# Deconvolution of spherical data corrupted with unknown noise

Jérémie Capitao-Miniconi, Elisabeth Gassiat
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay,
91405 Orsay, France

###### Abstract

We consider the deconvolution problem for densities supported on a
$(d-1)$-dimensional sphere with unknown center and unknown radius, in the
situation where the distribution of the noise is unknown and without any other
observations. We propose estimators of the radius, of the center, and of the
density of the signal on the sphere that are proved consistent without further
information. The estimator of the radius is proved to have almost parametric
convergence rate for any dimension $d$. When $d=2$, the estimator of the
density is proved to achieve the same rate of convergence over Sobolev
regularity classes of densities as when the noise distribution is known.

##  1 Introduction

In this paper, we study the deconvolution problem of random data on a sphere
corrupted by independent additive noise. The observations are

$Y_{i}=X_{i}+\varepsilon_{i},\;i=1,\ldots,n$ (1)

where $(X_{i})_{i\geq 1}$ (the signal) is a sequence of independent
identically distributed (i.i.d.) random variables taking values on a
$(d-1)$-dimensional sphere (for $d\geq 2$) with unknown center $C^{\star}$ and
unknown radius $R^{\star}$, $(\varepsilon_{i})_{i\geq 1}$ (the noise) is a
sequence of i.i.d. random variables independent of the signal and with totally
unknown distribution. The distribution of the signal is also unknown, it is
only known that it is spherically supported. To solve the deconvolution
problem and estimate the structural parameters $C^{\star}$ and $R^{\star}$,
the only assumption we shall put on the noise is that its $d$ coordinates are
independently distributed.

The statistical estimation of the center and of the radius of the sphere is of
interest in various applications such as object tracking, robotics, pattern
recognition, see for instance [5], [6], [13], among others, see also [3] and
references therein. Several methods have been proposed based on least squares,
maximum likelihood, see [11] for a recent likelihood based algorithm, most of
them modeling the noise distribution with a Gaussian distribution.

The deconvolution problem of the distribution of the signal when the radius
and the center are known is studied for circular signals (that is when $d=2$)
in [10]. The author proves that the minimax rate of convergence of the
estimator over a wide collection of smoothness classes of the density of the
signal on the circle does not depend on the (known) noise distribution, for a
variety of different noise distributions, contrasting with the situation where
the signal has a density with respect to Lebesgue over the whole space.

Recently, it has been proved in [7] that deconvolution with unknown noise
distribution is possible for multivariate signals, as soon as the signal can
be decomposed in two components that satisfy a mild dependence assumption,
that its distribution has light enough tails, and without any assumption on
the noise distribution except that its two corresponding components are
independently distributed. The authors of [7] then consider the situation
where the probability of $X_{1}$ has a density with respect to Lebesgue
measure, and they prove that not knowing the noise distribution does not
deteriorate the estimation rate of the density on Sobolev regularity classes
for compactly supported signals.

Here, the probability distribution of the signal is singular with respect to
Lebesgue measure on $\mathbb{R}^{d}$ and their convergence results do not
apply. However, we prove that the general conditions they propose under which
deconvolution with unknown noise is possible is satisfied for spherical
signals, this is our first main identifiability result Theorem 2. The main
contribution of our work is then to exhibit estimators that achieve remarkable
properties:

  * •

We propose estimators of the radius, the center, and the distribution of the
signal, which are proved consistent whatever the noise distribution, see
Proposition 2.

  * •

Under the mild assumption that the noise has finite variance, we prove that
the radius of the sphere can be estimated at almost parametric rate with
totally unknown noise distribution, see Theorem 3.

  * •

When $d=2$, that is for circular signals, we prove that the center can be
estimated at almost parametric rate and that the density of the signal
distribution on the circle can be estimated at the same rate as when the
distribution of the noise is known on some Sobolev regularity classes, with a
rate which is minimax as proved in [10], see Theorem 4 and Theorem 5.

In Section 2, we first recall general results of [7] and we prove in
Proposition 1 a strengthened version of the local $L^{2}$-consistency of the
general estimator of the characteristic function of the signal that will be a
basic stone for all our convergence rates theorems. We then state our
identifiability theorem, give the definition of the estimators and prove their
consistency. Section 3 studies the rates of convergence of our estimators, and
in section 4 we study the situation where radius and center of the sphere
together with the noise distribution are unknown, though the distribution of
the angles of the random signal is known. Simulations illustrating our
findings are given in Section 5. We discuss possible further work and related
questions in Section 6. Proofs of propositions and lemmas are detailed in
Section 7.

