))^{2}d{\bf
t}\Big{]}\Big{)}^{\beta/2}$ $\displaystyle=$
$\displaystyle\Big{(}{\mathbb{E}}\Big{[}\mbox{$\sum^{\prime}$}(\\#\Delta_{j})^{-1}\sum_{k:{\bf
U}_{k}\in\Delta_{j}}\int_{\Delta_{j}}{\mathbb{E}}\big{[}(Z({\bf U}_{k})-Z({\bf
t}))^{2}\mid{\bf U}_{k}\big{]}d{\bf t}\Big{]}\Big{)}^{\beta/2}$
$\displaystyle\leq$ $\displaystyle c\,\max_{j=1,\ldots,\tilde{p}}\max_{{\bf
v},{\bf t}\in\Delta_{j}}\,\big{(}{\mathbb{E}}\big{[}(X({\bf v})-X({\bf
t}))^{2}\big{]}\big{)}^{\beta/2}$ $\displaystyle\leq$ $\displaystyle c\,|{\bf
1}/\tilde{p}^{1/d}|^{\gamma_{X}\beta/2}=(\sqrt{d/\tilde{p}^{2/d}})^{\gamma_{X}\beta/2}=c\,\tilde{p}^{-\beta\gamma_{X}/(2d)}\,.$

We conclude that

$\displaystyle{\mathbb{E}}[I_{11}]\leq c\,(J_{1}+J_{2})\leq
c\,\big{(}p^{-1}+\tilde{p}^{-\gamma_{X}/d}\big{)}^{\beta/2}\,,$

and in turn

(9.3) $\displaystyle{\mathbb{E}}[|I_{1}|]\leq
c\,\big{(}p^{-1}+\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d}\big{)}^{\beta/2}\,.$

Assume $\beta\in(1,2)$. Applying $|x^{\beta}-y^{\beta}|\leq(x\vee
y)^{\beta-1}|x-y|$ for positive $x,y$ and Hölder’s inequality, we obtain

$\displaystyle{\mathbb{E}}\big{[}|D_{p}^{(\beta)}(Z)|\big{]}$
$\displaystyle\leq$ $\displaystyle
c\,{\mathbb{E}}\big{[}\big{(}\|Z^{(p)}\|_{2}^{\beta-1}\vee\|Z\|_{2}^{\beta-1}\big{)}\,|D_{p}^{(1)}(Z)|\big{]}$
$\displaystyle\leq$ $\displaystyle
c\,\big{(}{\mathbb{E}}\big{[}\|Z^{(p)}\|_{2}^{2}\vee\|Z\|_{2}^{2}|\big{]}\big{)}^{(\beta-1)/2}\times\big{(}{\mathbb{E}}\big{[}\|Z^{(p)}-Z\|_{2}^{2/(3-\beta)}\big{]}\big{)}^{(3-\beta)/2}=:c\,P_{1}P_{2}.$

Since we assume finite second moments via (A1) we have $P_{1}<\infty$.
Moreover,

$\displaystyle P_{2}^{2/(3-\beta)}$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\big{[}\|Z^{(p)}-Z\|_{2}^{2/(3-\beta)}\big{]}={\mathbb{E}}\Big{[}\Big{(}\sum_{j=1}^{\tilde{p}}\int_{\Delta_{j}}(\overline{Z}_{j}-Z({\bf
t}))^{2}\,d{\bf t}\big{)}\Big{)}^{1/(3-\beta)}\Big{]}\,.$

Observing that $(3-\beta)^{-1}<1$ we can use the same arguments as for
$\beta\in(0,1]$ to obtain the bound

$\displaystyle P_{2}^{2/(3-\beta)}\leq
c\,\big{(}p^{-1}+\tilde{p}^{-\gamma_{X}/d}\big{)}^{1/(3-\beta)}.$

Under (A1), we conclude that we have the following upper bound for
$\beta\in(1,2)$:

$\displaystyle{\mathbb{E}}[|I_{1}|]\leq
c\,(\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d}+p^{-1})^{1/2}.$

Combining this bound with (9.3), we finally have for any $\beta\in(0,2)$,

$\displaystyle{\mathbb{E}}[|I_{1}|]\leq\,c\,(\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d}+p^{-1})^{(1\wedge\beta)/2}\,.$

