Strong uniform convergence rates of the linear wavelet estimator of a
multivariate copula density

Cheikh Tidiane Seck, Salha Mamane
Département de mathématiques, Université Alioune Diop, Bambey, Sénégal
School of Statistics and Actuarial Science, University of the Witwatersrand,
Johannesburg, South-Africa

Abstract
In this paper, we investigate the almost sure convergence, in supremum norm,
of the rank-based linear wavelet estimator for the multivariate copula
density. Based on empirical process tools, we prove a uniform limit law for
the deviation, from its expectation, of an oracle estimator (obtained for
known margins), from which we derive the exact convergence rate for the rank-
based linear estimator. This rate reveals to be optimal in a minimax sense
over Besov balls for the supremum norm loss, whenever the resolution level is
suitably chosen.

Keywords : Copula density, Nonparametric estimation, Wavelet methods, Almost
sure uniform convergence rates.

Mathematics Subject Classification (2010): 62G07, 62G20

##  1 Introduction

A copula is a multivariate distribution function $C$ defined on
$[0,1]^{d},d\geq 2$, with uniform margins. Unlike the linear correlation
coefficient, it gives a full characterization of the dependence between random
variables, be it linear or nonlinear. Given a vector $(X_{1},\ldots,X_{d})$ of
continuous random variables with marginal distribution functions
$F_{1},\cdots,F_{d}$, the copula $C$ may be defined as the joint cumulative
distribution function of the random vector
$(F_{1}(X_{1}),\ldots,F_{d}(X_{d}))$ . If it exists, the copula density is
defined as the derivative, $c$, of the copula distribution function $C$ with
respect to the Lebesgue measure :

$c(u_{1}\ldots,u_{d})=\frac{\partial^{d}}{\partial u_{1},\ldots\partial
u_{d}}C(u_{1},\ldots,u_{d}),\enskip\forall\;(u_{1},\ldots,u_{d})\in(0,1)^{d}.$

Nonparametric estimation of copula density is an active reseach domain that
has been investigated by many authors. For instance, [12] and [8] used
convolution kernel methods to construct consistent estimators for the copula
density, while [21] employed techniques based on Bernstein polynomials. A
drawback of kernel methods is the existence of boundary effects due to the
compact support of the copula function. To overcome this difficulty, some
approaches have been proposed. For example [12] used a mirror-reflexion
technique, while [2] employed a local linear kernel procedure. In the same
vein [20] proposed some improved copula kernel estimators in order to face the
boundary bias. Recently, [10] introduced kernel-type estimators for the copula
density, based on a probit transformation method that can take care of the
boundary effects.

In this paper, we deal more neatly with the boundary bias problem by using
wavelet methods, which are very convenient to describe features of functions
at the edges and corners of the unit cube, because of their good localization
properties. Indeed, wavelet bases automatically handle the boundary effects by
locally adapting to the properties of the curve being estimated. The use of
wavelet methods in density and regression estimation problems is surveyed in
[15], where approximation properties of wavelets are discussed at length. For
more details on wavelet theory we refer to [19], [5], [18] and [23] and
references therein.

Wavelet methods have been already used in nonparametric copula density
estimation. For instance, [11] dealt with a rank-based linear wavelet
estimator of the bivariate copula density and established, under certain
conditions, its optimality in the minimax sense on Besov-balls for the
$L_{2}$-norm loss, as well as on H$\ddot{o}$lder-balls for the pointwise-norm
loss. [1] extended these results to the nonlinear thresholded estimators of
multivariate copula densities. These nonlinear estimates are near optimal (up
to a logarithmic factor) for the $L_{2}$-norm loss, and have the advantage of
being adaptive to the regularity of the copula density function. In the same
spirit, [9] established an upper bound on $L_{p}$-losses, $2\leq p<\infty$,
for linear wavelet-based estimators of the bivariate copula density, when the
latter is bounded.

Our goal in this paper, is to establish the exact almost sure convergence
rate, in supremum norm loss, for the rank-based linear wavelet estimator of
the multivariate copula density. Our methodology is largely inspired by [14],
who established almost sure convergence rates, in supremum norm loss, for the
linear wavelet estimator of a univariate density function on $\mathbb{R}$.
Here, we want to extend this result to multivariate copula densities on
$(0,1)^{d}$. We prove that if copula density $c$ is regular enough (i.e. $c$
belongs to a Besov space of regularity $t$, corresponding to the
H$\ddot{o}$lder space of order $t$) and the resolution level, say $j_{n}$,
satisfies : $2^{j_{n}}\simeq(n/\log n)^{1/(2t+d)}$, then the rank-based linear
wavelet estimator achieves the optimal minimax rate, for supremum norm loss,
over Besov-balls.

The rest of the paper is organized as follows. In Section 2, we recall some
facts on wavelet theory and define the rank-based linear wavelet estimator of
the multivariate copula density as in [1]. Section 3 presents the main
theoretical results along with some comments. In appendix A, we recall some
useful facts on empirical process theory. Appendix B contains the proof of the
uniform limit law given in Proposition 3.1.

##  2 Wavelet theory and Estimation procedure

Let $\phi$ be a father wavelet and $\psi$ its associated mother wavelet, which
are both assumed compactly supported. [4] proposed orthonormal wavelet bases
for $L_{2}([0,1])$, the space of all measurable and square integrable
functions on $[0,1]$. Precisely, for all fixed $j_{0}\in\mathbb{N}$, the
family $\\{\phi_{j_{0},l}:l=1,\ldots,2^{j_{0}}\\}\bigcup\\{\psi_{j,l}:j\geq
j_{0},l=1,\ldots,2^{j}\\}$ is an orthonormal basis for $L_{2}([0,1])$, where
$\phi_{j,l}(u)=2^{j/2}\phi(2^{j}u-l)$ and
$\psi_{j,l}(u)=2^{j/2}\psi(2^{j}u-l),\;\forall j,l\in\mathbb{Z}$, $u\in[0,1]$
. Using the tensorial product, one can construct a multivariate wavelet basis
for $L_{2}([0,1]^{d}),d\geq 2$. For
$\textbf{k}=(k_{1},\ldots,k_{d})\in\mathbb{Z}^{d}$, define the following
functions of $\textbf{u}=(u_{1},\ldots,u_{d})\in[0,1]^{d}$ :

$\displaystyle\phi_{j_{0},\textbf{k}}(u_{1}\ldots,u_{d})=\prod_{m=1}^{d}\phi_{j_{0},k_{m}}(u_{m}),$
$\displaystyle\psi^{\epsilon}_{j,\textbf{k}}(u_{1},\ldots,u_{d})=\prod_{m=1}^{d}\phi^{1-\epsilon_{m}}_{j,k_{m}}(u_{m})\psi^{\epsilon_{m}}_{j,k_{m}}(u_{m}),$

where
$\epsilon=(\epsilon_{1},\ldots,\epsilon_{d})\in\mathcal{S}_{d}=\\{0,1\\}^{d}\setminus\\{(0,\ldots,0)\\}$.
Then the family
$\\{\phi_{j_{0},\textbf{k}},\psi^{\epsilon}_{j,\textbf{h}}:j\geq
j_{0},\textbf{k}\in\\{1,\ldots,2^{j_{0}}\\}^{d},\textbf{h}\in\\{1,\ldots,2^{j}\\}^{d},\epsilon\in\mathcal{S}_{d}\\}$
is an orthonormal basis for $L_{2}([0,1]^{d})$, for any fixed
$j_{0}\in\mathbb{N}$. Thus, assuming that the copula density $c$ belongs to
$L_{2}([0,1]^{d})$, we have the following representation :

$c(\textbf{u})=\sum_{\textbf{k}\in\\{1,\ldots,2^{j_{0}}\\}^{d}}\alpha_{j_{0},\textbf{k}}\phi_{j_{0},\textbf{k}}(\textbf{u})+\sum_{j\geq
j_{0}}\sum_{\textbf{k}\in\\{1,\ldots,2^{j}\\}^{d}}\sum_{\epsilon\in\mathcal{S}_{d}}\beta^{\epsilon}_{j,\textbf{k}}\psi^{\epsilon}_{j,\textbf{k}}(\textbf{u}),$
(1)

for all $\textbf{u}\in[0,1]^{d},$ where the scaling coefficients
$\alpha_{j_{0},\textbf{k}}$ and wavelet coefficients
$\beta^{\epsilon}_{j,\textbf{k}}$ are respectively defined as

$\alpha_{j_{0},\textbf{k}}=\int_{[0,1]^{d}}c(\textbf{u})\phi_{j_{0},\textbf{k}}(\textbf{u})d\textbf{u}\quad\text{
and
}\quad\beta^{\epsilon}_{j,\textbf{k}}=\int_{[0,1]^{d}}c(\textbf{u})\psi^{\epsilon}_{j,\textbf{k}}(\textbf{u})d\textbf{u}.$

Now, let $(\textbf{X}_{1},\cdots,\textbf{X}_{n})$ be an independent and
identically distributed (i.i.d) sample of the random vector
$\textbf{X}=(X_{1},\ldots,X_{d})$, with continuous marginal distribution
functions $F_{1},\ldots,F_{d}$, and where
$\textbf{X}_{i}=(X_{i1},\ldots,X_{id}),i=1,\ldots,n$. The distribution
function of the random vector
$\textbf{U}_{i}=(F_{1}(X_{i1}),\ldots,F_{d}(X_{id}))$ is the copula $C$ and
its density, if it exists, is $c$. Denoting the expectation operator by
$\mathbb{E}$, the coefficients $\alpha_{j_{0},\textbf{k}}$ and
$\beta^{\epsilon}_{j,\textbf{k}}$ can be rewritten as follows :

$\alpha_{j_{0},\textbf{k}}=\mathbb{E}[\phi_{j_{0},\textbf{k}}(\textbf{U}_{i})],\qquad\enskip\beta^{\epsilon}_{j,\textbf{k}}=\mathbb{E}[\psi^{\epsilon}_{j,\textbf{k}}(\textbf{U}_{i})].$

If the margins $F_{1},\ldots,F_{d}$ were known, natural estimators for
$\alpha_{j_{0},\textbf{k}}$ and $\beta^{\epsilon}_{j,\textbf{k}}$ would be
given by,

$\tilde{\alpha}_{j_{0},\textbf{k}}=\frac{1}{n}\sum_{i=1}^{n}\phi_{j_{0},\textbf{k}}(\textbf{U}_{i})\enskip,\enskip\tilde{\beta}^{\epsilon}_{j,\textbf{k}}=\frac{1}{n}\sum_{i=1}^{n}\psi^{\epsilon}_{j,\textbf{k}}(\textbf{U}_{i}).$
(2)

But, usually the marginal ditribution functions $F_{1},\ldots,F_{d}$ are
unknown ; and it is customary to replace them by their empirical counterparts
$F_{1n},\ldots,F_{dn}$ (or rescaled versions thereof), with

$F_{jn}(x_{j})=\sum_{i=1}^{n}\mathbb{I}(X_{ij}\leq x_{j}),\quad
j=1,\ldots,d,\quad x_{j}\in\mathbb{R},$

where $\mathbb{I}(\cdot)$ denotes the indicator function. Then, putting
$\hat{\textbf{U}}_{i}=(\hat{U}_{i1},\ldots,\hat{U}_{id})$, where
$\hat{U}_{ij}=F_{jn}(X_{ij}),j=1,\ldots,d$ ; $i=1,\ldots,n$, the modified
empirical coefficients are

$\hat{\alpha}_{j_{0},\textbf{k}}=\frac{1}{n}\sum_{i=1}^{n}\phi_{j_{0},\textbf{k}}(\textbf{U}_{i})\enskip,\enskip\hat{\beta}^{\epsilon}_{j,\textbf{k}}=\frac{1}{n}\sum_{i=1}^{n}\psi^{\epsilon}_{j,\textbf{k}}(\textbf{U}_{i}).$
(3)

Now, choosing a suitable resolution level $j_{n}\geq j_{0}$ and considering
the orthogonal projection of $c$ onto the sub-space $V_{j_{n}}$ of the
underlying multiresolution analysis on $L_{2}([0,1]^{d})$, we obtain the rank-
based linear wavelet estimator of $c$ :

$\hat{c}_{j_{n}}(\textbf{u})=\sum_{\textbf{k}\in\\{1,\ldots,2^{j_{n}}\\}^{d}}\hat{\alpha}_{j_{n},\textbf{k}}\phi_{j_{n},\textbf{k}}(\textbf{u}),\qquad\textbf{u}\in(0,1)^{d}.$
(4)

As remarked in [11], the estimator $\hat{c}_{j_{n}}$ is not necessarily a
density, because it can take negative values on parts of its domain and fails
to integrate to one. In practice, some truncations and normalizations are
necessary for its use.

To obtain the exact rate of convergence of the linear estimator
$\hat{c}_{j_{n}}$, our methodology of proof follows the empirical process
approach developped in [7] (see also [14],[13]). In fact, we can rewrite
$\hat{c}_{j_{n}}$ in terms of the empirical measure. For $j_{n}$ fixed, define
the following kernel functions :

$\widetilde{K}(x,y)=\sum_{l=1}^{2^{j_{n}}}\phi(x-l)\phi(y-l),\qquad(x,y)\in\mathbb{R}$
(5)
$\quad\widetilde{K}_{j_{n}}(x,y)=2^{j_{n}}\widetilde{K}(2^{j_{n}}x,2^{j_{n}}y),\qquad(x,y)\in\mathbb{R}.$

For
$\textbf{x}=(x_{1},\ldots,x_{m})\in{\mathbb{R}}^{d},\;\textbf{y}=(y_{1},\ldots,y_{m})\in{\mathbb{R}}^{d}$,
set :

$\textbf{K}(\textbf{x},\textbf{y})=\prod_{m=1}^{d}\widetilde{K}(x_{m},y_{m}),$
(6)

$\textbf{K}_{j_{n}}(\textbf{x},\textbf{y})=\prod_{m=1}^{d}\widetilde{K}_{j_{n}}(x_{m},y_{m}).$
(7)

Then, the linear wavelet estimator $\hat{c}_{j_{n}}$ can be transformed into

$\hat{c}_{j_{n}}(\textbf{u})=\frac{1}{n}\sum_{i=1}^{n}\textbf{K}_{j_{n}}(\hat{\textbf{U}}_{i},\textbf{u})=\frac{2^{dj_{n}}}{n}\sum_{i=1}^{n}\textbf{K}(2^{j_{n}}\hat{\textbf{U}}_{i},2^{j_{n}}\textbf{u}).$
(8)

##  3 Asymptotic behaviour of the estimator

Let’s introduce an auxillary estimator $\tilde{c}_{j_{n}}$ corresponding to
the case where the marginal distribution functions $F_{1},\ldots,F_{d}$ are
known. In this situation,
$(U_{i1},\ldots,U_{id})=(F_{1}(X_{i1}),\ldots,F_{d}(X_{id})),i=1,\ldots,n$ are
direct observations of the copula $C$, and $\tilde{c}_{j_{n}}$ may be defined
as

$\tilde{c}_{j_{n}}(\textbf{u})=\sum_{\textbf{k}\in\\{1,\ldots,2^{j_{n}}\\}^{d}}\tilde{\alpha}_{j_{n},\textbf{k}}\phi_{j_{n},\textbf{k}}(\textbf{u}),$
(9)

where

$\tilde{\alpha}_{j_{n},\textbf{k}}=\frac{1}{n}\sum_{i=1}^{n}\phi_{j_{n},\textbf{k}}(F_{1}(X_{i1}),\ldots,F_{d}(X_{id}))$
(10)

is an unbiased estimator of $\alpha_{j_{n},\textbf{k}}$.