##  2 Identifiability and estimation method

In this section, we prove that model (1) is identifiable with no more
assumptions. We then explain the estimation method and define the estimators
which will be studied in Section 3.

###  2.1 Preliminaries: deconvolution with unknown noise

We first recall general results in [7]. Then, we prove a proposition which
will be used to obtain the nearly parametric rate of our estimators of the
radius and the center. In [7], the authors consider the situation where the
observations $Y_{i}\in\mathbb{R}^{d}$ come from the model

$\begin{pmatrix}Y^{(1)}\\\ Y^{(2)}\end{pmatrix}=\begin{pmatrix}X^{(1)}\\\
X^{(2)}\end{pmatrix}+\begin{pmatrix}\varepsilon^{(1)}\\\
\varepsilon^{(2)}\end{pmatrix}$

in which $Y^{(1)}\in\mathbb{R}^{d_{1}}$, $d_{1}\geq 1$, and
$Y^{(2)}\in\mathbb{R}^{d_{2}}$, $d_{2}\geq 1$, with $d_{1}+d_{2}=d$, and where
$\varepsilon^{(1)}$ is independent of $\varepsilon^{(2)}$. They prove
identifiability under very mild assumptions on the signal distribution. The
first one is about the tail of its distribution.

  * A($\rho$)

There exists $\rho<2$, $a>0$ and $b>0$ such that for all
$\lambda\in\mathbb{R}^{d}$,
$E\left[\exp\left(\lambda^{\top}X\right)\right]\leq
a\exp\left(b\|\lambda\|_{2}^{\rho}\right)$.

Here, $X=\begin{pmatrix}X^{(1)}\\\ X^{(2)}\end{pmatrix}$ and $\|\cdot\|_{2}$
is the Euclidian norm.
Under A($\rho$), the characteristic function of the signal can be extended
into a multivariate analytic function denoted by

$\displaystyle\Phi:\mathbb{C}^{d_{1}}\times\mathbb{C}^{d_{2}}$
$\displaystyle\longrightarrow$ $\displaystyle\mathbb{C}$
$\displaystyle(z_{1},z_{2})$ $\displaystyle\longmapsto$ $\displaystyle
E\left[\exp\left(iz_{1}^{\top}X^{(1)}+iz_{2}^{\top}X^{(2)}\right)\right].$

The second assumption is a mild dependence assumption (see the discussion
after Theorem 2.1 in [7]).

  * A(dep)

For any $z_{0}\in\mathbb{C}^{d_{1}}$, $z\mapsto\Phi(z_{0},z)$ is not the null
function and for any $z_{0}\in\mathbb{C}^{d_{2}}$, $z\mapsto\Phi(z,z_{0})$ is
not the null function.

Obviously, if no centering constraint is put on the signal or on the noise, it
is possible to translate the signal by a fixed vector $m\in\mathbb{R}^{d}$ and
the noise by $-m$ without changing the observation. The model can thus be
identifiable only up to translation.

######  Theorem 1 (from [7]).

If the distribution of the signal satisfies A($\rho$) and A(dep), then the
distribution of the signal and the distribution of the noise can be recovered
from the distribution of the observation, up to translation.

The first step in the estimation procedure is the estimation of the
characteristic function of the signal by a method inspired by the proof of the
identifiability theorem. For any $S>0$, let $\Upsilon_{\rho,S}$ be the subset
of multivariate analytic functions from $\mathbb{C}^{d}$ to $\mathbb{C}$
defined as follows.