Now we turn to $I_{2},I_{3}$. We have

$\displaystyle|I_{2}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{n^{2}}\sum_{k,l=1}^{n}|D_{p}^{(\beta)}(X_{k}-X_{l})|\,\frac{1}{n^{2}}\sum_{k,l=1}^{n}\|Y_{k}^{(p)}-Y_{l}^{(p)}\|_{2}^{\beta}+\frac{1}{n^{2}}\sum_{k,l=1}^{n}\|X_{k}-X_{l}\|_{2}^{\beta}\frac{1}{n^{2}}\sum_{k,l=1}^{n}|D_{p}^{(\beta)}(Y_{k}-Y_{l})|$

and a similar bound holds for $|I_{3}|$. The same arguments as for
${\mathbb{E}}[|I_{1}|]$ yield

${\mathbb{E}}[|I_{2}+I_{3}|]\leq
c\big{(}\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d}+p^{-1}\big{)}^{(1\wedge\beta)/2}.$

We omit further details.
2. Now we assume that $X,Y$ have finite $\beta$th moment for some $\beta\in(0,2)$ and (B1), (B2) hold.
First we study $I_{1}$ and focus on $I_{11}$. We follow the patterns of the
proof under condition (A1) in the finite variance case.
Assume $\beta\in(0,1]$. We again write $Z=X_{1}-X_{2}$. In view of (9.1) and
[Lemma 9.1 (1)] it suffices to bound $J_{1},J_{2}$. If (B1) holds we can
proceed as in part 1.:

$\displaystyle J_{2}$ $\displaystyle\leq$
$\displaystyle\Big{(}{\mathbb{E}}\Big{[}\tilde{p}^{-1}\sum_{j=1}^{\tilde{p}}{\mathbf{1}}(\\#\Delta_{j}=0)\Big{]}\Big{)}^{\beta/2}\,{\mathbb{E}}\Big{[}\max_{{\bf
t}\in B}|Z({\bf t})|^{\beta}\Big{]}\leq c\,p^{-\beta/2}\,.$

We also have by (B2)

$\displaystyle J_{1}$ $\displaystyle\leq$ $\displaystyle
c\,\tilde{p}^{-\beta/2}\,\sum_{j=1}^{\tilde{p}}{\mathbb{E}}\Big{[}\max_{{\bf
v},{\bf t}\in\Delta_{j}}|X({\bf t})-X({\bf v})|^{\beta}\Big{]}\leq
c\,\widetilde{p}^{1-\gamma_{X}/d-\beta/2}\,.$

We conclude that

(9.5) $\displaystyle{\mathbb{E}}[|I_{1}|]\leq
cp^{1-\beta/2}(\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d}+p^{-1}).$

Similar bounds hold for $I_{2},I_{3}$. We omit details.
Assume $\beta\in(1,2)$. We start from (9):

$\displaystyle{\mathbb{E}}[|D_{p}^{(\beta)}(Z)|]$ $\displaystyle\leq$
$\displaystyle
c\,{\mathbb{E}}\big{[}\big{(}\|Z^{(p)}\|_{2}^{\beta-1}\vee\|Z\|_{2}^{\beta-1}\big{)}|D_{p}^{(1)}(Z)|\big{]}$
$\displaystyle\leq$ $\displaystyle
c\,\big{(}{\mathbb{E}}\big{[}\|Z^{(p)}\|_{2}^{\beta}\vee\|Z\|_{2}^{\beta}\big{]}\big{)}^{(\beta-1)/\beta}\big{(}{\mathbb{E}}\big{[}\|Z^{(p)}-Z\|_{2}^{\beta}\big{]}\big{)}^{1/\beta}=:c\,\widetilde{P}_{1}\widetilde{P}_{2}.$

The quantity $\widetilde{P}_{1}$ is bounded by a constant for all $p$.
Proceeding as for $\beta\in(0,1]$, we obtain

$\displaystyle\widetilde{P}_{2}\leq
c\big{(}p^{1-\beta/2}(\tilde{p}^{-\gamma_{X}/d}+p^{-1})\big{)}^{1/\beta},$

and we finally have

${\mathbb{E}}[|I_{1}|]\leq
c\big{(}p^{1-\beta/2}(\tilde{p}^{-(\gamma_{X}\wedge\gamma_{Y})/d}+p^{-1})\big{)}^{1/\beta}.$

The quantities ${\mathbb{E}}[|I_{i}|],i=2,3$ can be bounded in a similar way.
This proves part 2.
3. We will show that ${\mathbb{E}}[|T_{n,\beta}(X,X)-T_{n,\beta}(X^{(p)},X^{(p)})|]\to 0$ as $n\to\infty$. We again use the decomposition (8.2) for $X=Y$.
The finite variance case. Assume (A1), (A2).
Assume $\beta\in(0,1]$. We observe that