For all $\textbf{u}\in(0,1)^{d}$, we can decompose the estimation error
$\hat{c}_{j_{n}}-c$ as

$\displaystyle\hat{c}_{j_{n}}(\textbf{u})-c(\textbf{u})$ $\displaystyle=$
$\displaystyle[\hat{c}_{j_{n}}(\textbf{u})-\tilde{c}_{j_{n}}(\textbf{u})]+[\tilde{c}_{j_{n}}(\textbf{u})-\mathbb{E}\tilde{c}_{j_{n}}(\textbf{u})]+[\mathbb{E}\tilde{c}_{j_{n}}(\textbf{u})-c(\textbf{u})]$
(11) $\displaystyle=:$ $\displaystyle
R_{n}(\textbf{u})+D_{n}(\textbf{u})+B_{n}(\textbf{u}).$

To obtain the almost sure convergence rate of $\hat{c}_{j_{n}}$ uniformly in
$\textbf{u}\in(0,1)^{d}$, we have to investigate the limiting behavior of each
of the three above terms. We need the following hypotheses in the sequel :

  * (H.1)

The father wavelet $\phi\in L^{2}(\mathbb{R})$ is bounded, compactly supported
and admits a bounded derivative $\phi^{\prime}$.

  * (H.2)

There exists a bounded and compactly supported function
$\Phi:\mathbb{R}\rightarrow\mathbb{R}_{+}$ such that
$|\widetilde{K}(x,y)|\leq\Phi(x-y)$ and the function
$\theta_{\phi}(x)=\sum_{k=1}^{2^{j_{n}}}|\phi(x-k)|$ is bounded.

  * (H.3)

The kernel $\widetilde{K}$ satisfies : for all $y\in\mathbb{R},$
$\int_{-\infty}^{\infty}\widetilde{K}(x,y)dx=1.$

  * (H.4)

As $n\rightarrow\infty,$ the sequence $(j_{n})_{n\geq 0}$ satisfies :

$j_{n}\nearrow\infty,\qquad\frac{n}{j_{n}2^{(d+1)j_{n}}}\rightarrow\infty,\qquad\frac{j_{n}}{\log\log
n}\rightarrow\infty.$

###### Remark. 1

Hypotheses (H.1), (H.2) and (H.3) are usual conditions that are satisfied by
many wavelets bases, for example the Daubechies wavelets and the Haar wavelet
$\phi(u)=1_{[0,1]}(u)$. The conditions in Hypothesis (H.4) are analogous to
some conditions imposed on the bandwidth parameter in convolution-kernel
estimation methods.

The following proposition gives the asymptotic behavior of the second term
$D_{n}(\textbf{u})$, corresponding to the deviation of the auxillary estimator
$\tilde{c}_{j_{n}}$ from its expectation. In the sequel, we denote $I=(0,1)$,
$\|c\|_{\infty}=\sup_{\textbf{u}\in I^{d}}|c(\textbf{u})|$ and for any real
function $\varphi$ defined on $\mathbb{R}^{d},d\geq 1$,
$\|\varphi\|_{\infty}=\sup_{\textbf{x}\in\mathbb{R}^{d}}|\varphi(\textbf{x})|$.

###### Proposition 3.1

Suppose that the father wavelet $\phi$ is uniformly continuous, with compact
support $[0,B]$, where $B$ is a positive integer. Further, assume that the
copula density $c$ is continuous and bounded on $I^{d}$ and that Hypotheses
(H.1), (H.2), (H.3) and (H.4) hold. Then, almost surely (a.s.),

$\lim_{n\rightarrow\infty}r_{n}\sup_{\textbf{u}\in
I^{d}}\frac{|\tilde{c}_{j_{n}}(\textbf{u})-\mathbb{E}\tilde{c}_{j_{n}}(\textbf{u})|}{\sqrt{\int_{{\mathbb{R}}^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n}}\textbf{u})d\textbf{x}}}=\sqrt{{\|c\|_{\infty}}},$
(12)

with

$r_{n}=\sqrt{\frac{n}{(2d\log 2)j_{n}2^{dj_{n}}}}.$ (13)

Proof : It is largely inspired by [14] and is postponed to Appendix B. It will
consist in establishing a lower bound and an upper bound for the limit in
(12), a methodology borrowed from [7] (see also [13]).

###### Remark. 2

Proposition 3.1 gives the exact almost sure convergence rate, in supremum
norm, of the deviation $D_{n}(\textbf{u})$ to zero, which is of order
$O(\sqrt{j_{n}2^{dj_{n}}/n})$. In fact, by hypotheses (H.1-2-3) the quantity
$\int_{{\mathbb{R}}^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n}}\textbf{u})d\textbf{x}$
can be bounded above and below : there exist two positive constants $D_{1}$
and $D_{2}$ independent of u and $n$ such that :

$D_{1}\leq\int_{{\mathbb{R}}^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n}}\textbf{u})d\textbf{x}\leq
D_{2}.$ (14)

This readily implies

$\sup_{\textbf{u}\in
I^{d}}|D_{n}(\textbf{u})|=O\left(\sqrt{\frac{j_{n}2^{dj_{n}}}{n}}\right),\quad
a.s.$ (15)

The following theorem constitutes our principal result. We need some notation
before stating it. Let $N$ be a positive integer and $t=N+\alpha,0<\alpha\leq
1$. For any bounded real function $f$ defined on $I^{d}$ and possessing
derivatives up to order $N$, set

$\|f\|_{t,\infty,\infty}=\|f\|_{\infty}+\sum_{k=0}^{N}\sup_{u\neq v,u,v\in
I^{d}}\frac{|f^{(k)}(u)-f^{(k)}(v)|}{|u-v|^{\alpha}}.$ (16)

We say that $f$ belongs to the Besov space of regularity $t$,
$B^{t}_{\infty,\infty}$, if and only if $\|f\|_{t,\infty,\infty}<\infty$.
The following condition is also needed for the proof :
Condition $1(N):$ the father wavelet $\phi$ is compactly supported and admits
weak derivatives up to order $N,N\in\mathbb{N}$, that are all in
$\mathcal{L}^{p}$ for some $1\leq p\leq\infty$.

###### Theorem 3.1

Suppose that the assumptions of Proposition 3.1 are fulfilled. If, moreover,
$c$ belongs to $B_{\infty,\infty}^{t}$ and $\phi$ satisfies Condition $1(N)$,
with $0<t<N+1$, then, as $n\rightarrow\infty$, we have

$\sup_{\textbf{u}\in
I^{d}}|\hat{c}_{j_{n}}(\textbf{u})-c(\textbf{u})|=O\left(\sqrt{\frac{j_{n}2^{dj_{n}}}{n}}+2^{-j_{n}t}\right)+o(1),\quad
a.s.$ (17)

Proof. In view of decomposition (11), it suffices to handle the first and the
last term. The behavior of the second term is given by the previous
Proposition 3.1. Let us begin with the first term $R_{n}(\textbf{u})$. We have
for $\textbf{k}=(k_{1},\ldots,k_{d})\in\mathbb{Z}^{d}$

$\displaystyle\hat{\alpha}_{j_{n},\textbf{k}}-\tilde{\alpha}_{j_{n},\textbf{k}}$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\phi_{j_{n},\textbf{k}}(F_{1n}(X_{i1}),\ldots,F_{dn}(X_{id}))-\phi_{j_{n},\textbf{k}}(F_{1}(X_{i1}),\ldots,F_{d}(X_{id}))\right]$
$\displaystyle=:$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\xi_{\textbf{k}}(X_{i1},\ldots,X_{id}),$

where, we set

$\xi_{\textbf{k}}(X_{i1},\ldots,X_{id})=\phi_{j_{n},\textbf{k}}(F_{1n}(X_{i1}),\ldots,F_{dn}(X_{id}))-\phi_{j_{n},\textbf{k}}(F_{1}(X_{i1}),\ldots,F_{d}(X_{id})).$
(18)

For $d=2$, [11] observes that, with $k=(k_{1},k_{2}),$

$\xi_{k}(X_{i1},X_{i2})=\xi_{k_{1}}(X_{i1})\xi_{k_{2}}(X_{i2})+\xi_{k_{1}}(X_{i1})\phi_{j_{n}k_{2}}(F_{2}(X_{i2}))+\xi_{k_{2}}(X_{i2})\phi_{j_{n}k_{1}}(F_{1}(X_{i1})),$
(19)

where
$\xi_{k_{m}}(X_{im})=\phi_{j_{n}k_{m}}(F_{mn}(X_{im}))-\phi_{j_{n}k_{m}}(F_{m}(X_{im})),$
for $m=1,2$.
By induction of (19), we obtain for all fixed $d\geq 2$ that

$\xi_{\textbf{k}}(X_{i1},\ldots,X_{id})=\sum_{q=0}^{d-1}\sum_{\epsilon_{1}+\ldots+\epsilon_{d}=q}\prod_{m=1}^{d}\left[\phi_{j_{n}k_{m}}(F_{m}(X_{im}))\right]^{\epsilon_{m}}\left[\xi_{k_{m}}(X_{im})\right]^{1-\epsilon_{m}},$
(20)

where $(\epsilon_{1},\ldots\epsilon_{d})\in\\{0,1\\}^{d}$. Recall that
$\phi_{jl}(u)=2^{j/2}\phi(2^{j}u-l),\forall j,l\in\mathbb{Z}$. By using the
derivability of $\phi$ by hypothesis, we can write, for all $m=1,\ldots,d$

$\displaystyle\xi_{k_{m}}(X_{im})$ $\displaystyle=$ $\displaystyle
2^{\frac{j_{n}}{2}}\phi(2^{j_{n}}F_{mn}(X_{im})-k_{m})-2^{\frac{j_{n}}{2}}\phi(2^{j_{n}}F_{m}(X_{im})-k_{m})$
$\displaystyle=$ $\displaystyle
2^{\frac{3}{2}j_{n}}\left[F_{mn}(X_{im})-F_{m}(X_{im})\right]\phi^{\prime}(\zeta_{im}),$

where $\zeta_{im}$ lies between $F_{mn}(X_{im})$ and $F_{m}(X_{im})$. Now,
combining Chung’s (1949) law of the iterated logarithm (LIL) with the
boundedness of $\phi$ and $\phi^{\prime}$, we obtain, for all $m=1,\ldots,d$

$|\xi_{k_{m}}(X_{im})|\leq 2^{\frac{3}{2}j_{n}}\times\sqrt{\frac{\log\log
n}{2n}}\|\phi^{\prime}\|_{\infty},\quad a.s.$ (21)

Thus, for $d=2$, the expression in (20) can be bounded above ; that is

$|\xi_{k}(X_{i1},X_{i2})|\leq 2^{3j_{n}}\left(\frac{\log\log
n}{2n}\right)\|\phi^{\prime}\|^{2}_{\infty}+2.2^{2j_{n}}\sqrt{\frac{\log\log
n}{2n}}\|\phi^{\prime}\|_{\infty}\|\phi\|_{\infty},\quad a.s.$ (22)

Since

$\frac{2^{3j_{n}}\left(\frac{\log\log
n}{2n}\right)}{2^{2j_{n}}\sqrt{\frac{\log\log
n}{2n}}}=\frac{1}{\sqrt{2}}\left(\frac{j_{n}2^{2j_{n}}}{n}\right)^{1/2}\left(\frac{\log\log
n}{j_{n}}\right)^{1/2},$

which, by hypothesis (H.4) converges to 0, as $n\rightarrow\infty$. Then
$\;2^{3j_{n}}\left(\frac{\log\log
n}{2n}\right)=o\left(2^{2j_{n}}\sqrt{\frac{\log\log n}{2n}}\right)$. That is,
for $d=2$,

$|\xi_{k}(X_{i1},X_{i2})|=O\left(2^{2j_{n}}\sqrt{\frac{\log\log
n}{n}}\right).$

By induction of formula (22), we have for all $d\geq 2$,

$|\xi_{\textbf{k}}(X_{i1},\ldots,X_{id})|\leq
2^{\frac{3}{2}dj_{n}}\left(\frac{\log\log
n}{n}\right)^{\frac{d}{2}}\|\phi^{\prime}\|^{d}_{\infty}+\cdots+d.2^{\frac{3}{2}j_{n}}\sqrt{\frac{\log\log
n}{n}}2^{\frac{d-1}{2}j_{n}}\|\phi^{\prime}\|_{\infty}\|\phi\|^{d-1}_{\infty}.$
(23)

Note that the number of terms in the summation in the right hand side of
inequality (23) is finite. Moreover, as we observe for the case $d=2$, all
these terms are dominated (small-o’s) by the last one, which is of order
$O\left(2^{\frac{3}{2}j_{n}}\sqrt{\frac{\log\log
n}{n}}2^{\frac{d-1}{2}j_{n}}\right)$. Then

$|\xi_{\textbf{k}}(X_{i1},\ldots,X_{id})|=O\left(2^{\frac{3}{2}j_{n}}\sqrt{\frac{\log\log
n}{n}}2^{\frac{d-1}{2}j_{n}}\right).$

and

$\displaystyle|\hat{\alpha}_{j_{n},\textbf{k}}-\tilde{\alpha}_{j_{n},\textbf{k}}|$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}|\xi_{\textbf{k}}(X_{i1},\ldots,X_{id})|$
$\displaystyle=$ $\displaystyle
O\left(2^{(\frac{2+d}{2})j_{n}}\sqrt{\frac{\log\log n}{n}}\right)$

Finally, by using the boundedness of the function
$\theta_{\phi}(x)=\sum_{l=1}^{2^{j_{n}}}|\phi(x-l)|$, we obtain