$\Upsilon_{\rho,S}=\left\\{\phi\text{ analytic }\text{s.t. }\forall
z\in\mathbb{R}^{d},\overline{\phi(z)}=\phi(-z),\phi(0)=1\text{ and }\forall
j\in\mathbb{N}^{d}\setminus\\{0\\},\left|\frac{\partial^{j}\phi(0)}{\prod_{a=1}^{d}j_{a}!}\right|\leq\frac{S^{\|j\|_{1}}}{\|j\|_{1}^{\|j\|_{1}/\rho}}\right\\}$

where $\|j\|_{1}=\sum_{a=1}^{d}i_{a}$. For all $\Phi$ satisfying A($\rho$),
there exists $S$ such that $\Phi\in\Upsilon_{\rho,S}$. Let
$\Phi_{\varepsilon^{(p)}}$ be the characteristic function of
$\varepsilon^{(p)}$, $p=1,2$, and define for all $\phi\in\Upsilon_{\rho,S}$
and any $\nu>0$,

$M(\phi;\nu|\Phi)=\int_{B_{\nu}^{d_{1}}\times
B_{\nu}^{d_{2}}}|\phi(t_{1},t_{2})\Phi(t_{1},0)\Phi(0,t_{2})-\Phi(t_{1},t_{2})\phi(t_{1},0)\phi(0,t_{2})|^{2}|\Phi_{\varepsilon^{(1)}}(t_{1})\Phi_{\varepsilon^{(2)}}(t_{2})|^{2}dt_{1}dt_{2},$

where $B_{\nu}=[-\nu,\nu]$. It is proved in [7] that if
$\phi\in\Upsilon_{\rho,S}$ satisfies A(dep), $M(\phi;\nu|\Phi)=0$ for a fixed
$\nu$ if and only if $\phi=\Phi$ (up to translation). The estimator of the
characteristic function of the signal can then be defined as a minimizer of
the empirical estimator of $M$.

Fix some $\nu_{\text{est}}>0$. Let $\cal H$ be a subset of functions from
$\mathbb{R}^{d}$ to $\mathbb{C}^{d}$ such that all elements of $\cal H$
satisfy A(dep) and which is closed in $L^{2}(B_{\nu_{\text{est}}}^{d})$.
Define $\widehat{\phi}_{n}$ as a (up to $1/n$) measurable minimizer of the
functional $M_{n}$ over $\Upsilon_{\rho,S}\cap{\cal H}$, where $M_{n}$ is
defined as

$M_{n}(\phi)=\int_{B_{\nu_{\text{est}}}^{d_{1}}\times
B_{\nu_{\text{est}}}^{d_{2}}}|\phi(t_{1},t_{2})\tilde{\phi}_{n}(t_{1},0)\tilde{\phi}_{n}(0,t_{2})-\tilde{\phi}_{n}(t_{1},t_{2})\phi(t_{1},0)\phi(0,t_{2})|^{2}dt_{1}dt_{2},$

where for all $(t_{1},t_{2})\in\mathbb{C}^{d_{1}}\times\mathbb{C}^{d_{2}}$,

$\tilde{\phi}_{n}(t_{1},t_{2})=\frac{1}{n}\sum_{\ell=1}^{n}\exp\left\\{it_{1}^{\top}Y_{\ell}^{(1)}+it_{2}^{\top}Y_{\ell}^{(2)}\right\\}.$

It appears that, for any $\nu>0$, $\widehat{\phi}_{n}$ is a consistent
estimator of $\Phi$ in $L^{2}([-\nu,\nu]^{d})$ at almost parametric rate. The
constants will depend on the signal through $\rho$ and $S$, and on the noise
through its second moment and the following quantity:

$c_{\nu}=\inf\\{|\Phi_{\varepsilon^{(1)}}(t)|,\;t\in
B_{\nu}^{d_{1}}\\}\wedge\inf\\{|\Phi_{\varepsilon^{(2)}}(t)|,\;t\in
B_{\nu}^{d_{2}}\\}.$ (2)

For any noise distribution, for small enough $\nu$, $c_{\nu}$ is a positive
real number. We prove the following.

######  Proposition 1.

Assume $\Phi\in\Upsilon_{\rho,S}\cap{\cal H}$ and $\varepsilon_{1}$ has finite
variance. Fix some $\nu\in[(d+4/3)e/S,\nu_{\text{est}}]$ such that
$c_{\nu}>0$. For all $\delta>0$, there exist positive constants $c_{1},c_{2}$,
$c_{3}$ which depend on $\delta$, $\nu_{\text{est}}$, $\nu$, $c_{\nu}$,$\rho$,
$S$, $\cal H$, $d$ and $E(\|Y_{1}\|^{2})$ such that for all $x\geq 1$ and
$n\geq(1\vee xc_{1})/c_{2}$, with probability at least $1-e^{-x}$,

$\int_{B_{\nu}^{d}}|\widehat{\phi}_{n}(t)-\Phi(t)|^{2}dt\leq
c_{3}\left(\frac{x}{n^{1-\delta}}\vee\frac{x^{2}}{n^{2-2\delta}}\right).$

Proposition 1 improves on Proposition A.3 in [7] and is proved in Section 7.1.