$\displaystyle{\mathbb{E}}[|I_{1}|]$
$\displaystyle\leq{\mathbb{E}}\big{[}\|Z^{(p)}-Z\|_{2}^{\beta}(\|Z^{(p)}\|_{2}^{\beta}+\|Z\|_{2}^{\beta})\big{]}$
(9.6)
$\displaystyle\leq\big{(}{\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{2\beta}]\big{)}^{1/2}\big{(}({\mathbb{E}}[\|Z^{(p)}\|_{2}^{2\beta}])^{1/2}+({\mathbb{E}}[\|Z\|_{2}^{2\beta}])^{1/2}\big{)}.$

We can re-use the argument for bounding
${\mathbb{E}}[\|Z^{(p)}-Z\|^{\beta}_{2}]$, replacing $\beta/2$ by $\beta$ in
the derivation. Thus the first expectation in (9.6) is bounded by
$c\big{(}\tilde{p}^{-\gamma_{X}/d}+p^{-1}\big{)}^{\beta}$. The remaining two
expectations are finite thanks to [Lemma 9.1 (1)]. Thus we have

${\mathbb{E}}[|I_{1}|]\leq
c\big{(}\tilde{p}^{-\gamma_{X}/d}+p^{-1}\big{)}^{\beta/2}\to 0\,,\qquad
n\to\infty\,.$

Assume $\beta\in(1,2)$. We may proceed similarly as for the distance
covariance. We have

$\displaystyle{\mathbb{E}}[|I_{1}|]$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}[\big{|}\|Z^{(p)}\|_{2}^{2\beta}-\|Z\|_{2}^{2\beta}\big{|}]$
$\displaystyle\leq$ $\displaystyle
c{\mathbb{E}}\big{[}(\|Z^{(p)}\|_{2}^{2\beta-1}\vee\|Z\|_{2}^{2\beta-1})\,\|Z^{(p)}-Z\|_{2}\big{]}$
$\displaystyle\leq$ $\displaystyle
c\big{(}{\mathbb{E}}[\|Z^{(p)}\|_{2}^{2(2\beta-1)}\vee\|Z\|_{2}^{2(2\beta-1)}]\big{)}^{1/2}\big{(}{\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{2}]\big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle
c\,\big{(}{\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{2}]\big{)}^{1/2}\,.$

In the last step we used [Lemma 9.1 (2)] and (A2). Now we may proceed as for
the bound of (9) to obtain

${\mathbb{E}}[|I_{1}|]\leq
c\big{(}\tilde{p}^{-\gamma_{X}/d}+p^{-1}\big{)}^{1/2}\to 0\,,\qquad
n\to\infty\,.$

We can deal with ${\mathbb{E}}[|I_{2}|],{\mathbb{E}}[|I_{3}|]$ in the same way
by observing that

$\displaystyle|I_{2}|\leq$
$\displaystyle\frac{1}{n^{4}}\sum_{j,k,l,m=1}^{n}|D_{p}^{(\beta)}(X_{j}-X_{k})|\|X_{l}^{(p)}-X_{m}^{(p)}\|_{2}^{\beta}+\frac{1}{n^{4}}\sum_{j,k,l,m=1}^{n}\|X_{j}-X_{l}\|_{2}^{\beta}|D_{p}^{(\beta)}(X_{l}-X_{m})|$
$\displaystyle=:I_{21}+I_{22}$

and

$\displaystyle|I_{3}|\leq$
$\displaystyle\frac{1}{n^{3}}\sum_{k,l,m=1}^{n}|D_{p}^{(\beta)}(X_{k}-X_{l})|\|X_{l}^{(p)}-X_{m}^{(p)}\|_{2}^{\beta}+\frac{1}{n^{3}}\sum_{k,l,m=1}^{n}\|X_{k}-X_{l}\|_{2}^{\beta}|D_{p}^{(\beta)}(X_{l}-X_{m})|,$
$\displaystyle=:I_{31}+I_{32}.$

This means that
${\mathbb{E}}[I_{21}],{\mathbb{E}}[I_{22}],{\mathbb{E}}[I_{31}],{\mathbb{E}}[I_{32}]$
can be bounded in a similar way and these expectations converge to zero as
$n\to\infty$. We illustrate this for $I_{32}$.
Assume $\beta\in(0,1]$. By the Cauchy-Schwarz inequality and using similar
bounds as above, we have