$\displaystyle|\hat{c}_{j_{n},\textbf{k}}(\textbf{u})-\tilde{c}_{j_{n},\textbf{k}}(\textbf{u})|$
$\displaystyle\leq$
$\displaystyle\sum_{\textbf{k}\in\\{1,\ldots,2^{j_{n}}\\}^{d}}|\hat{\alpha}_{j_{n},\textbf{k}}-\tilde{\alpha}_{j_{n},\textbf{k}}|2^{\frac{d}{2}j_{n}}\prod_{m=1}^{d}\phi(2^{j_{n}}u_{m}-k_{m})$
$\displaystyle=$ $\displaystyle
O\left[\|\theta_{\phi}\|_{\infty}^{d}2^{\frac{2+2d}{2}j_{n}}\sqrt{\frac{\log\log
n}{n}}\right]$ $\displaystyle=$ $\displaystyle
O\left[\left(\frac{j_{n}2^{(1+d)j_{n}}}{n}\right)^{1/2}\left(\frac{\log\log
n}{j_{n}}\right)^{1/2}\right]$

which, by hypothesis (H.4), converges to 0, as $n\rightarrow\infty$. Hence

$\sup_{\textbf{u}\in I^{d}}|R_{n}(\textbf{u})|\longrightarrow
0,n\rightarrow\infty,\quad a.s.$ (24)

To handle the last term $B_{n}(\textbf{u})$ corresponding to the bias of
$\tilde{c}_{j_{n}}$, we make use of approximation properties in Besov spaces.
Let $K_{j_{n}}$ denote the orthogonal projection kernel onto the sub-space
$V_{j_{n}}$. That is

$K_{j_{n}}(c)(\textbf{u})=\int_{I^{d}}K_{j_{n}}(\textbf{u},\textbf{v})c(\textbf{v})d\textbf{v},\quad\textbf{u}\in
I^{d}.$

Then, we can write

$B_{n}(\textbf{u})=\mathbb{E}\tilde{c}_{j_{n}}(\textbf{u})-c(\textbf{u})=K_{j_{n}}(c)(\textbf{u})-c(\textbf{u}).$

Since $\phi$ satisfies Condition $1(N)$ and $c\in B^{t}_{\infty,\infty}$,
$0<t<N+1$, then applying Theorem 9.4 in [15] gives :

$\|K_{j_{n}}(c)-c\|_{\infty}\leq A2^{-j_{n}t},$

where $A$ is a positive constant depending on the Besov norm of $\|c\|$. Hence

$\sup_{\textbf{u}\in I^{d}}|B_{n}(\textbf{u})|=O(2^{-j_{n}t}).$ (25)

Combining (15), (24) and (25) gives the proof of the theorem. $\square$

###### Remark. 3

Theorem 3.1 implies that if $2^{j_{n}}\simeq(n/\log n)^{\frac{1}{2t+d}}$, then
the rank-based linear estimator $\hat{c}_{j_{n}}$ achieves the optimal rate of
convergence in supremum norm, $(\log n/n)^{\frac{t}{2t+d}}$, over Besov-balls
in $B^{t}_{\infty,\infty}$. This rate is the best possible, as far as the
supremum norm loss is concerned $(p=\infty)$, and the estimated density is
defined on a compact set [see, e.g., [16] for optimality in minimax theory].
However, notice that this rate is slower than that obtained in the case of
quadratic and pointwise loss functions and established in [11] and [1] for the
wavelet linear estimators.

###### Remark. 4

For copula densities in general Besov spaces $B^{t}_{pq}$ with $t>1/p$, we
also have optimal rates for the wavelet linear estimator $\hat{c}_{j_{n}}$.
Indeed, if $c\in B^{t}_{pq}$ with $t>1/p$, the Sobolev embedding properties
entails $B^{t}_{pq}\subset B^{t-1/p}_{\infty,\infty}$. Thus, if
$2^{j_{n}}\simeq(n/\log n)^{\frac{1}{2(t-1/p)+d}}$, then $\hat{c}_{j_{n}}$
attains the optimal rate :
$\mathbb{E}(\|\hat{c}_{j_{n}}-c\|_{\infty})=O\left((\log
n/n)^{\frac{t-1/p}{2(t-1/p)+d}}\right)$ [see, e.g., [6], Theorem 1].

###### Remark. 5

As established in [11] for the quadratic norm, we have proved by another
approach using Chung’s (1949) LIL, that the error term associated with the use
of ranks (coming from the pseudo-observations) is also negligible for the
supremum norm case. That is resorting to pseudo-observations instead of
genuine observations does not affect the convergence rate of the linear
wavelet estimators of the copula density.

## Appendix A : Useful results on empirical process

Bernstein’s inequality (maximal version):
Let $Z_{1},\ldots,Z_{n}$ be independent random variables with
$\mathbb{E}(Z_{i})=0,i=1,\ldots,n$ and ${\rm
Var}(\sum_{i=1}^{n}Z_{i})\leq\nu$. Assume further that for some constant
$M>0,\;|Z_{i}|<M$, $i=1,...,n$. Then for all $t>0$

$\mathbb{P}\left(\max_{q\leq n}\left|\sum_{i=1}^{q}Z_{i}\right|>t\right)\leq
2\exp\left\\{\frac{-t^{2}}{2\nu+(2/3)Mt}\right\\},$ (26)

Lemma A.1 [Einmahl and mason(2000)]:
Let $\mathcal{F}$ and $\mathcal{G}$ be two classes of real-valued measurable
functions on $\mathcal{X}$ satisfying

$|f(x)|\leq F(x),\quad f\in\mathcal{F},\quad x\in\mathcal{X},$

where $F$ is a finite valued measurable envelope function on $\mathcal{X}$;

$\|g\|\leq M\quad g\in\mathcal{G},$

where $M>0$ is a finite constant. Assume that for all probability measure $Q$
with $0<Q(F^{2})<\infty$,

$N(\varepsilon(Q(F^{2}))^{1/2},\mathcal{F},d_{Q})\leq
C_{1}\varepsilon^{-\nu_{1}},\qquad 0<\varepsilon<1$

and

$N(\varepsilon M,\mathcal{G},d_{Q})\leq C_{2}\varepsilon^{-\nu_{2}},\qquad
0<\varepsilon<1$

where $\nu_{1},\nu_{1},C_{1},C_{2}\geq 1$ are suitable constants. Then we have
for all probability measure $Q$ with $0<Q(F^{2})<\infty$,

$N(\varepsilon M(Q(F^{2}))^{1/2},\mathcal{F}\mathcal{G},d_{Q})\leq
C_{3}\varepsilon^{-\nu_{1}-\nu_{2}},\qquad 0<\varepsilon<1$

for some finite constant $0<C_{3}<\infty$.

Proposition 2 [Einmahl and Mason (2000)]:
Let $Z,Z_{1},Z_{2},\ldots$ , be a sequence of i.i.d. random vectors taking
values in $\mathbb{R}^{m},m\geq 1$. For each $n\geq 1$, consider the empirical
distribution function based on the first $n$ of these random vectors, defined
by

$G_{n}(s)=\frac{1}{n}\sum_{i=1}^{n}1_{Z_{i}\leq s},\quad s\in\mathbb{R}^{m},$

where as usual $z\leq s$ means that each component of $z$ is less than or
equal to the corresponding component of $s$. For any measurable real valued
function $g$ defined on $\mathbb{R}^{m}$, set

$G_{n}(g)=\int_{\mathbb{R}^{m}}g(s)dG_{n}(s),\qquad\mu(g)=\mathbb{E}g(Z)\qquad\text{and}\quad\sigma(g)=Var(g(Z)).$

Let ${a_{n}:n\geq 1}$ denote a sequence of positive constants converging to
zero. Consider a sequence $\mathcal{G}_{n}=\\{g^{(n)}_{i}:i=1,...,k_{n}\\}$ of
sets of real-valued measurable functions on $\mathbb{R}^{2}$, satisfying,
whenever $g^{(n)}_{i}\in\mathcal{G}_{n}$:

$\mathbb{P}(g^{(n)}_{i}(Z)=0,\;g^{(n)}_{j}(Z))=0,\quad i\neq
j\quad\text{and}\quad\sum_{i=1}^{k_{n}}\mathbb{P}({g^{(n)}_{i}(Z)\neq 0})\leq
1/2.$

Further assume that :
For some $0<r<\infty$, $a_{n}k_{n}\rightarrow r$, as $n\rightarrow\infty.$
For some $-\infty<\mu_{1},\mu_{2}\infty$, uniformly in $i=1,\ldots,k_{n}$, for
all large $n$, $a_{n}\mu_{1}\leq\mu(g^{(n)}_{i})\leq a_{n}\mu_{2}$
For some $0<\sigma_{1}<\sigma_{2}<\infty,$ uniformly in $i=1,\ldots k_{n}$,
for all large $n$,
$\sigma_{1}\sqrt{n}a_{n}\leq\bar{\sigma}(g^{(n)}_{i})\leq\sigma_{2}\sqrt{n}a_{n}$
For some $0<B<\infty$, uniformly in $i=1,\ldots,k_{n}$, for all large $n$,
$|g^{(n)}_{i}|\leq B$

###### Proposition 3.2

Under these assumptions, with probability one, for each $0<\varepsilon<1$,
there exists $N_{\varepsilon}$ such that for $n\geq N_{\varepsilon}$,

$\max_{1\leq i\leq
k_{n}}\frac{\sqrt{n}\\{G_{n}(g^{(n)}_{i})-\mu(g^{(n)}_{i})\\}}{\bar{\sigma}(g^{(n)}_{i})\sqrt{2|\log
a_{n}|}}\geq 1-\varepsilon.$

Talagrand’s inequality:
Let $X_{i},i=1,\ldots,n$ be an independent and identically distributed random
samples of $X$ with probability law $P$ on $\mathbb{R}$, and $\mathcal{G}$ a
$P$-centered (i.e.,$\int gdP=0$ for all $g\in\mathcal{G}$) countable class of
real-valued functions on $\mathbb{R}$, uniformly bounded by the constant U.
Let $\sigma$ be any positive number such that
$\sigma^{2}\geq\sup_{g\in\mathcal{G}}\mathbb{E}(g^{2}(X))$. Then, Talagrand’s
(1996) inequality implies that there exists a universal constant $L$ such that
for all $t>0$,

$\mathbb{P}\left(\max_{q\leq
n}\left\|\sum_{i=1}^{q}(g(X_{i})\right\|_{\mathcal{G}}>E+t\right)\leq
L\exp\left\\{\frac{-t}{L{\rm U}}\log(1+\frac{t{\rm U}}{V})\right\\},$ (27)

with

$E=\mathbb{E}\left\|\sum_{i=1}^{n}g(X_{i})\right\|_{\mathcal{G}}\;\text{ and
}\;V=\mathbb{E}\left\|\sum_{i=1}^{n}(g(X_{i}))^{2}\right\|_{\mathcal{G}}.$

Further, if $\mathcal{G}$ is a VC-type class of functions, with
characteristics $A,\,v$ then, there exist a universal constant $B$ such that :
[see, e.g., Giné and Guillou (2001)]

$E\leq B\left[v{\rm U}\log\frac{A{\rm
U}}{\sigma}+\sqrt{v}\sqrt{n\sigma^{2}}\frac{A{\rm U}}{\sigma}\right]$ (28)

Next, if $\sigma<\frac{{\rm U}}{2}$, the constant $A$ may be replaced by 1 at
the price of changing the constant $B$, and then if, moreover,
$n\sigma^{2}>C_{0}\log\left(\frac{{\rm U}}{\sigma}\right)$, we have

$E\leq C_{1}\sqrt{n\sigma^{2}\log\left(\frac{{\rm U}}{\sigma}\right)},\;\text{
and }\;V\leq L^{\prime}n\sigma^{2},$ (29)

where $C_{1},L^{\prime}$ are constants depending only on $A,v,C_{0}$. Finally,
it follows from (42) and (29) that, for all $t>0$ satisfying :
$C_{1}\sqrt{n\sigma^{2}\log\left(\frac{{\rm U}}{\sigma}\right)}\leq t\leq
C_{2}\frac{n\sigma^{2}}{{\rm U}}$ for all constant $C_{2}\geq C_{1},$

$\mathbb{P}\left(\max_{n_{k-1}\leq n\leq
n_{k}}\left\|\sum_{i=1}^{n}g(X_{i})\right\|_{\mathcal{G}}>t\right)\leq
R\exp\left\\{\frac{-1}{C_{3}}\frac{t^{2}}{n\sigma^{2}}\right\\},$ (30)

where $C_{3}=\log(1+C_{2}/L^{\prime})/RC_{2}$ and $R$ a constant depending
only on $A$ and $v$.