###  2.2 Identifiability theorem

For any $Z\in\mathbb{R}^{d}$, denote $Z^{(1)},\ldots,Z^{(d)}$ its $d$
coordinates. We shall parametrize a vector on a sphere through angles. For any
$u\in[0,1]^{d-1}$, define $S(u)$ on the unit $d$-dimensional sphere as

$S(u)=\begin{pmatrix}\cos(2\pi u^{(1)})\\\ \sin(2\pi u^{(1)})\cos(\pi
u^{(2)})\\\ \sin(2\pi u^{(1)})\sin(\pi u^{(2)})\cos(\pi u^{(3)})\\\ \vdots\\\
\sin(2\pi u^{(1)})\sin(\pi u^{(2)})\cdots\sin(\pi u^{(d-2)})\cos(\pi
u^{(d-1)})\\\ \sin(2\pi u^{(1)})\sin(\pi u^{(2)})\cdots\sin(\pi
u^{(d-2)})\sin(\pi u^{(d-1)})\end{pmatrix}.$

Then for a sequence $(U_{i})_{i\geq 1}$ of i.i.d random vectors taking values
in $[0,1]^{d-1}$, we have for all $i\geq 1$,

$X_{i}=C^{\star}+R^{\star}S(U_{i}),$ (3)

with $C^{\star}$ the center of the sphere and $R^{\star}$ its radius.

We shall also make the following assumptions.

  * (H1)

The coordinates $\varepsilon_{1}^{(1)},\ldots,\varepsilon_{1}^{(d)}$ of the
noise are independently distributed. We denote
$\mathbb{Q}^{\star}=\otimes_{j=1}^{d}\mathbb{Q}^{\star}_{j}$ the distribution
of $\varepsilon_{1}$, with $\mathbb{Q}_{j}^{\star}$ the distribution of
$\varepsilon_{1}^{(j)}$, $j=1,\ldots,d$.

  * (H2)

The distribution of $U_{1}$ has a density $f^{\star}$ with respect to Lebesgue
measure on $[0,1]^{d-1}$. When $d>2$, we assume $f^{\star}$ positive on
$(0,\gamma)^{d-1}$ for some $\gamma>0$.

We shall sometimes call $f^{\star}$ exploration density of the angles or
exploration density. For any $\mathbb{Q}=\otimes_{j=1}^{d}\mathbb{Q}_{j}$,
with $\mathbb{Q}_{j}$, $j=1,\ldots,d$, probability distributions on
$\mathbb{R}$, any probability density $f$ on $[0,1]^{d-1}$, any
$C\in\mathbb{R}^{d}$ and any $R\geq 0$, let $\mathbb{P}_{C,R,f,\mathbb{Q}}$ be
the distribution of $Y_{1}$ when $X_{1}$ lies on the sphere with center $C$,
radius $R$, and $U_{1}$ has density $f$.

######  Theorem 2.

Assume (H1) and (H2). For any $\mathbb{Q}=\otimes_{j=1}^{d}\mathbb{Q}_{j}$,
any probability density $f$ on $[0,1]^{d-1}$, any $C\in\mathbb{R}^{d}$ and any
$R\geq 0$,
$\mathbb{P}_{C,R,f,\mathbb{Q}}=\mathbb{P}_{C^{\star},R^{\star},f^{\star},\mathbb{Q}^{\star}}$
if and only if $R=R^{\star}$, $f=f^{\star}$, and there exists
$m\in\mathbb{R}^{d}$ such that $C=C^{\star}+m$ and
$\mathbb{Q}(\cdot)=\mathbb{Q}^{\star}(\cdot+m)$. If moreover $\mathbb{Q}$ and
$\mathbb{Q}^{\star}$ have finite first moment and are centered distributions,
then $m=0$, that is $C=C^{\star}$ and $\mathbb{Q}=\mathbb{Q}^{\star}$.