(9.7)
$\displaystyle\begin{split}{\mathbb{E}}[I_{32}]&\leq\big{(}{\mathbb{E}}[\|X_{1}-X_{2}\|_{2}^{2\beta}]\big{)}^{1/2}\big{(}{\mathbb{E}}[|D_{p}^{(\beta)}|^{2}]\big{)}^{1/2}\\\
&\leq\big{(}{\mathbb{E}}[\|X_{1}-X_{2}\|_{2}^{2\beta}]\big{)}^{1/2}\big{(}{\mathbb{E}}\big{[}\|X_{1}^{(p)}-X_{3}^{(p)}-(X_{1}-X_{3})\|_{2}^{2\beta}\big{]}\big{)}^{1/2}\\\
&\leq c\big{(}\tilde{p}^{-\gamma_{X}/d}+p^{-1}\big{)}^{\beta/2}\to 0\,,\qquad
n\to\infty\,.\end{split}$

For $\beta\in(1,2)$, multiple use of Hölder’s inequality yields

$\displaystyle{\mathbb{E}}[I_{32}]$ $\displaystyle\leq$ $\displaystyle
c{\mathbb{E}}[\|(X_{1}^{(p)}-X_{2}^{(p)})-(X_{1}-X_{2})\|_{2}\big{(}\|X_{1}^{(p)}-X_{2}^{(p)}\|_{2}^{\beta-1}\vee\|X_{1}-X_{2}\|_{2}^{\beta-1}\big{)}\|X_{1}-X_{3}\|_{2}^{\beta}]$
$\displaystyle\leq$ $\displaystyle
c\big{(}{\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{2}]\big{)}^{1/2}\big{(}{\mathbb{E}}[\|Z\|_{2}^{2(2\beta-1)}]\big{)}^{\frac{\beta}{2(2\beta-1)}}\,\big{(}{\mathbb{E}}[\|Z^{(p)}\|_{2}^{2(2\beta-1)}\vee\|Z\|_{2}^{2(2\beta-1)}]\big{)}^{\frac{\beta-1}{2(2\beta-1)}}$
$\displaystyle\leq$ $\displaystyle
c\,\big{(}{\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{2}]\big{)}^{1/2}\big{(}{\mathbb{E}}[\|Z\|_{2}^{2(2\beta-1)}]\big{)}^{\frac{\beta}{2(2\beta-1)}}\,$
$\displaystyle\times\Big{(}\big{(}{\mathbb{E}}[\|Z^{(p)}\|_{2}^{2(2\beta-1)}]\big{)}^{\frac{\beta-1}{2(2\beta-1)}}+\big{(}{\mathbb{E}}[\|Z\|_{2}^{2(2\beta-1)}]\big{)}^{\frac{\beta-1}{2(2\beta-1)}}\Big{)}.$

The first quantity is bounded by
$c\big{(}\tilde{p}^{-\gamma_{X}/d}+p^{-1}\big{)}^{1/2}$ and the other
quantities are finite by virtue of [Lemma 9.1 (2)]. Hence
${\mathbb{E}}[I_{32}]\to 0$, $n\to\infty$.
The infinite variance case. We assume a finite $2\beta$th moment of $X$,
$\beta\in(0,1)$, and (B3).
Let $2\beta\in(0,1]$. First we follow the inequality (9.6) of distance
variance and then we use the bound of
${\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{2\beta}]$ for the distance covariance in the
infinite variance case; see (9.5) with $\beta$ replaced by $2\beta$. By [Lemma
9.1 (1)], ${\mathbb{E}}[\|Z^{(p)}\|_{2}^{2\beta}]$ and
${\mathbb{E}}\|Z\|_{2}^{2\beta}]$ are bounded by a constant. As a result we
have by (B3),

(9.8) $\displaystyle{\mathbb{E}}[|I_{1}|]\leq
c\big{(}{\mathbb{E}}[\|Z^{(p)}-Z\|_{2}^{2\beta}]\big{)}^{1/2}\leq
c\big{(}p^{1-\beta}\big{(}\tilde{p}^{-\gamma_{X}^{\prime}/d}+p^{-1}\big{)}\big{)}^{1/2}.$

The right-hand side converges to zero as $n\to\infty$ provided
$1<\beta+\gamma_{X}^{\prime}/d$. For $2\beta\in(1,2)$ we have by Hölder’s
inequality