## Appendix B : Proof of Proposition 3.1

### UPPER BOUND

###### Lemma 3.2

Under the assumptions of Proposition 3.1, one has almost surely

$\limsup_{n\rightarrow\infty}r_{n}\sup_{\textbf{u}\in
I^{d}}\frac{|D_{n}(\textbf{u})|}{\sqrt{{\int_{{\mathbb{R}}^{d}}K^{2}(\textbf{x},2^{j_{n}}\textbf{u})d\textbf{x}}}}\leq\sqrt{\|c\|_{\infty}}.$
(31)

Proof : Given $\lambda>1$, define $n_{k}=[\lambda^{k}],k\in\mathbb{N}$, where
$[a]$ denotes the integer part of a real $a$. Let $\delta_{m}=1/m$, $m\geq 1$
integer, then we can cover the set $I^{d}$ by a number $l_{k}$ of small cubes
$S_{k,r}$, each of side length $\delta_{m}2^{-j_{n_{k}}}$, with

$l_{k}\leq\left(\frac{1}{\delta_{m}2^{-j_{n_{k}}}}+1\right)^{d}\leq\left(\frac{2}{\delta_{m}2^{-j_{n_{k}}}}\right)^{d},$
(32)

for $k$ large enough. Let us choose points $\textbf{u}_{k,r}\in S_{k,r}\cap
I^{d},\;r=1,\ldots,l_{k}$. We want to prove Lemma 3.2 over the discrete grid
of points $\\{\textbf{u}_{k,r}:r=1,\ldots,l_{k}\\}$. For all $\eta\in(0,1)$ we
claim that

$\limsup_{k\rightarrow\infty}\sqrt{\frac{n_{k}}{(2d\log
2)j_{n_{k}}2^{dj_{n_{k}}}}}\max_{1\leq r\leq l_{k}}\max_{n_{k-1}\leq n\leq
n_{k}}|D_{n}(\textbf{u}_{k,r})|\leq(1+\eta)\sqrt{\|c\|_{\infty}[K^{2}]},$ (33)

where we note

$[K^{2}]=\int_{{\mathbb{R}}^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n}}\textbf{u})d\textbf{x}.$

To prove (33), we apply the maximal version of Bernstein inequality [see,
Appendix A above]. Given $\textbf{u}\in I^{d}$ and $k\in\mathbb{N}$, for all
$n$ satisfying : $n_{k-1}\leq n\leq n_{k}$ let

$Z_{i}(\textbf{u})=\textbf{K}(2^{j_{n}}\textbf{U}_{i},2^{j_{n}}\textbf{u})-\mathbb{E}\textbf{K}(2^{j_{n}}\textbf{U}_{i},2^{j_{n}}\textbf{u}),\;\;i=1,\ldots,n.$

Observe that for each $n$, the $Z_{i}(\textbf{u})^{\prime}s$ are independent
and identically distributed zero-mean random variables, and for all
$\textbf{u}\in I^{d}$,

$D_{n}(\textbf{u})=\frac{2^{dj_{n}}}{n}\sum_{i=1}^{n}Z_{i}(\textbf{u}).$ (34)

By hypothesis (H.2), we have

$\displaystyle\left|\textbf{K}(2^{j_{n}}\textbf{U}_{i},2^{j_{n}}\textbf{u})\right|$
$\displaystyle=$
$\displaystyle\prod_{m=1}^{d}\left|\tilde{K}(2^{j_{n}}U_{im},2^{j_{n}}u_{m})\right|\leq\prod_{m=1}^{d}\Phi(2^{j_{n}}(U_{im}-u_{m}))\leq\|\Phi\|^{d}_{\infty},$

where $\|\Phi\|_{\infty}=\sup_{x\in\mathbb{R}}|\Phi(x)|$. This implies

$\left|\mathbb{E}\textbf{K}[(2^{j_{n}}\textbf{U}_{i},2^{j_{n}}\textbf{u})]\right|\leq\mathbb{E}\|\Phi\|^{2}_{\infty}=\|\Phi\|^{d}_{\infty}.$

Thus, for all $\textbf{u}\in I^{d}$,

$|Z_{i}(\textbf{u})|\leq 2\|\Phi\|^{d}_{\infty}:=M.$

Since $Z_{i}(\textbf{u})^{\prime}s$ are independent and centered, we can write
for $n=n_{k}$

$Var\left(\sum_{i=1}^{n_{k}}Z_{i}(\textbf{u})\right)={n_{k}}Var(Z_{1}(\textbf{u}))={n_{k}}\mathbb{E}(Z_{1}^{2}(\textbf{u})).$

Then using the change of variables
$\textbf{s}=2^{-j_{n_{k}}}\textbf{x},\textbf{s}=(s_{1},\ldots,s_{d})$,
$\textbf{x}=(x_{1},\ldots,x_{d})$, we obtain

$\displaystyle\mathbb{E}(Z_{1}^{2}(\textbf{u}))$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\textbf{K}^{2}[(2^{j_{n_{k}}}\textbf{U}_{1},2^{j_{n_{k}}}\textbf{u})$
$\displaystyle\leq$
$\displaystyle\int_{I^{d}}\textbf{K}^{2}(2^{j_{n_{k}}}\textbf{s},2^{j_{n_{k}}}\textbf{u})c(\textbf{s})d\textbf{s}$
$\displaystyle\leq$ $\displaystyle
2^{-dj_{n_{k}}}{\|c\|_{\infty}}\int_{[0,2^{j_{n_{k}}}]^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n_{k}}}\textbf{u})d\textbf{x}$

which yields

$Var\left(\sum_{i=1}^{n_{k}}Z_{i}(\textbf{u})\right)\leq{n_{k}}2^{-dj_{n_{k}}}\|c\|_{\infty}\int_{\mathbb{R}^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n}}\textbf{u})d\textbf{x}:=\sigma^{2}_{k}.$

Now, applying the maximal version Bernstein’s inequality, for each point
$\textbf{u}_{k,r}$, we obtain for all $t>0$,

$\mathbb{P}\left(\max_{n_{k-1}\leq n\leq
n_{k}}\left|\sum_{i=1}^{n}Z_{i}(\textbf{u}_{k,r})\right|>t\right)\leq
2\exp\left\\{\frac{-t^{2}}{2\sigma^{2}_{k}+(2/3)Mt}\right\\},$ (35)

which yields

$\displaystyle\mathbb{P}\left(\max_{1\leq r\leq l_{k}}\max_{n_{k-1}\leq n\leq
n_{k}}\left|\sum_{i=1}^{n}Z_{i}(\textbf{u}_{k,r})\right|>t\right)$
$\displaystyle=$
$\displaystyle\mathbb{P}\left(\bigcup_{r=1}^{l_{k}}\left\\{\max_{n_{k-1}\leq
n\leq
n_{k}}\left|\sum_{i=1}^{n}Z_{i}(\textbf{u}_{k,r})\right|>t\right\\}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{r=1}^{l_{k}}\mathbb{P}\left(\max_{n_{k-1}\leq n\leq
n_{k}}\left|\sum_{i=1}^{n}Z_{i}(\textbf{u}_{k,r})\right|>t\right)$
$\displaystyle\leq$ $\displaystyle
l_{k}2\exp\left\\{\frac{-t^{2}}{2\sigma^{2}_{k}+(2/3)Mt}\right\\}.$

Let $t=\sqrt{2(1+\eta)n_{k}2^{-dj_{n_{k}}}\log
2^{dj_{n_{k}}}{\|c\|_{\infty}[K^{2}]}}$. Then, for $k$ large enough,
$t\rightarrow\infty$. Combining this with (32), we obtain after some little
algebra,

$\displaystyle\mathbb{P}\left(\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq
n\leq
n_{k}}\left|\sum_{i=1}^{n}Z_{i}(\textbf{u}_{k,r})\right|}{\sqrt{2n_{k}2^{-dj_{n_{k}}}\log
2^{dj_{n_{k}}}{\|c\|_{\infty}[K^{2}]}}}>\sqrt{1+\eta}\right)$
$\displaystyle\leq$ $\displaystyle
2l_{k}\exp\left\\{\frac{-t^{2}}{\frac{t^{2}}{(1+\eta)\log
2^{dj_{n}}}+\frac{4}{3}\|\Phi\|^{2}t}\right\\}$ $\displaystyle\leq$
$\displaystyle 2l_{k}\exp\left\\{-(1+\eta)\log 2^{dj_{n_{k}}}\right\\}$
$\displaystyle\leq$ $\displaystyle 2^{d}\delta^{-d}_{m}2^{-d\eta j_{n_{k}}}.$

Since the series $\sum_{k\geq 0}2^{-d\eta j_{n_{k}}}<\infty$, Borel-Cantelli
lemma yields

$\mathbb{P}\left(\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq n\leq
n_{k}}\left|\sum_{i=1}^{n}Z_{i}(\textbf{u}_{k,r})\right|}{\sqrt{(2d\log
2)n_{k}j_{n_{k}}2^{-dj_{n_{k}}}{\|c\|_{\infty}}[K^{2}]}}>\sqrt{1+\eta}\right)=o(1),$
(36)

That is

$\limsup_{k\rightarrow\infty}\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq
n\leq n_{k}}\left|\sum_{i=1}^{n}Z_{i}(\textbf{u}_{k,r})\right|}{\sqrt{(2d\log
2)n_{k}j_{n_{k}}2^{-dj_{n_{k}}}{\|c\|_{\infty}}[K^{2}]}}\leq\sqrt{1+\eta},\quad
a.s.$ (37)

Since the function $x\mapsto x2^{-2x}$ is decreasing for $x>2\log 2$, we have
for $n_{k-1}\leq n\leq n_{k}$, and $k$ large enough,

$\sqrt{\frac{n_{k}j_{n_{k}}2^{-dj_{n_{k}}}}{nj_{n}2^{-dj_{n}}}}\leq\sqrt{\frac{n_{k}}{n}}\leq\sqrt{\frac{n_{k}}{n_{k-1}}}\leq\sqrt{\lambda}.$
(38)

In view of inequality (38), Statement (37) yields

$\limsup_{k\rightarrow\infty}\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq
n\leq n_{k}}\left|\sum_{i=1}^{n}Z_{i}(\textbf{u}_{k,r})\right|}{\sqrt{(2d\log
2)nj_{n}2^{-dj_{n}}{\|c\|_{\infty}}[K^{2}]}}\leq\sqrt{\lambda(1+\eta)},\quad
a.s.$ (39)

Now, multiplying the numerator and the denominator of the fraction in (39) by
the factor $2^{dj_{n}}/n$, an recalling the expression of $D_{n}(\textbf{u})$
in (34), we finally get for all $\eta\in(0,1)$,

$\limsup_{k\rightarrow\infty}\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq
n\leq n_{k}}\sqrt{n}\left|D_{n}(\textbf{u}_{k,r})\right|}{\sqrt{(2d\log
2)j_{n}2^{dj_{n}}}}\leq\sqrt{\lambda(1+\eta)\|c\|_{\infty}[K^{2}]},$ (40)

which proves Lemma 3.2 over the discrete grid.

Next, to prove Lemma 3.2 between the grid points, we shall make use of
Talagrand’s (1996) inequality (27). Let us introduce the sequence of functions
defined as follows : for all $n\geq 1$, $k\geq 1$, $1\leq r\leq l_{k}$ and any
fixed $\textbf{u}\in S_{k,r}$, define

$g_{k,r}^{(n)}(\textbf{s},\textbf{u})=\textbf{K}(2^{j_{n_{k}}}\textbf{s},2^{j_{n_{k}}}\textbf{u}_{k,r})-\textbf{K}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{u}),\qquad\textbf{s}\in
I^{d}.$ (41)

and set, for all $\lambda>1$,

$\mathcal{G}_{k,r}(\lambda)=\left\\{g:\textbf{s}\mapsto
g_{k,r}^{(n)}(\textbf{s},\textbf{u}):\textbf{u}\in S_{k,r}\cap
I^{d},\;n_{k-1}\leq n\leq n_{k}\right\\}.$

Let $\textbf{S}=(S_{1},\ldots,S_{d})$ be a vector of $[0,1]-$uniform random
variables, now we have to check the following conditions in order to apply
Talagrand’s (1996) inequality :

  * i)

The classes $\mathcal{G}_{k,r}(\lambda),1\leq r\leq l_{k}$, are of VC-type
with characteristics $A$ and $v$ ;

  * ii)

$\forall g\in\mathcal{G}_{k,r}(\lambda)$, $\|g\|_{\infty}\leq{\rm U};$

  * iii)

$\forall g\in\mathcal{G}_{k,r}(\lambda)$, ${\rm
Var}[g(\textbf{S})]\leq\sigma^{2}_{k},$

  * iv)

$\sigma_{k}<\frac{{\rm U}}{2}$ and
$n_{k}\sigma^{2}_{k}>C_{0}\log\left(\frac{{\rm U}}{\sigma_{k}}\right)$,
$\;C_{0}>0$.

These conditions will be checked below.
Recall that
$(\textbf{U}_{i}=(F_{1i}(X_{i1}),\ldots,F_{di}(X_{id})),i,\ldots,n$ is a
sequence of independent and identically distributed vectors of $[0,1]-$uniform
components. We have shown (see below) that each class
$\mathcal{G}_{k,r}(\lambda)$ satisfies all the conditions i), ii), iii) and
iv) for ${\rm U}=2\|\Phi\|_{\infty}^{d}$ and
$\sigma^{2}_{k}=D_{0}2^{-dj_{n_{k}}}{\|c\|_{\infty}}\omega^{2}_{\phi}(\delta_{m})$,
where $\omega_{\phi}$ is the modulus of continuity of $\phi$ defined below
(49) and $D_{0}$ is a positive constant depending on $\|\Phi\|_{\infty}$ and
$d$. Then, Talagrand’s (1996) inequality gives : for all $t>0$,

$\mathbb{P}\left(\max_{n_{k-1}\leq n\leq
n_{k}}\left\|\sum_{i=1}^{n}(g(\textbf{U}_{i})-\mathbb{E}g(\textbf{U}_{i}))\right\|_{\mathcal{G}_{k,r}(\lambda)}>t\right)\leq
R\exp\left\\{\frac{-1}{C_{3}}\frac{t^{2}}{n_{k}\sigma^{2}_{k}}\right\\},$ (42)

which yields, by taking the maximum over $r$ and
$t=C_{1}\sqrt{n_{k}\sigma^{2}_{k}\log\left(\frac{{\rm
U}}{\sigma_{k}}\right)}$,

$\mathbb{P}\left(\max_{1\leq r\leq l_{k}}\max_{n_{k-1}\leq n\leq
n_{k}}\left\|\sum_{i=1}^{n}(g(\textbf{U}_{i})-\mathbb{E}g(\textbf{U}_{i}))\right\|_{\mathcal{G}_{k,r}(\lambda)}>t\right)\leq
Rl_{k}\exp\left\\{\frac{-C_{1}^{2}}{C_{3}}\log\left(\frac{{\rm
U}}{\sigma_{k}}\right)\right\\}.$ (43)

Whenever $m\rightarrow\infty$, $\omega_{\phi}(\delta_{m})\rightarrow 0$.
Hence, for any $\varepsilon>0$, there exists $m_{0}\in\mathbb{N}$ such that
$\omega_{\phi}(\delta_{m})<\varepsilon$ for $m\geq m_{0}$. Using this fact, we
can replace $\sigma_{k}^{2}$ by
$4D2^{-dj_{n_{k}}}\varepsilon{\|c\|_{\infty}}$, for $m$ large enough. We also
have, for $k$ large enough,

$\log\left(\frac{{\rm U}}{\sigma_{k}}\right)=\log\left(\frac{{\rm
U}}{4D\varepsilon\|c\|_{\infty}}\right)+j_{n_{k}}\log 2\sim j_{n_{k}}\log 2$

and thus, for $k,m$ large enough,

$t=C_{1}\sqrt{n_{k}\sigma^{2}_{k}\log\left(\frac{{\rm
U}}{\sigma_{k}}\right)}\sim\sqrt{4DC_{1}^{2}n_{k}2^{-dj_{n_{k}}}d\,j_{n_{k}}\log
2\,\varepsilon\|c\|_{\infty}}.$

By combining these facts with (32) we obtain, with $A_{0}=\sqrt{2DC_{1}}$,

$\mathbb{P}\left(\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq n\leq
n_{k}}\left\|\sum_{i=1}^{n}(g(\textbf{U}_{i})-\mathbb{E}g(\textbf{U}_{i}))\right\|_{\mathcal{G}_{k,r}(\lambda)}}{\sqrt{(2d\log
2)n_{k}j_{n_{k}}2^{-dj_{n_{k}}}}}>A_{0}\sqrt{\varepsilon\|c\|_{\infty}}\right)\leq
4R\delta^{-2}2^{-[C_{1}^{2}/2C_{3}-1]2j_{n_{k}}}.$ (44)