Proof of Theorem 2.
We shall apply Theorem 1. For any probability density $f$ on $[0,1]^{d-1}$,
$C\in\mathbb{R}^{d}$ and $R\geq 0$, A($\rho$) holds with $\rho=1$ and with the
constants $a=1$ and $b=\|C\|_{2}+R$. To verify A(dep), since all coordinates
of the noise are independently distributed, we first choose a decomposition of
the signal in two components. We define $\widetilde{X}^{(1)}=X_{1}^{(1)}$, and
$\widetilde{X}^{(2)}=(X_{1}^{(2)},\ldots,X_{1}^{(d)})^{T}$, and we prove in
Section 7.2 the following Lemma from which A(dep) follows.

######  Lemma 1.

Assume (H2). Then for all $z\in\mathbb{C}$,
$E\left[\exp\left(iz\widetilde{X}^{(1)}\right)|\widetilde{X}^{(2)}\right]$ is
not $P_{\widetilde{X}^{(2)}}$-a.s. the null random variable, and for all
$z\in\mathbb{C}^{d-1}$,
$E\left[\exp\left(iz^{T}\widetilde{X}^{(2)}\right)|\widetilde{X}^{(1)}\right]$
is not $P_{\widetilde{X}^{(1)}}$-a.s. the null random variable. Here,
$P_{\widetilde{X}^{(p)}}$ denotes the distribution of ${\widetilde{X}^{(p)}}$,
$p=1,2$.

Then, translation of the spherical signal does not change the radius of the
sphere and the exploration density of the angles on the sphere, but only the
centering of the sphere and correspondingly the distribution of the noise.
Applying Theorem 1 leads then to the conclusion of Theorem 2.

The proof of Lemma 1 proceeds by computing explicitly the conditional
expectation, and then to give an argument why it can not be the null random
variable. The argument for $d=2$ does not apply to $d>2$, in which case we use
another argument needing the positivity of $f^{\star}$ near the origin. Since
the choice of the positive first coordinate to define the angles and the
density is arbitrary, the proof still holds under the assumption that the
density is positive near the point of the sphere at the intersection with one
of the $2d$ axis directions.

###  2.3 Estimation method and consistency

We shall apply the method described in Section 2.1 to estimate $R^{\star}$ and
$f^{\star}$. For any positive real number $R$ and any probability density $f$
on $[0,1]^{d-1}$, define $\Psi_{f,R}$ the characteristic function of the
random variable with distribution on the centered sphere with radius $R$, and
exploration density of the angles $f$, that is, for all $t\in\mathbb{R}^{d}$,

$\Psi_{f,R}(t)=\int_{(0,1)^{d-1}}\exp\left\\{iRt^{T}S(u)\right\\}f(u)du.$ (4)

We shall consider functions $\Psi_{f,R}$ for any function $f$ on $(0,1)^{d-1}$
(not only probability densities) as defined by (4). Notice that $\Psi_{f,R}$
can be extended to $\mathbb{C}^{d}$.
Since all components of $\epsilon_{1}$ are independent, we have to make a
choice of $d_{1}$ and $d_{2}$ for the definition of $M_{n}$ and $M$, thus, in
the following, we assume to have $d_{1}=1$ and $d_{2}=d-1$, as in the proof of
Theorem 2. For any $\nu>0$, define

${M}(f,R)=\int_{B_{\nu}\times
B_{\nu}^{d-1}}|\Psi_{f,R}(t_{1},t_{2})\Psi_{f^{\star},R^{\star}}(t_{1},0)\Psi_{f^{\star},R^{\star}}(0,t_{2})-\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\Psi_{f,R}(t_{1},0)\Psi_{f,R}(0,t_{2})|^{2}|\Phi_{\epsilon}(t_{1},t_{2})|^{2}dt_{1}dt_{2}.$

The parameter $\nu$ does not appear in the notation of $M$ and can be chosen
as needed.
Fix some $\nu_{\text{est}}>0$, and define