$\displaystyle{\mathbb{E}}[|I_{1}|]$
$\displaystyle\leq{\mathbb{E}}\big{[}\big{(}\|Z^{(p)}\|_{2}^{2\beta-1}\vee\|Z\|_{2}^{2\beta-1}\big{)}\|Z^{(p)}-Z\|_{2}\big{]}$
$\displaystyle\leq
c\big{(}{\mathbb{E}}[\|Z^{(p)}\|_{2}^{2\beta}\vee\|Z\|_{2}^{2\beta}]\big{)}^{(2\beta-1)/(2\beta)}\big{(}{\mathbb{E}}[\|Z-Z^{(p)}\|_{2}^{2\beta}]\big{)}^{1/(2\beta)}$
$\displaystyle=c\widehat{P}_{1}\widehat{P}_{2}.$

We can bound $\widehat{P}_{2}$ in the same way as for $\beta\in(0,1/2]$:

$\displaystyle\widehat{P}_{2}\leq
c\big{(}p^{1-\beta}\big{(}\tilde{p}^{-\gamma_{X}^{\prime}/d}+p^{-1}\big{)}\big{)}^{1/(2\beta)}.$

The quantity $\widehat{P}_{1}$ is finite by virtue of [Lemma 9.1 (1)] and
(B3). Thus we have the bound for general $\beta\in(0,1)$:

$\displaystyle{\mathbb{E}}[|I_{1}|]\leq
c\big{(}p^{1-\beta}(\tilde{p}^{-\gamma_{X}^{\prime}/d}+p^{-1})\big{)}^{(1\wedge\beta^{-1})/2}\to
0\,,\qquad n\to\infty.$

For $I_{2}$ and $I_{3}$, recall the bounds $|I_{2}|\leq I_{21}+I_{22}$ and
$|I_{3}|\leq I_{31}+I_{32}$, and take expectations on both sides of the
inequalities. We illustrate how to deal with ${\mathbb{E}}[I_{32}]$. Since
$2\beta\in(0,2)$ inequality (9.7) holds. Then for $2\beta\in(0,1]$ it follows
from (9.8) that ${\mathbb{E}}[I_{32}]\leq
c\big{(}p^{1-\beta}(\tilde{p}^{-\gamma_{X}^{\prime}/d}+p^{-1})\big{)}^{1/2}$,
and for $2\beta\in(1,2)$ we obtain the same bound as for $\widehat{P}_{2}$ by
the similar calculations.
This finishes the proof.

##  Appendix A Negative moments of a Poisson random variable

We consider a family $(N_{p})_{p>0}$ of Poisson random variables with
${\mathbb{E}}[N_{p}]=p$ and are interested in the asymptotic behavior of
${\mathbb{E}}[N_{p}^{-\gamma}{\mathbf{1}}(N_{p}>0)]$ for $\gamma>0$ as
$p\to\infty$.

######  Lemma A.1.

We consider a family $(N_{p})_{p>0}$ of Poisson random variables with
${\mathbb{E}}[N_{p}]=p$. Then for any $\gamma>0$,
$p^{\gamma}{\mathbb{E}}[N_{p}^{-\gamma}{\mathbf{1}}(N_{p}>0)]\to 1$.

###### Proof.

We observe that

$\displaystyle{\mathbb{E}}[N_{p}^{-\gamma}{\mathbf{1}}(N_{p}>0)]$
$\displaystyle=$ $\displaystyle\Big{(}\sum_{k\in A_{p}}+\sum_{k\in
A_{p}^{c}}\Big{)}k^{-\gamma}{\rm e}\,^{-p}\dfrac{p^{k}}{k!}\,,$

where $A_{p}=\\{k\geq 1:|k-p|>\varepsilon_{p}\,p\\}$ and $A_{p}^{c}=\\{k\geq
1:|k-p|\leq\varepsilon_{p}\,p\\}$ and $\varepsilon_{p}=\sqrt{(C\log p)/p}$ for
some constant $C=C(\gamma)>0$. We will show that

(A.1) $\displaystyle p^{\gamma}\,{\mathbb{P}}(|N_{p}-p|>\varepsilon_{p}\,p)\to
0\,,\qquad p\to\infty\,.$