Now, we can choose the constant $C_{1}$ in such a way that
$C_{1}^{2}/2C_{3}-1>0$ ; in which case the series $\sum_{k\geq
0}2^{-[C_{1}^{2}/2C_{3}-1]2j_{n_{k}}}$ converges. Thus, Borel-Cantelli lemma
implies

$\mathbb{P}\left(\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq n\leq
n_{k}}\left\|\sum_{i=1}^{n}(g(\textbf{U}_{i})-\mathbb{E}g(\textbf{U}_{i}))\right\|_{\mathcal{G}_{k,r}(\lambda)}}{\sqrt{(2d\log
2)n_{k}j_{n_{k}}2^{-dj_{n_{k}}}}}>A_{0}\sqrt{\varepsilon\|c\|_{\infty}}\right)=o(1),$
(45)

that is

$\limsup_{k\rightarrow\infty}\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq
n\leq
n_{k}}\left\|\sum_{i=1}^{n}(g(\textbf{U}_{i})-\mathbb{E}g(\textbf{U}_{i}))\right\|_{\mathcal{G}_{k,r}(\lambda)}}{\sqrt{(2d\log
2)n_{k}j_{n_{k}}2^{-dj_{n_{k}}}}}\leq
A_{0}\sqrt{\varepsilon\|c\|_{\infty}},\quad a.s.$ (46)

Arguing as in the discrete case, with Statement (38) in view, we conclude that

$\limsup_{k\rightarrow\infty}\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq
n\leq
n_{k}}\sqrt{n}\left\|D_{n}(\textbf{u}_{k,r})-D_{n}(\textbf{u}))\right\|_{\mathcal{G}_{k,r}(\lambda)}}{\sqrt{(2d\log
2)j_{n}2^{dj_{n}}}}\leq A_{0}\sqrt{\lambda\varepsilon\|c\|_{\infty}},\quad
a.s.,$ (47)

which completes the proof of Lemma (3.2) between the grid-points.

Now recapitulating, we can infer from (40) and (47) that

$\limsup_{k\rightarrow\infty}\max_{1\leq r\leq l_{k}}\frac{\max_{n_{k-1}\leq
n\leq n_{k}}\sqrt{n}\left|D_{n}(\textbf{u})\right|}{\sqrt{(4\log
2)j_{n}2^{-2j_{n}}}}\leq\sqrt{\lambda(1+\eta)\|c\|_{\infty}[K^{2}]}+A_{0}\sqrt{\lambda\varepsilon\|c\|_{\infty}},\quad
a.s.$ (48)

Since $\eta$ and $\varepsilon$ are arbitrary, letting $\lambda\rightarrow 1$
completes the proof of Lemma 3.2.$\square$

##  Checking conditions i), ii), iii), iv)

Checking i): Observe that the elements of the class
$\mathcal{G}_{k,r}(\lambda)$ may be rewritten as

$g_{k,r}^{(n)}(\textbf{s},\textbf{u})=\prod_{m=1}^{d}\widetilde{K}(2^{j_{n_{k}}}s_{m},2^{j_{n_{k}}}u_{k,r,m})-\prod_{m=1}^{d}\widetilde{K}(2^{j_{n}}s_{m},2^{j_{n}}u_{m}),$

where $\widetilde{K}(x,y)=\sum_{l=1}^{2^{j_{n}}}\phi(x-l)\phi(y-l),$ with
$\phi$ compactly supported and is of bounded variation. For $m=1,\ldots,d$,
define the classes of functions :
$\mathcal{F}_{m}=\\{v\mapsto\sum_{l\in\mathbb{Z}}\phi(2^{j}w-l)\phi(2^{j}v-l):w\in[0,1],j\in\mathbb{N}\\}$.
By Lemma 2 in [14], $\mathcal{F}_{1},\ldots,\mathcal{F}_{m}$ are VC-type
classes of functions. Moreover, $\mathcal{F}_{1},\ldots,\mathcal{F}_{m}$ are
uniformly bounded. Indeed, for all $w\in[0,1],\;j\in\mathbb{N}$, we have
$\left|\sum_{l=1}^{2^{j_{n}}}\phi(2^{j}w-l)\phi(2^{j}\cdot-l)\right|\leq\|\phi\|_{\infty}\|\theta_{\phi}\|_{\infty}$,
as the function $\theta_{\phi}(x)=\sum_{l=1}^{2^{j_{n}}}|\phi(x-l)|$ is
bounded. By Lemma A.1 in [7], this implies that the product
$\mathcal{F}_{1}\cdots\mathcal{F}_{m}$ is also a VC-type class of functions.
Now, using properties (iv) and (v) of Lemma 2.6.18 in [24], we can infer that
the classes of functions $\mathcal{G}_{k,r}(\lambda)$ are of VC-type for all
$k,r$ fixed.

Checking ii): For all $k\geq 1,\;0\leq r\leq l_{k},\;n_{k-1}\leq n\leq n_{k}$,
using hypothesis (H.2), we can write

$\displaystyle\left|g_{{k},r}^{(n)}(\cdot,\textbf{u})\right|$
$\displaystyle\leq$
$\displaystyle\left|\textbf{K}(2^{j_{n_{k}}}\cdot,2^{j_{n_{k}}}\textbf{u}_{{k},r})\right|+\left|\textbf{K}(2^{j_{n}}\cdot,2^{j_{n}}\textbf{u})\right|$
$\displaystyle\leq$
$\displaystyle\prod_{m=1}^{d}\widetilde{K}(2^{j_{n_{k}}}\cdot,2^{j_{n_{k}}}u_{{k},r,m})+\prod_{m=1}^{d}\widetilde{K}(2^{j_{n}}\cdot,2^{j_{n}}u_{m})\leq
2\|\Phi\|^{d}_{\infty}$

and ii) holds with ${\rm U}=2\|\Phi\|^{d}_{\infty}$.

Checking iii): For all $k\geq 1,\;0\leq r\leq l_{k},\;n_{k-1}\leq n\leq
n_{k}$. As in [14] we choose $\lambda\in(0,1),$ such that $j_{n_{k}}=j_{n}$.
By a change of variable
$\textbf{s}=\textbf{u}+2^{-j_{n_{k}}}\textbf{x},\;\textbf{s}=(s_{1},\ldots,s_{d}),\;\textbf{x}=(x_{1},\ldots,x_{d}),\;\textbf{u}=(u_{1},\ldots,u_{d})$,
we have

$\displaystyle\mathbb{E}\left[\left(g_{{k},r}^{(n)}(\textbf{S},\textbf{u})\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(\textbf{K}(2^{j_{n_{k}}}\textbf{S},2^{j_{n_{k}}}\textbf{u}_{k,r})-\textbf{K}(2^{j_{n_{k}}}\textbf{S},2^{j_{n_{k}}}\textbf{u})\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\int_{I^{d}}\left(\textbf{K}(2^{j_{n_{k}}}\textbf{s},2^{j_{n_{k}}}\textbf{u}_{k,r})-\textbf{K}(2^{j_{n_{k}}}\textbf{s},2^{j_{n_{k}}}\textbf{u})\right)^{2}c(\textbf{s})d\textbf{s}$
$\displaystyle\leq$
$\displaystyle\frac{\|c\|_{\infty}}{2^{-dj_{n_{k}}}}\int_{[-2^{j_{n_{k}}},2^{j_{n_{k}}}]^{d}}\left(\textbf{K}(2^{j_{n_{k}}}\textbf{u}+\textbf{x},2^{j_{n_{k}}}\textbf{u}_{k,r})-\textbf{K}(2^{j_{n_{k}}}\textbf{u}+\textbf{x},2^{j_{n_{k}}}\textbf{u})\right)^{2}d\textbf{x}.$
$\displaystyle\leq$
$\displaystyle{2^{-dj_{n_{k}}}}{\|c\|_{\infty}}\int_{\mathbb{R}^{d}}\left(\textbf{K}(2^{j_{n_{k}}}\textbf{u}+\textbf{x},2^{j_{n_{k}}}\textbf{u}_{k,r})-\textbf{K}(2^{j_{n_{k}}}\textbf{u}+\textbf{x},2^{j_{n_{k}}}\textbf{u})\right)^{2}d\textbf{x}.$

To simplify, let us take $\textbf{w}=2^{j_{n_{k}}}\textbf{u}+\textbf{x}$, then
$d\textbf{x}=d\textbf{w}$, $w=(w_{1},\ldots,w_{d})\in\mathbb{R}^{d}$.
Put
$A(\textbf{w})=\textbf{K}(\textbf{w},2^{j_{n_{k}}}\textbf{u}_{k,r})-\textbf{K}(\textbf{w},2^{j_{n_{k}}}\textbf{u})$
; using the multiplicativity of the kernel K, we can rewrite $A(\textbf{w})$
as

$\displaystyle A(\textbf{w})$ $\displaystyle=$
$\displaystyle\prod_{l=1}^{d}\widetilde{K}(w_{l},2^{j_{n_{k}}}u_{k,r,l})-\prod_{l=1}^{d}\widetilde{K}(w_{l},2^{j_{n_{k}}}u_{k,r,l})$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{d}\left[\widetilde{K}(w_{l},2^{j_{n_{k}}}u_{k,r,l})-\widetilde{K}(w_{l},2^{j_{n_{k}}}u_{l})\right]\prod_{p=1,p\neq
l}^{d}\widetilde{K}(w_{p},2^{j_{n_{k}}}u_{k,r,p})$

For any $\delta>0$, the modulus of continuity of $\phi$ is defined as

$\omega_{\phi}(\delta)=\\{\sup|\phi(x)-\phi(y)|:|x-y|\leq\delta\\}.$ (49)

Recall that $\widetilde{K}(x,y)=\sum_{h=1}^{2^{j_{n}}}\phi(x-h)\phi(y-h).$
Combining these facts with the inequality $(a_{1}+\cdots+a_{d})^{2}\leq
d(a_{1}^{2}+\cdots+a_{d}^{2})$, and Fubini’s Theorem, we get

$\displaystyle\int_{\mathbb{R}^{d}}|A(\textbf{w})|^{2}d\textbf{w}$
$\displaystyle\leq$ $\displaystyle
d\int_{\mathbb{R}^{d}}\sum_{l=1}^{d}\left[\widetilde{K}(w_{l},2^{j_{n_{k}}}u_{k,r,l})-\widetilde{K}(w_{l},2^{j_{n_{k}}}u_{l})\right]^{2}\prod_{p=1,p\neq
l}^{d}\widetilde{K}^{2}(w_{l},2^{j_{n_{k}}}u_{k,r,p})d\textbf{w}.$

Then

$\displaystyle\int_{\mathbb{R}^{d}}|A(\textbf{w})|^{2}d\textbf{w}$
$\displaystyle\leq$ $\displaystyle
d\sum_{l=1}^{d}\int_{\mathbb{R}^{d}}\left[\sum_{h=1}^{2^{j_{n}}}\phi(2^{j_{n_{k}}}w_{l}-h)[\phi(2^{j_{n_{k}}}u_{k,r,l}-h)-\phi(2^{j_{n_{k}}}u_{l}-h)]\right]^{2}$
$\displaystyle\qquad\qquad\times\prod_{p=1,p\neq
l}^{d}\widetilde{K}^{2}(w_{p},2^{j_{n_{k}}}u_{k,r,p})d\textbf{w}$
$\displaystyle\leq$ $\displaystyle
d\omega_{\phi}^{2}(\delta_{m})\sum_{l=1}^{d}\int_{\mathbb{R}^{d}}\left[\sum_{h}^{2^{j_{n}}}\phi(2^{j_{n_{k}}}w_{l}-h)\right]^{2}\prod_{p=1,p\neq
l}^{d}\widetilde{K}^{2}(w_{p},2^{j_{n_{k}}}u_{k,r,p})d\textbf{w}$
$\displaystyle\leq$ $\displaystyle
d\omega_{\phi}^{2}(\delta_{m})\sum_{l=1}^{d}\int_{\mathbb{R}}\left[\sum_{h=1}^{2^{j_{n}}}\phi(2^{j_{n_{k}}}w_{l}-h)\right]^{2}dw_{l}\prod_{p=1,p\neq
l}^{d}\int_{\mathbb{R}}\widetilde{K}^{2}(w_{p},2^{j_{n_{k}}}u_{k,r,p})d{w_{p}}.$

Now, since the family $\\{\phi(\cdot-h):h=1,\ldots,{2^{j_{n}}}\\}$ is an
orthonormal basis, the quantity
$\int_{\mathbb{R}}\left(\sum_{h=1}^{2^{j_{n}}}\phi(w_{l}-h)\right)^{2}dw_{l}$
can be bounded by a constant $M_{0}$ ; thus

$\displaystyle\int_{\mathbb{R}^{d}}|A(\textbf{w})|^{2}d\textbf{w}$
$\displaystyle\leq$ $\displaystyle
M_{0}d\omega_{\phi}^{2}(\delta_{m})\sum_{l=1}^{d}\prod_{p=1,p\neq
l}^{d}\int_{\mathbb{R}}\widetilde{K}^{2}(w_{p},2^{j_{n_{k}}}u_{k,r,p})d{w_{p}}$
$\displaystyle\leq$ $\displaystyle M_{0}d^{2}\omega^{2}_{\phi}(\delta_{m})D,$

where we use Hypothesis (H.2) for the last inequality, with $D$ a positive
constant depending on $\|\Phi\|_{\infty}$. Finally, we obtain

$\mathbb{E}\left[\left(g_{{k},r}^{(n)}(\textbf{S},\textbf{u})\right)^{2}\right]\leq
M_{0}d^{2}D2^{-dj_{n_{k}}}{\|c\|_{\infty}}\omega^{2}_{\phi}(\delta_{m}),$ (50)

and iii) holds with

$\sigma^{2}_{k}=D_{0}2^{-dj_{n_{k}}}{\|c\|_{\infty}}\omega^{2}_{\phi}(\delta_{m}),\qquad
D_{0}=M_{0}d^{2}D.\\\ $ (51)