${M_{n}}(f,R)=\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}|\Psi_{f,R}(t_{1},t_{2})\tilde{\psi}_{n}(t_{1},0)\tilde{\psi}_{n}(0,t_{2})-\tilde{\psi}_{n}(t_{1},t_{2})\Psi_{f,R}(t_{1},0)\Psi_{f,R}(0,t_{2})|^{2}dt_{1}dt_{2},$

with

$\tilde{\psi}_{n}(t_{1},t_{2})=\frac{1}{n}\sum_{\ell=1}^{n}\exp\left\\{it_{1}\tilde{Y}_{\ell}^{(1)}+it_{2}^{\top}\tilde{Y}_{\ell}^{(2)}\right\\}.$

where for all $\ell$, $\tilde{Y}_{\ell}^{(1)}=Y_{\ell}^{(1)}$, and
$\tilde{Y}_{\ell}^{(2)}=(Y_{\ell}^{(2)},\ldots,Y_{\ell}^{(d)})^{T}$.
We need to fix the compact subset on which we minimize $M_{n}$. We choose
${\cal F}$ a compact subset of ${\mathbb{L}}^{2}[(0,1)^{d-1}]$ such that for
all $f\in{\cal F}$, $\int_{(0,1)^{d-1}}f(u)du=1$, and real numbers
$R_{\text{min}}$ and $R_{\text{max}}$ such that
$0<R_{\text{min}}<R_{\text{max}}<+\infty$. Since we shall study minimax rates
in Section 3, we shall fix later ${\cal F}$ to include all Sobolev classes of
interest in that paper. Then we define $(\widehat{f},\widehat{R})$ as any
measurable random variable such that

$M_{n}\left(\widehat{f},\widehat{R}\right)\leq\inf_{(f,R)\in{\cal
F}\times[R_{\text{min}};R_{\text{max}}]}M_{n}\left(f,R\right)+\frac{1}{n}.$
(5)

Notice that we do not constrain functions in ${\cal F}$ to be non negative,
that is we do not constrain $\widehat{f}$ to be a probability density. Using
Proposition 1 we get the following corollary which will be the basic stone to
obtain estimation rates of our estimators. For any $\nu>0$, define
$c_{\nu}^{\star}$ as in (2) with $\mathbb{Q}=\mathbb{Q}^{\star}$.

######  Corollary 1.

Assume $f^{\star}\in{\cal F}$, $R^{\star}\in(R_{\text{min}};R_{\text{max}})$
and $\varepsilon_{1}$ has finite variance. For all
$\nu\in(0,\nu_{\text{est}}]$ such that $c_{\nu}^{\star}>0$, for all
$\delta>0$, there exist positive constants $c_{1},c_{2},c_{3}$ which depend on
$\delta$, $\nu$, $c_{\nu}^{\star}$, $d$ and $E(\|Y\|^{2})$ such that for all
$x\geq 1$ and $n\geq(1\vee xc_{1})/c_{2}$, with probability at least
$1-e^{-x}$,

$\int_{[-\nu,\nu]^{d}}|\Psi_{\widehat{f},\widehat{R}}(t)-\Psi_{f^{\star},R^{\star}}(t)|^{2}dt\leq
c_{3}\left(\frac{x}{n^{1-\delta}}\vee\frac{x^{2}}{n^{2-2\delta}}\right).$

We insist on the fact that the quantity $c_{\nu}^{\star}$ is unknown, and that
its knowledge is not needed to construct the estimators and to get asymptotic
rates, since there always exists a small enough $\nu$ such that
$c_{\nu}^{\star}>0$.

When $\mathbb{Q}^{\star}$ has a finite first moment and is centered, we can
estimate the center of the sphere. We define

$\widehat{C}=\frac{1}{n}\sum_{i=1}^{n}Y_{i}-\widehat{R}\int_{(0,1)^{d-1}}S(u)\widehat{f}(u)du.$
(6)

The estimators of the radius and of the exploration density can be proved to
be consistent by applying $M$-estimator general results. Then consistency of
the estimator of the radius follows. We give a detailed proved of the
following proposition in Section 7.3. Here, it is not needed that the noiuse
has finite variance.

######  Proposition 2.

Assume $f^{\star}\in{\cal F}$ and
$R^{\star}\in(R_{\text{min}};R_{\text{max}})$. Then
$\widehat{R}=R^{\star}+o_{\mathbb{P}_{C^{\star},R^{\star},f^{\star},\mathbb{Q}^{\star}}}(1)$
and
$\int_{(0,1)^{d-1}}(\widehat{f}(u)-f^{\star}(u))^{2}du=o_{\mathbb{P}_{C^{\star},R^{\star},f^{\star},\mathbb{Q}^{\star}}}(1)$.
If moreover $\mathbb{Q}^{\star}$ has finite first moment and is a centered
distribution, then also
$\widehat{C}=C^{\star}+o_{\mathbb{P}_{C^{\star},R^{\star},f^{\star},\mathbb{Q}^{\star}}}(1)$.