Then ${\mathbb{P}}(|N_{p}-p|\leq\varepsilon_{p}\,p)\to 1$ and

$\displaystyle p^{\gamma}\,\sum_{k\in A_{p}^{c}}k^{-\gamma}{\rm
e}\,^{-p}\dfrac{p^{k}}{k!}$ $\displaystyle=$ $\displaystyle\sum_{k\in
A_{p}^{c}}(p/k)^{\gamma}{\rm e}\,^{-p}\dfrac{p^{k}}{k!}$ $\displaystyle=$
$\displaystyle(1+o(1))\,{\mathbb{P}}(|N_{p}-p|\leq\varepsilon_{p}\,p)\to 1\,.$

Next we turn to the proof of (A.1). By Markov’s inequality and a Taylor
expansion we have for $h=\varepsilon_{p}$,

$\displaystyle p^{\gamma}\,{\mathbb{P}}(N_{p}>(1+\varepsilon_{p})\,p)$
$\displaystyle\leq$ $\displaystyle p^{\gamma}\,{\rm
e}\,^{-h\,(1+\varepsilon_{p})\,p}\,{\mathbb{E}}[{\rm
e}\,^{h\,N_{p}}]=p^{\gamma}\,\exp(-h\,(1+\varepsilon_{p})\,p-p(1-{\rm
e}\,^{h}))$ $\displaystyle=$ $\displaystyle
p^{\gamma}\,\exp\big{(}-(\varepsilon_{p}\,p+\varepsilon_{p}^{2}\,p)+(\varepsilon_{p}\,p+0.5\varepsilon_{p}^{2}\,p(1+o(1)))\big{)}$
$\displaystyle=$
$\displaystyle\exp\big{(}-0.5\varepsilon_{p}^{2}\,p(1+o(1))+\gamma\,\log
p\big{)}$ $\displaystyle=$ $\displaystyle\exp\big{(}-0.5C\log
p(1+o(1))+\gamma\,\log p\big{)}\to 0\,,\qquad p\to\infty,$

provided $C>2\gamma$. On the other hand, by monotonicity of the Poisson
probabilities for $k\leq p$ we have

$\displaystyle{\rm e}\,^{-p}\dfrac{p^{k}}{k!}\leq{\rm
e}\,^{-p}\dfrac{p^{[p(1-\varepsilon_{p})]}}{[p(1-\varepsilon_{p})]!}\,,\qquad
k\leq[p(1-\varepsilon_{p})]\,.$

Hence by Stirling’s formula, $(\sqrt{2\pi n}(n/{\rm e}\,)^{n}\leq n!)$,

$\displaystyle p^{\gamma}\,{\mathbb{P}}(p-N_{p}>\varepsilon_{p}\,p)$
$\displaystyle=$ $\displaystyle
p^{\gamma}\,{\mathbb{P}}(N_{p}<p(1-\varepsilon_{p}))\leq p^{\gamma+1}{\rm
e}\,^{-p}\dfrac{p^{[p(1-\varepsilon_{p})]}}{[p(1-\varepsilon_{p})]!}$
$\displaystyle\leq$ $\displaystyle c\,p^{\gamma+0.5}{\rm
e}\,^{-p+[p\,(1-\varepsilon_{p})]}\Big{(}\dfrac{p}{[p(1-\varepsilon_{p})]}\Big{)}^{[p(1-\varepsilon_{p})]}$
$\displaystyle\leq$ $\displaystyle c\exp\Big{(}(\gamma+0.5)\log
p-p\,\varepsilon_{p}+[p(1-\varepsilon_{p})]\,\log\dfrac{p}{[p(1-\varepsilon_{p})]}\Big{)}=:c\,{\rm
e}\,^{v(p)}\,.$

We have by a Taylor expansion, for large $p$,

$\displaystyle v(p)-(\gamma+0.5)\,\log p$ $\displaystyle=$
$\displaystyle-p\,\varepsilon_{p}+[p(1-\varepsilon_{p})]\Big{(}\dfrac{p}{[p(1-\varepsilon_{p})]}-1-\dfrac{1}{2}\Big{(}\dfrac{p}{[p(1-\varepsilon_{p})]}-1\Big{)}^{2}(1+o(1))\Big{)}$
$\displaystyle\leq$ $\displaystyle-0.25\,p\,\varepsilon_{p}^{2}=-0.25\,C\log
p\,.$

Choosing $C>4\,(\gamma+0.5)$, we have ${\rm e}\,^{v(p)}\to 0$. Combining these
bounds, we proved the lemma. ∎

Acknowledgments Muneya Matsui’s research is partly supported by the JSPS
Grant-in-Aid for Scientific Research C (19K11868). Thomas Mikosch’s research
is partially supported by Danmarks Frie Forskningsfond Grant No 9040-00086B

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