Checking iv): For $m>0$ fixed, we have

$\frac{\sigma_{k}}{\rm
U}=\frac{D_{0}^{1/2}2^{-\frac{d}{2}j_{n_{k}}}{\|c\|_{\infty}^{1/2}}\omega_{\phi}(\delta_{m})}{2\|\Phi\|^{d}_{\infty}}\rightarrow
0,k\rightarrow\infty,$

which implies that $\frac{\sigma_{k}}{\rm U}<\varepsilon$, for all
$\varepsilon>0$ and $k$ large enough. Hence, for $\varepsilon=1/2$, we have
$\sigma_{k}<\frac{\rm U}{2}$. We also have, for all large $k$,

$\frac{n_{k}\sigma_{k}^{2}}{\log\left(\frac{\rm
U}{\sigma_{k}}\right)}=\frac{D_{0}n_{k}{\|c\|_{\infty}}\omega^{2}_{\phi}(\delta_{m})}{j_{n_{k}}2^{dj_{n_{k}}}\,\log
2}\longrightarrow\infty,$

by Hypothesis (H.4). This readily implies that, for any constant $C_{0}>0$,
$n_{k}\sigma^{2}_{k}>C_{0}\log\left(\frac{\rm U}{\sigma_{k}}\right)$ for all
large $k$, and iv) holds. $\square$

### LOWER BOUND

###### Lemma 3.3

Under the assumptions of Proposition 3.1, one has almost surely

$\liminf_{n\rightarrow\infty}r_{n}\sup_{\textbf{u}\in
I^{d}}\frac{|D_{n}(\textbf{u})|}{\sqrt{{\int_{\mathbb{R}^{d}}K^{2}[(\textbf{x},2^{j_{n}}\textbf{u})]d\textbf{x}}}}\geq\sqrt{\|c\|_{\infty}}.$
(52)

Proof : It is an adaptation of the proof of Proposition 2 in [14], which is,
itself, inspired by Proposition 2 in [7]. According to this latter
proposition, (52) holds if and only if for all $\tau>0$, and all large $n$,
there exists $k_{n}=:k_{n}(\tau)$ points
$\textbf{z}_{i,n}=(z_{1,i,n},\ldots,z_{d,i,n})\in I^{d},\;i=1,\ldots,k_{n}$
such that, for functions
$g_{i}^{(n)}(\textbf{s})=\textbf{K}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n}),\;\textbf{s}\in
I^{d},$ and for $\textbf{U}=(U_{1},\ldots,U_{d})$ a random vector with joint
density $c$, the following conditions hold :

  * C.1)

$\mathbb{P}(g_{i}^{(n)}(\textbf{U})\neq
0,\;g_{i^{\prime}}^{(n)}(\textbf{U})\neq 0)=0,\quad\forall i\neq i^{\prime};$

  * C.2)

$\sum_{i=1}^{k_{n}}\mathbb{P}(g_{i}^{(n)}(\textbf{U})\neq 0)\leq 1/2;$

  * C.3)

$2^{-j_{n}}k_{n}\longrightarrow r\in]0,\infty[;$

  * C.4)

$\exists\;\mu_{1},\mu_{2}\in\mathbb{R}$ :
$2^{-dj_{n}}\mu_{1}\leq\mathbb{E}g_{i}^{(n)}(\textbf{U})\leq
2^{-dj_{n}}\mu_{2},\quad\forall i=1,\ldots,k_{n};$

  * C.5)

$\exists\;\sigma_{1},\sigma_{2}>0$ : $2^{-dj_{n}}\sigma_{1}^{2}\leq{\rm
Var}[g_{i}^{(n)}(\textbf{U})]\leq 2^{-dj_{n}}\sigma_{2}^{2},\quad\forall
i=1,\ldots,k_{n};$

  * C.6)

$\|g_{i}^{(n)}\|_{\infty}<\infty,\quad\forall i=1,\ldots,k_{n};\;\forall n\geq
1;$

Now, we have to check these conditions. By hypothesis the copula density $c$
is continuous and bounded on $I^{d}$, then there exists some orthrotope
$D\subset I^{d}$ such that $\max_{\textbf{s}\in
D}c(\textbf{s})=\|c\|_{\infty}$. Thus, for all $\tau>0$ there exists
$\textbf{s}_{0}\in D$ such that $c(\textbf{s}_{0})\geq(1-\tau)\|c\|_{\infty}$.
Let

$D_{\tau}=\\{\textbf{s}\in D:c(\textbf{s})\geq(1-\tau)\|c\|_{\infty}\\},$ (53)

and choose a subset $D_{0}\subset D_{\tau}\;\text{ such that
}\;\mathbb{P}(\textbf{U}\in D_{0})\leq\frac{1}{2}$. Suppose that
$D_{0}=\prod_{j=1}^{d}[a_{j},b_{j}],$ with $0\leq a_{j}<b_{j}\leq 1,\;\;\text{
and }b_{j}-a_{j}=\ell,\;\forall j=1,\ldots,d.$
Set $\delta=3B$ and define

$z_{j,i,n}=a+i\delta 2^{-j_{n}},\quad i=1,\ldots,\left[\frac{b-a}{\delta
2^{-j_{n}}}\right]-1:=k_{n},\quad j=1,\ldots,d$

where $[x]$ designs the integer part of a real $x$.

Checking C.1) : Recall that $\phi$ is supported on $[0,B]$, then

$g_{i}^{(n)}(\textbf{U})\neq 0\iff\forall
k,l\in\mathbb{Z}\quad\left\\{\begin{array}[]{ccc}0\leq&2^{j_{n}}U_{j}-l&\leq
B,\;j=1,\ldots,d\quad(1)\\\ 0\leq&2^{j_{n}}z_{j,i,n}-l&\leq
B,\;j=1,\ldots,d\quad(2)\end{array}\right.$

and

$g_{i^{\prime}}^{(n)}(\textbf{U})\neq 0\iff\forall
k,l\in\mathbb{Z}\quad\left\\{\begin{array}[]{ccc}0\leq&2^{j_{n}}U_{j}-l&\leq
B,\;j=1,\ldots,d\quad(1)^{\prime}\\\ 0\leq&2^{j_{n}}z_{j,i^{\prime},n}-l&\leq
B,\;j=1,\ldots,d\quad(2)^{\prime}\end{array}\right.$

Combining (2) and $(2)^{\prime}$ gives, for every $j=1,\ldots,d$,

$|z_{j,i,n}-z_{j,i^{\prime},n}|\leq 2^{-j_{n}}B.\quad(3)$

But, by definition, for all $i\neq i^{\prime}$,
$|z_{j,i,n}-z_{j,i^{\prime},n}|>\delta 2^{-j_{n}}=3B2^{-j_{n}}$, which
contradicts (3). Hence, the event $\\{g_{i}^{(n)}(\textbf{U})\neq
0,g_{i^{\prime}}^{(n)}(\textbf{U})\neq 0\\}$ is empty for $i\neq i^{\prime}$
and condition C.1) holds.

Checking C.2) : For all $n\geq 1$, the sets $\\{g_{i}^{(n)}(\textbf{U})\neq
0\\},\,i=1,\ldots,k_{n}$ are disjoint in view of Condition C.1). Then, we have

$\sum_{i=1}^{k_{n}}\mathbb{P}(\\{g_{i}^{(n)}(\textbf{U})\neq
0\\})=\mathbb{P}\left(\bigcup_{i=1}^{k_{n}}\\{g_{i}^{(n)}(\textbf{U})\neq
0\\}\right).$

Now, it suffices to show that
$\bigcup_{i=1}^{k_{n}}\\{g_{i}^{(n)}(\textbf{U})\neq
0\\}\subset\\{\textbf{U}\in D_{0}\\}$. From statements (1) and (3) above, we
can write, fro all $j=1,\ldots,d$

$\displaystyle-B\leq 2^{j_{n}}(U_{j}-u_{j,i,n})\leq B$ $\displaystyle
u_{j,i,n}-2^{-j_{n}}B\leq U_{j}\leq u_{j,i,n}+2^{-j_{n}}B$ $\displaystyle
a_{j}\leq a_{j}+(3i-1)2^{-j_{n}}B\leq U_{j}\leq a_{j}+(3i+1)2^{-j_{n}}B\leq
b_{j}.$

That is $U_{j}\in[a_{j},b_{j}]$, and hence
$\textbf{U}=(U_{1},\ldots,U_{d})\in\prod_{j=1}^{d}[a_{j},b_{j}]=D_{0}.$ It
follows that,

$\displaystyle\forall\;i=1,\ldots,k_{n},\quad\\{g_{i}^{(n)}(\textbf{U})\neq
0\\}\subset\\{\textbf{U}\in D_{0}\\}$
$\displaystyle\bigcup_{i=1}^{k_{n}}\\{g_{i}^{(n)}(\textbf{U})\neq
0\\}\subset\\{\textbf{U}\in D_{0}\\}$
$\displaystyle\mathbb{P}\left(\bigcup_{i=1}^{k_{n}}\\{g_{i}^{(n)}(\textbf{U})\neq
0\\}\right)\leq\mathbb{P}(\\{\textbf{U}\in D_{0}\\})\leq\frac{1}{2}.$

Hence, C.2) is fulfilled.

Checking C.3): It is immediate, since

$2^{-j_{n}}k_{n}=2^{-j_{n}}\left(\left[\frac{b-a}{\delta
2^{-j_{n}}}\right]-1\right)=\left[\frac{b-a}{\delta}\right]-2^{-j_{n}}\rightarrow\left[\frac{b-a}{\delta}\right]=:r>0,\;n\rightarrow\infty.$

Checking C.4) : Using a change of variables
$\textbf{s}=2^{-j_{n}}\textbf{x},\;\textbf{s}=(s_{1},\ldots,s_{d}),\;\textbf{x}=(x_{1},\ldots,x_{d})$,
we have

$\displaystyle|\mathbb{E}g_{i}^{(n)}(\textbf{U})|$ $\displaystyle\leq$
$\displaystyle\int_{I^{d}_{\epsilon}}\left|\textbf{K}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n})\right|c(\textbf{s})d\textbf{s}$
$\displaystyle\leq$ $\displaystyle
2^{-dj_{n}}\|c\|_{\infty}\int_{\mathbb{R}^{d}}\left|\textbf{K}(\textbf{x},2^{j_{n}}\textbf{z}_{i,n})\right|d\textbf{x}$
$\displaystyle\leq$ $\displaystyle
2^{-dj_{n}}\|c\|_{\infty}\int_{\mathbb{R}^{d}}|\prod_{j=1}^{d}\widetilde{K}(x_{j},2^{j_{n}}u_{j,i,n})|dx_{j}$
$\displaystyle\leq$ $\displaystyle
2^{-dj_{n}}\|c\|_{\infty}\int_{\mathbb{R}^{d}}\prod_{j=1}^{d}\Phi(x_{j}-2^{j_{n}}u_{j,i,n})dx_{j}$
$\displaystyle\leq$ $\displaystyle 2^{-2j_{n}}\mu,$

where
$\mu=\|c\|_{\infty}\int_{\mathbb{R}^{d}}\prod_{j=1}^{d}\Phi(x_{j}-2^{j_{n}}u_{j,i,n})dx_{j}$
exists, because the function $\Phi$ is integrable by hypothesis (H.2). The
last inequality is equivalent to

$-2^{-2j_{n}}\mu\leq\mathbb{E}g_{i}^{(n)}(U,V)\leq 2^{-2j_{n}}\mu,\;\forall
i=1,\cdots,k_{n}.$

That is C.4) holds.

Checking C.5) : For $n\geq 1,\;i=1,\ldots,k_{n}$, using a change of variables
$\textbf{s}=2^{-j_{n}}\textbf{x}+\textbf{z}_{i,n},\;\textbf{s}=(s_{1},\ldots,s_{d}),\;\textbf{x}=(x_{1},\ldots,x_{d}),\;\textbf{z}_{i,n}=(z_{1,i,n},\ldots,z_{d,i,n})$,
we can write

$\displaystyle{\rm Var}[g_{i}^{(n)}(\textbf{U})]$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\left[\left(g_{i}^{(n)}(\textbf{U})\right)^{2}\right]$
$\displaystyle\leq$
$\displaystyle\int_{I^{d}}\textbf{K}^{2}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n})c(\textbf{s})d\textbf{s}$
$\displaystyle\leq$ $\displaystyle
2^{-dj_{n}}{\|c\|_{\infty}}\int_{\mathbb{R}^{d}}\textbf{K}^{2}(\textbf{x}+2^{j_{n}}\textbf{z}_{i,n},2^{j_{n}}\textbf{z}_{i,n})d\textbf{x}.$

Putting
$\sigma^{2}_{2}:={\|c\|_{\epsilon}}\int_{\mathbb{R}^{d}}\textbf{K}^{2}(\textbf{x}+2^{j_{n}}\textbf{z}_{i,n},2^{j_{n}}\textbf{z}_{i,n})d\textbf{x}$
yields

${\rm Var}[g_{i}^{(n)}(\textbf{U})]\leq 2^{-dj_{n}}\sigma^{2}_{2},$

which is the upper bound in condition C.5). For the lower bound, we have

$\displaystyle{\rm
Var}[g_{i}^{(n)}(\textbf{U})]=\mathbb{E}\left[\left(g_{i}^{(n)}(\textbf{U})\right)^{2}\right]-\left[\mathbb{E}g_{i}^{(n)}(\textbf{U})\right]^{2}$
$\displaystyle=$
$\displaystyle\int_{I^{d}}\textbf{K}^{2}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n})c(\textbf{s})d\textbf{s}-\left(\int_{I^{d}}\textbf{K}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n})c(\textbf{s})d\textbf{s}\right)^{2}.$

Put
$\mu_{n}^{2}=\left(\int_{I^{d}}\textbf{K}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n})c(\textbf{s})d\textbf{s}\right)^{2}.$
Noting that $D_{\tau}\subset I^{d}$, by a change of variables
$\textbf{x}=2^{j_{n}}\textbf{s}$, we obtain

$\displaystyle{\rm Var}[g_{i}^{(n)}(\textbf{U})]$ $\displaystyle\geq$
$\displaystyle\int_{D_{\tau}}\textbf{K}^{2}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n})c(\textbf{s})d\textbf{s}-\mu_{n}^{2}$
$\displaystyle\geq$
$\displaystyle(1-\tau)\|c\|_{\infty}\int_{D_{\tau}}\textbf{K}^{2}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n})d\textbf{s}-\mu_{n}^{2}$
$\displaystyle\geq$
$\displaystyle(1-\tau)\|c\|_{\infty}2^{-dj_{n}}\int_{\mathbb{R}^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n}}\textbf{z}_{i,n})d\textbf{x}-\mu_{n}^{2}.$

Proceeding again to the same change of variables, and observing from
hypothesis (H.3) that
$\int_{\mathbb{R}^{d}}\textbf{K}(\textbf{x},2^{j_{n}}\textbf{z}_{i,n})d\textbf{x}=1$,
we can write

$\displaystyle\mu_{n}^{2}$ $\displaystyle\leq$
$\displaystyle\left(\|c\|_{\infty}2^{-dj_{n}}\int_{\mathbb{R}^{d}}\textbf{K}(\textbf{x},2^{j_{n}}\textbf{z}_{i,n})d\textbf{x}\right)^{2}\leq\|c\|^{2}_{\infty}2^{-2dj_{n}},$

which implies $-\mu_{n}^{2}\geq-\|c\|^{2}_{\epsilon}2^{-4j_{n}}.$ Thus, for
$n$ large enough, we obtain the lower bound in condition C.5), i.e.

$\displaystyle{\rm Var}[g_{i}^{(n)}(\textbf{U})]$ $\displaystyle\geq$
$\displaystyle
2^{-dj_{n}}(1-\tau)\|c\|_{\infty}\int_{\mathbb{R}^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n}}\textbf{z}_{i,n})d\textbf{x}-\|c\|_{\infty}^{2}2^{-2dj_{n}}$
$\displaystyle\geq$ $\displaystyle 2^{-dj_{n}}\sigma_{1}^{2}+o(1),$

with
$\sigma_{1}^{2}:=(1-\tau)\|c\|_{\infty}\int_{\mathbb{R}^{d}}K^{2}(\textbf{x},2^{j_{n}}\textbf{z}_{i,n}]d\textbf{x}.$
Finally, C.5) holds.
Moreover, letting $\tau\rightarrow 0$, we get
$\sigma_{2}^{2}=\sigma_{1}^{2}=\|c\|_{\infty}\int_{\mathbb{R}^{d}}\textbf{K}^{2}(\textbf{x},2^{j_{n}}\textbf{u}d\textbf{x}$.

Checking C.6) : For all $\textbf{s}\in I^{d}$,$n\geq 1,\quad
i=1,\ldots,k_{n}$, by using hypotheses (H.1-2) and the multiplicativity of
kernel K, we have

$\displaystyle|g_{i}^{(n)}(\textbf{s})|$ $\displaystyle=$
$\displaystyle|\textbf{K}(2^{j_{n}}\textbf{s},2^{j_{n}}\textbf{z}_{i,n})|=\prod_{m=1}^{d}|\widetilde{K}(2^{j_{n}}s_{m},2^{j_{n}}z_{i,n,m})|$
$\displaystyle\leq$
$\displaystyle\prod_{m=1}^{d}\sum_{l=1}^{2^{j_{n}}}|\phi(2^{j_{n}}s_{m}-l)\phi(2^{j_{n}}z_{i,n,m}-l)|$
$\displaystyle\leq$
$\displaystyle\|\phi\|_{\infty}^{d}\prod_{m=1}^{d}\sum_{l=1}^{2^{j_{n}}}|\phi(2^{j_{n}}s_{m}-l)|$
$\displaystyle\leq$
$\displaystyle\|\phi\|_{\infty}^{d}\|\theta_{\phi}\|_{\infty}^{d}.$

Hence, $\sup_{n\geq 1,1\leq i\leq
k_{n}}\|g_{i}^{(n)}\|\leq\|\phi\|_{\infty}^{d}\|\theta_{\phi}\|_{\infty}^{d}$,
and C.6) holds.
Since Conditions C.1-2-3-4-5-6) are fulfilled, we can now apply Proposition 2
in [7] to complete the proof of Lemma 3.3. $\square$

Finally, Lemma 3.2 and Lemma 3.3 give the proof of Proposition (3.1).$\square$

## Conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

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# RWN: Robust Watermarking Network for Image Cropping Localization
††thanks: This work is supported by National Natural Science Foundation of
China under Grant U20B2051, U1936214. Qichao Ying and Xiaoxiao Hu contribute
equally. ⋆ Corresponding author: Zhenxing Qian (zxqian@fudan.edu.cn). We are
grateful to the anonymous reviewers for their constructive and insightful
suggestions on improving this paper.

Qichao Ying  School of Computer Science
Fudan University, Shanghai, China
<EMAIL_ADDRESS>Xiaoxiao Hu  School of Computer Science
Fudan University, Shanghai, China
<EMAIL_ADDRESS>Xiangyu Zhang  School of Computer Science
Fudan University, Shanghai, China
<EMAIL_ADDRESS>Sheng Li  School of Computer Science
Fudan University, Shanghai, China
<EMAIL_ADDRESS>Zhenxing Qian⋆ School of Computer Science
Fudan University, Shanghai, China
<EMAIL_ADDRESS>Xinpeng Zhang  School of Computer Science
Fudan University, Shanghai, China
<EMAIL_ADDRESS>
###### Abstract

Image cropping can be maliciously used to manipulate the layout of an image
and alter the underlying meaning. Previous image crop detection schemes only
predicts whether an image has been cropped, ignoring which part of the image
is cropped. This paper presents a novel robust watermarking network (RWN) for
image crop localization. We train an anti-crop processor (ACP) that embeds a
watermark into a target image. The visually indistinguishable protected image
is then posted on the social network instead of the original image. At the
recipient’s side, ACP extracts the watermark from the attacked image, and we
conduct feature matching on the original and extracted watermark to locate the
position of the crop in the original image plane. We further extend our scheme
to detect tampering attack on the attacked image. Besides, we explore a simple
yet efficient method (JPEG-Mixup) to improve the generalization of JPEG
robustness. According to our comprehensive experiments, RWN is the first to
provide high-accuracy and robust image crop localization. Besides, the
accuracy of tamper detection is comparable with many state-of-the-art passive-
based methods.

###### Index Terms:

image crop localization, image tamper detection, robustness, image forensics

##  I Introduction

Image manipulation leads to severe security threats, where misleading
photojournalism, copyright violation or fabricating stories can be a means for
some writers or politicians to potentially influence public opinion.
Researchers have developed a number of schemes to detect various kinds of
digital attacks, e.g., tampering [1, 2, 3], DeepFake [4] and cropping [5, 6,
7, 8]. Among them, cropping is an extremely cheap and effective way to
manipulate the layout of an image and alter the underlying meaning. Existing
crop detection algorithms [5, 6] mainly focus on predicting whether an image
is cropped. They are represented by detecting the exposing evidences of
asymmetrical image cropping, e.g., the shift of the image center [5], and the
inconsistency of JPEG blocking artifacts [6, 7]. Despite the effectiveness of
these works, different cropping behaviors in the real world usually lead to
varied intentions. Crop detection is only a binary classification from this
perspective, unable to distinguish benign crop behaviors from those malicious
attacks such as discarding a visible watermark or removing a person. Van et
al. [8] proposes an image crop localization scheme to study how image cropping
causes chromatic and vignetting aberration. But the tiny traces only exists in
high-definition images and can be easily destroyed by image processing, which
makes the scheme hard to be applicable. Experiments show that the scheme
cannot be applied on highly-compressed or low-quality images. Therefore, image
cropping localization remains a big issue.

Watermarking [9, 10, 11] aims at hiding information imperceptibly into the
host data for covert purposes. The technology focuses on robustness against
possible digital attacks, such as image compression, noise adding, filtering,
etc. Watermarking has been widely used in copyright protection and content
authentication of images in multimedia. Recently, many novel watermarking
schemes for image protection is proposed. Khachaturov el al. [12] proposes a
watermarking-like adversarial method that prevents inpainting systems to
perform normally on a protected image. In [13], the authors propose an
adversarial attack method to cause errors in super-resolution model, including
making the SR image has undesired style, be incorrectly classified in
classification task, or generates wrong words in image caption task. The myth
behind these schemes is that once the secret information successfully travel
through the Online Social Network (OSN), the recipient can decode the
information to administrate a number of tasks.

This paper explores the potential of robust watermarking on image cropping
localization. We propose a robust watermarking network (RWN) for image crop
localization by protecting the original image through watermarking. If the
protected image is cropped, RWN is expected to locate the cropped region at
the recipient’s side. Watermarking is utilized to hide imperceptible traces
for cropping localization which is resilient against common attacks, namely,
JPEG compression, etc. We further introduce a tamper detector to deal with
tampering attack. Therefore, compared to previous works [8, 5], the tough goal
of finding natural-yet-fragile traces for image cropping can be circumvented
by maintaining the overall image quality after watermark embedding.

We train an anti-crop processor (ACP) based on invertible neural network [14]
to embed a watermark into a original image. The watermark is shared with the
recipient. ACP produces a visually indistinguishable protected image, which is
then posted on the social network instead of the unprotected version. On
receiving the attacked version of the protected image, we use ACP again to
extract the watermark, and conduct an efficient and classical feature matching
algorithms (SURF [15]) between the original and extracted watermark to
determine where the crop was positioned in the original image plane. We
further use the tamper detector to predict the tamper mask, and the features
within the predicted tamper mask is discarded to prevent mismatching. Besides,
we explore a simple yet efficient method (JPEG-Mixup) to improve the
generalization of JPEG robustness. We test our scheme by introducing man-made
hybrid attacks. The results demonstrate that our scheme can accurately locate
the cropped region. We also show the effectiveness of tamper detection by
comparison with some state-of-the-art image forgery detection schemes [1, 2,
3].

The highlights of this paper are three-folded.

  1. 1)

This paper presents the first high-accuracy robust image crop localization
scheme. We innovatively use normalizing flows to build invertible function for
image forensic problems.

  2. 2)

With the embedded watermark, RWN can also conduct high-accuracy tamper
detection, which is comparable with the state-of-the-art works.

  3. 3)

We explore a simple yet efficient method (JPEG-Mixup) for improved JPEG
simulation. Experiments show that JPEG-Mixup can remarkably improve the
generalization of RWN on robustness against JPEG compression.

##  II Related Works

Crop Detection and Localization. Cropping is a simple yet powerful way to
maliciously alter the message of an image. The existing crop detection
algorithms [5, 6, 7] mainly focus on predicting whether an image is cropped.
For example, Fanfani et al. [5] exploits the camera principal point
insensitive to image processing operations. Yerushalmy et al. [6] detects
whether there are vanishing points and lines on structured image content. But
cropping different areas in a same image will definitely result in different
semantic changes. Therefore, we need to predict the position of the crop. In
[7], the block artifact grids (BAGs) are extracted blindly with a new
extraction algorithm, and then abnormal BAGs can be detected with a marking
procedure as a trail of cropping detection. For crop localization, Van et al.
[8] investigates the impact that cropping has on the image distribution, and
predicts the absolute location of image patches. However, [8] requires the
images to be untouched and uncompressed so that the tiny traces are preserved.
To address the issue of practical usability, this paper presents RWN for
robust image crop localization that does not restrict image format or quality.

Tamper Localization Great efforts have been made on combating daily image
forgeries. Kown et al. [2] propose to model quantized DCT coefficient
distribution to trace compression artifacts. Mantra-Net [1] uses fully
convolutional networks for feature extraction and further uses long short-term
memory (LSTM) cells for pixel-wise anomaly detection. In MVSS-Net [3], a
system with multi-view feature learning and multi-scale supervision is
developed to jointly exploit the noise view and the boundary artifact to learn
manipulation detection features. In this paper, we study the effectiveness of
using watermarking to aid image forensics where we forgo the idea of finding
universal traces. The shortcoming of these methods is that they only detect
limited types of attacks and cannot generalize their performances well on
images with a chain of combined attacks. We robustly hide traces into the
original image instead of discovering a universal trace for a much easier
forgery localization.

Robust Watermarking.  The uprising of deep networks have given birth to a
series of novel watermarking methods with enhanced robustness [16, 10, 17].
Shin et al. [16] is the first in including a differentiable approximation to
JPEG in the watermarking model. Later, Zhu et al. [10] proposes a powerful and
comprehensive network that is robust against a variety kinds of attacks. From
then on, subsequent watermarking works [17] design more sophisticated and
accurate algorithm to simulate image post-processing attacks, especially the
JPEG compression. However, these methods uses fixed quantization table for
data compression. In contrast, the quantization table in real-world JPEG
images can be customized and flexibly controlled by the quality factor as well
as the image content. As a result, the neural networks can over-fit and lack
real-world robustness. We explore a simple yet efficient method for improved
JPEG simulation.

##  III Method

###  III-A Approach Overview

Pipeline Design.  The proposed RWN consists of five stages, namely, watermark
embedding, image redistribution, tamper detection, watermark extraction and
crop localization. We embed a watermark $\mathbf{W}$ into an original image
$\mathbf{I}$. The protected image $\mathbf{I}_{M}$ is generated and we upload
it onto the social cloud instead of the unprotected original image. The
attacker generates the attacked image $\mathbf{I}_{A}$ by freely adding three
kinds of attacks (benign attacks, cropping, tampering) on $\mathbf{I}_{M}$. On
the recipient’s side, the tamper detector $\mathcal{T}$ predicts the tamper
mask $\hat{\mathbf{M}}$ on the attacked image to see which parts of the image
are tampered, and we also extract the watermark as $\hat{\mathbf{W}}$ from the
received image. Afterwards, we rectify the extracted watermark by
$\hat{\mathbf{W}}^{\prime}=\hat{\mathbf{W}}\cdot(1-\hat{\mathbf{M}})$ to
discard the tampered contents. Finally, with the original watermark
$\mathbf{W}$ as reference, we uses a feature matching algorithm to locate the
position of the crop in the original image plane. An image is determined as
cropped if there is valid matching result.

Modeling.  We regard the embedding and extraction as the inverse problem, even
if the protected image is cropped and attacked. The formulation of the inverse
problem is:

$(\mathbf{I}_{M},\mathbf{R})=\mathcal{P}(\mathbf{I},\mathbf{W})$ (1)
$\hat{\mathbf{I}},\hat{\mathbf{W}}=\mathcal{P}^{-1}(\mathcal{A}(\mathbf{I}_{M}),\hat{\mathbf{R}}),$
(2)

subject to
$\mathbb{E}(\hat{\mathbf{I}})=\mathbf{I}_{G},\mathbb{E}(\hat{\mathbf{W}})=\mathbf{W}_{G}\text{
and }\mathbb{E}(\mathbf{G})=\hat{\mathbf{G}}$. Here, $\mathbf{R}$ and
$\hat{\mathbf{R}}$ are the additional output and input of the network to keep
the channel numbers as four. $\mathcal{A}$ denotes the function of image post-
processing that attacks the hidden watermark. We let
$\hat{\mathbf{R}}=0.5\cdot\mathbbmss{1}$ where $\mathbbmss{1}$ is a matrix
full of one. The ground-truth cropped watermark $\mathbf{W}_{G}$ and the
ground-truth cropped original image $\mathbf{I}_{G}$ can be generated by
sharing the crop mask used by $\mathcal{A}$. Besides the invertibility, we
regulate that $\mathbb{E}(\mathbf{I}_{M})=\mathbf{I}$ for the imperceptibility
of the embedding.

Fig. 1 shows the pipeline of our scheme. We employ an Anti-Cropping Processor
(ACP) $\mathcal{P}$ that jointly learns the paired function of both watermark
embedding and extraction. We implement $\mathcal{A}$ using a differentiable
attack layer that simulates the attacker’s behavior. The rest of the
components include a tamper detector $\mathcal{T}$, a discriminator
$\mathcal{D}$ and a feature matcher $\mathcal{M}$, which is the SURF
algorithm.

Figure 1: Pipeline design of RWN. The anti-crop processor (ACP) generates the
protected image and conversely in the back-propagation recovers the hidden
watermark. The attack layer simulates several kinds of digital attacks. The
tamper detector predicts the tamper mask and the feature detector locates the
cropped region of the attacked image. Figure 2: Watermark selection. We choose
textured watermarks (left) instead of ones with repeated patterns (right).
Figure 3: Results of watermark embedding, extraction and crop localization.
The attacks involved in each group are different, namely, (a) crop alone, (b)
crop & scaling, (c) crop & JPEG & tamper and (d) crop & Median blurring &
tamper.

###  III-B Network Implementation

Considering the efficiency of the invertible U-Net proposed in [18], we build
our ACP on top of this architecture. The network consists of six invertible
blocks each of which contains a Haar wavelet transformation and a double-side
affine coupling. The network ends with a conditional split layer. The number
of the input and output channels are four. The sizes of $\mathbf{R}$ and
$\hat{\mathbf{R}}$ are of the same as $\mathbf{W}$. $\mathbf{R}$ and
$\hat{\mathbf{R}}$ are not required to be the same.

In the attacking layer $\mathcal{A}$, we first convert the marked images into
the corresponding 8-bit RGB-formatted image. We adopt the Straight-Through
Estimator [19] on calculating the gradients. Then, we build differentiable
methods to simulate the hybrid cropping attacks, which include benign attacks,
cropping and tampering attack. The benign attacks are represented by JPEG
compression, scaling, etc., which do not alter the semantic meaning of the
image. We take the implementation from [10] except that we build our own JPEG
simulator. For cropping, we randomly crop a portion of the protected image
$\textbf{I}_{M}$. For tampering attack, we first randomly select random areas
using a binary matrix $\mathbf{M}$ inside $\mathbf{I}_{M}$, and generate the
tampered image by

$\mathbf{I}_{A}=\mathbf{I}_{\emph{irr}}\cdot\mathbf{M}+\mathbf{I}_{M}\cdot(1-\mathbf{M}_{R}),$
(3)

where $\mathbf{I}_{\emph{irr}}$ refers to the source of the tamper.

Although there are already many scheme which include a carefully-designed JPEG
simulator, e.g., JPEG-SS [17], JPEG-Mask [10], MBRS [20], the real-world JPEG
robustness of these schemes is still limited. We believe it mainly attribute
to that the networks are over-fitted to a fixed compression mode. In this
paper, we propose to apply the Mix-Up strategy [21] over these implementations
so that the generated results are more flexible. We randomly sample a quality
factor $QF_{1}$ that follows $QF_{1}\in[10,100]$, together with a previous
differentiable implementation of JPEG simulator $\mathcal{J}_{1}$, where
$\mathcal{J}_{1}\in${JPEG-SS, JPEG-Mask, MBRS}. We correspondingly generate a
pseudo JPEG image $\mathbf{I}_{\emph{jpg}}^{1}$ by
$\mathbf{I}_{\emph{jpg}}^{1}=\mathcal{J}_{1}(\mathbf{I}_{M},QF_{1})$. In the
same fashion, we generate $k-1$ more pseudo-JPEG images, denoted as
$\mathbf{I}_{\emph{jpg}}^{2},...,\mathbf{I}_{\emph{jpg}}^{k}$. Finally, the
output of JPEG-Mixup is produced by mixing the pseudo-JPEG images with
arbitrary contributing rates $\epsilon_{l}$.

$\mathbf{I}_{jpg}=\sum_{{l}\in[1,k]}\epsilon_{l}\cdot\mathcal{J}_{l}(\mathbf{I}_{M},QF_{l}).$
(4)

where $\sum\epsilon_{l}=1$. The proposed mix-up based JPEG simulation (JPEG-
Mixup) is fairly simple yet effective. Compared to previous schemes, the
interpolation technique can generates more diversed pseudo-JPEG images for
RWN. In the experiments, we show the performance gain in JPEG robustness using
JPEG-Mixup in comparison with the other simulators.

Finally, we build the tamper detector $\mathcal{T}$ upon U-Net [22], a
traditional image segmentation network, and implement the discriminator
$\mathcal{D}$ using Patch-GAN [23].

Watermark Selection. We choose gray-scaled natural images with rich texture as
valid watermark. Images with too much repetitive patterns will lead to
multiple matching solutions for the SURF detector. Also, considering that the
recipient can only blindly extract a partial watermark, we use those common
images available on the Internet that can be easily downloaded by the
recipient using image retrieval systems by referring to the extracted
watermark. Fig. 2 shows some examples of recommended/not recommended
watermarks.

###  III-C Objective Loss Function and Training Details

For ACP, the first part of the loss is reconstruction loss
$\mathcal{L}_{\emph{rec}}$ for $\mathbb{E}(\hat{\mathbf{W}})=\mathbf{W}_{G}$,
$\mathbb{E}(\hat{\mathbf{I}})=\mathbf{I}_{G}$ and
$\mathbb{E}(\mathbf{I}_{M})=\mathbf{I}$. $\mathbb{E}(\cdot)$ is the
expectation operator.
$\mathcal{L}_{\emph{rec}}=\mathcal{F}(\mathbf{I},\mathbf{I}_{M})+\mathcal{F}(\mathbf{W}_{G},\hat{\mathbf{W}})+\mathcal{F}(\mathbf{I}_{G},\hat{\mathbf{I}})$,
where $\mathcal{F}$ is the $\ell_{1}$ distance function. The second part is
the nullification loss for $\mathbb{E}(\mathbf{R})=\hat{\mathbf{R}}$.
$\mathcal{L}_{\emph{ran}}=\mathcal{F}(\mathbf{R},\hat{\mathbf{R}})$. The third
part is the adversarial loss, which further control the introduced distortion
by fooling the discriminator. We accept the least squared adversarial loss
(LS-GAN) [24]. The total loss for ACP is
$\mathcal{L}_{\mathcal{G}}=\mathcal{L}_{rec}+\alpha\cdot\mathcal{L}_{dis}+\beta\cdot\mathcal{L}_{ran}$,
where $\alpha$ and $\beta$ are hyper-parameters. For the tamper detector, we
minimize the binary cross entropy (BCE) loss between the predicted tamper mask
$\hat{\mathbf{M}}$ and the ground-truth mask $\mathbf{M}$.
$\mathcal{L}_{\mathcal{T}}=BCE(\mathbf{M},\hat{\mathbf{M}}).$

We resize the images to the size of $256\times~{}256$. The hyper-parameters
are set as $\alpha=1,\beta=8$. We set $k$ as 3, which is the generated pseudo-
JPEG images used in JPEG-Mixup. The batch size is set as 16, and we
empirically find that using Batch Normalization (BN) in our scheme outperforms
using Instance Normalization (IN). We use Adam optimizer [25] with the default
parameters. The learning rate is $1\times 10^{-4}$.

##  IV Experimental Results

###  IV-A Experiment Setup

We train the scheme on the COCO training/test set [26] with automatically
generated attacks. The dataset contains 117266 training samples and 40670
testing samples. The scheme is tested with human-participated attacks, where
we have invited some volunteers to crop the provided marked images and perform
further image post-processing attack on them. The crop rate is roughly
$\delta\in[0.5,1)$. During testing, we use real JPEG coder to compress the
protected images.

Evaluation Metric.  We employ peak signal-to-noise ratio (PSNR), structural
similarity (SSIM) [27] to evaluate the image quality, and F1 score [28] to
measure the accuracy of crop/tamper localization.

Benchmark. We compare RWN with Van et al. [8] for crop localization. In
addition, we compare RWN with Mantra-Net [1], MVSS-Net [3] and CAT-Net [2] for
tamper localization.

Figure 4: We compare the result of tamper localization on two test images with
several state-of-the-art image forgery detection schemes.

###  IV-B Real-World Performance of Crop Localization

Quality of the Protected Images. In Fig. 3, we randomly sample different pairs
of images as the original and watermark. From the figures, generally, little
detail of the watermark can be found. Though the magnified difference is
visible, it does not matter if the difference is visible, only that the marked
image is perceptually close to the original image. We have conducted more
embedding experiments over 1000 images from the test set, and the average PSNR
between the protected images and the original images is 36.23dB, and the
average SSIM [27] is 0.983.

Accuracy of the Crop Localization. Fig. 3 further shows the results of
watermark extraction, feature matching and crop localization. In row (a), we
only crop the protected image. We see that the crop mask is accurately
predicted. As a result, even without the prior knowledge of the original
image, we know the relative position of the attacked image in the original
image plane. We have conducted more experiments over 1000 images under
different crop rate and different kinds of benign attacks. Here for space
limit, we take $QF$=70 as an example. The average performances are reported in
Table I. Higher IoU (Intersection over Union) and SSIM [27] indicates more
accurate localization result. The average IoUs are above 0.8 where the
protected images do not undergo further attacks except cropping. The scheme is
proven to be agnostic to the crop size in that the performance does not
degrade significantly with larger crop rate.

TABLE I: Average performance of cropping localization under different cropping rate and attack. Rate | Index | NoAttack | JPEG | Scaling | MedianBlur | AWGN
---|---|---|---|---|---|---
90% | IoU | 0.919 | 0.895 | 0.803 | 0.838 | 0.784
SSIM | 0.949 | 0.896 | 0.957 | 0.953 | 0.832
70% | IoU | 0.858 | 0.813 | 0.878 | 0.915 | 0.763
SSIM | 0.942 | 0.878 | 0.940 | 0.940 | 0.804
50% | IoU | 0.821 | 0.706 | 0.603 | 0.540 | 0.513
SSIM | 0.914 | 0.7223 | 0.942 | 0.840 | 0.788
TABLE II: F1 score comparison for tamper detection among our scheme and the state-of-the-art methods. Method | NoAttack | JPEG | Blurring | Scaling | AWGN
---|---|---|---|---|---
Proposed | 0.773 | 0.736 | 0.695 | 0.745 | 0.573
Mantra-Net [1] | 0.566 | 0.480 | 0.557 | 0.540 | 0.347
MVSS-Net [3] | 0.545 | 0.364 | 0.399 | 0.485 | 0.323
CAT-Net [2] | 0.467 | 0.433 | 0.419 | 0.428 | 0.365

Robustness. We test the robustness of our scheme by conducting image post-
processing attack on the marked image. In the last two groups, tampering
attack is introduced. In Table I, we can observe that our scheme provides
high-accuracy crop localization despite the presence of the attacks along with
cropping. The results promote the real-world application of RWN. Thanks to the
robustness of feature matching of SURF, the experiments show that in most
cases our scheme do not require a precise watermark extraction.

Comparison.  Van et al. [8] is fragile where the targeted image cannot be
compressed or low-resolutioned. If the original images are in JPEG formats,
the required traces are much likely destroyed already. The F1 score of [8] are
0.575 on natural images where the resolution and image quality are randomized.
In contrast, RWN does not have any restriction on the original image. After
watermarking, we can robustly locate the crop with a much higher F1 score
0.867.

###  IV-C Accuracy of Tamper Detection.

Fig. 4 shows the tamper detection results on two test images. Note that for
fair comparison, we add the same attacks on the original images for [1, 2, 3].
We observe that although the images are tampered by a variety of random
attacks, we succeed in localizing the tampered areas. In Table II, we provide
the average results over 1000 images. The F1 score of RWN is 0.773 on
uncompressed images and 0.736 on JPEG attacked images ($QF=80$). The
performance of Mantra-Net on JPEG images is much worse than that on plain-text
images. We believe the reason is that less statistical trace is preserved in
the compressed version. In contrast, the embedded watermark signal serves as
the alternative trace for tamper detection, which is designed to resist benign
attacks. The performance on JPEG images does not drop too much. Therefore,
robust watermarking can successfully aid tamper detection by hiding crafted
traces similar to [13, 12].

Figure 5: Results of the ablation studies where we validate the network design
and the Mix-Up [21] strategy in JPEG simulation.

###  IV-D Ablation Study

We discuss the influences of the network design and the training strategies in
RWN. In each ablation test, we fine-tune the network till the accuracy of
cropping localization is close to the baseline. We also use fixed attacks
during testing. The experiments are conducted as follows.

Influence of INN architecture. In Test 1, we train two individual fully
convolutional networks to implement the hiding network and revealing network.
The normalizing-flow-based ACP is replaced in order to measure the
effectiveness brought by invertible problem formulation.

Influence of JPEG Simulator. In Test 2, we implement the JPEG attack with that
proposed in [20, 10, 17]. The JPEG QF and the crop ratio are kept the same for
fair comparison. We train the implementations together with the baseline under
the same losses and batch size.

Influence of Discriminator. In Test 3, we do not use the discriminator
$\mathcal{D}$ and train the pipeline with only the reconstruction and the BCE
loss.

Fig. 5 shows the detailed comparison results. First, the baseline results
outperform the encoder-decoder network design. Second, while MBRS [20] can
provide decent robustness, the extraction performance is even better using our
JPEG simulator. Specifically, the average SSIM between $\hat{\mathbf{W}}$ and
$\mathbf{W}_{G}$ using MBRS [20] is 0.797 compared to 0.878 reported in
Table.I. The Mix-Up strategy prevents the networks from being over-fitted to
any single JPEG simulator, which helps the scheme significantly improve its
real-world robustness. Third, without the discriminator, the extraction result
is also worse than the baseline.

##  V Conclusion

This paper presents a novel robust watermarking network for image crop
localization. We use an invertible pipeline for watermark embedding and
extraction, where the watermark is shared with the recipient. ACP produces a
visually indistinguishable protected image, which is then posted on the social
network instead of the unprotected version. On receiving the attacked version
of the protected image, we use ACP again to extract the watermark, and conduct
SURF feature matching algorithms to determine where the crop was positioned in
the original image plane. We also extend our scheme to detect tampering attack
on the attacked image. Experiments verify that RWN is effective in real-world
application. Finally, we prove the necessity of the progressive recovery as
well as the network design.

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