##  3 Convergence rates of the estimators

In this section, we prove that the estimator of the radius has almost
parametric rate of convergence, whatever the dimension $d$ of the sphere. We
then get rates of convergence for the estimator of the exploration density and
of the center in the case $d=2$ that is for circular signals. Our estimator of
the exploration density achieves the minimax rate on Sobolev regularity
classes and the estimator of the center can be proved to have almost
parametric rate.

###  3.1 The estimator of the radius

Our first main result is the fact that, without any knowledge of the noise
distribution and of the exploration density, the radius of the sphere can be
recovered at almost parametric rate.

######  Theorem 3.

Assume $f^{\star}\in{\cal F}$ and
$R^{\star}\in(R_{\text{min}};R_{\text{max}})$. Assume also that
$\varepsilon_{1}$ has finite variance. For all $\nu\in(0,\nu_{\text{est}}]$
such that $c_{\nu}^{\star}>0$, for all $\delta>0$, there exist positive
constants $c_{1},c_{2},c_{3}$ which depend on $\delta$, $\nu$,
$c_{\nu}^{\star}$, $d$ and $E(\|Y\|^{2})$ such that for all $x\geq 1$ and
$n\geq(1\vee xc_{1})/c_{2}$, with probability at least $1-e^{-x}$,

$|\widehat{R}-R^{\star}|^{2}\leq\frac{c_{3}}{R_{\text{min}}^{2}(1-\frac{(\nu
R_{\text{max}})^{2}}{2d+8})^{2}}\left(\frac{x}{n^{1-\delta}}\vee\frac{x^{2}}{n^{2-2\delta}}\right).$

###### Proof.

We denote by $\Delta$ the Laplacian operator in the Cartesian coordinate
system,

$\Delta=\sum_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}},$

and for all $k\geq 1$, $\Delta^{k}=\Delta^{k-1}\circ\Delta$ with $\Delta^{0}$
the identity operator.
Notice that $\Psi_{f,R}$ is an eigenfunction of the Laplacian, with eigenvalue
$R^{2}$,

$\Delta\Psi_{f,R}(x)+R^{2}\ \Psi_{f,R}(x)=0,$

so that for all $k\geq 1$,

$\Delta^{k}\Psi_{f,R}(x)+(-1)^{k-1}R^{2k}\ \Psi_{f,R}(x)=0.$

Define $\mathbb{D}_{d}(0,\nu)=\\{x\in\mathbb{R}^{d},\;\|x\|_{2}\leq\nu\\}$ the
$d$-dimensional disk centered at the origin and with radius $\nu$, $\Gamma$
the gamma function and $\lambda_{d}$ the $d$-dimensional Lebesgue measure.
Then, according to [12], for all $\nu>0$ and for all multivariate analytic
function $\psi$ on $\mathbb{C}^{d}$,

$\displaystyle\frac{1}{\lambda_{d}(\mathbb{D}_{d}(0,\nu))}\int_{\mathbb{D}_{d}(0,\nu)}\psi(x)dx$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\Delta^{k}\psi(0)}{2^{k}k!\prod_{j=1}^{k}(d+2j)}\nu^{2k}$
$\displaystyle=$
$\displaystyle\Gamma\left(\frac{d}{2}+1\right)\sum_{k=0}^{\infty}\frac{\Delta^{k}\psi(0)}{2^{2k}k!\Gamma(\frac{d}{2}+k+1)}\nu^{2k}.$

Applying this equality to ${\Psi}_{\widehat{f},\widehat{R}}$ and to
$\Psi_{f^{\star},R^{\star}}$ we get that

$\frac{1}{(\sqrt{\pi}\nu)^{d}}\int_{\mathbb{D}_{d}(0,\nu)}\left({\Psi}_{\widehat{f},\widehat{R}}(x)-\Psi_{f^{\star},R^{\star}}(x)\right)dx=\sum_{k=0}^{\infty}(-1)^{k}\frac{\widehat{R}^{2k}-(R^{\star})^{2k}}{2^{2k}k!\Gamma(\frac{d}{2}+k+1)}\nu^{2k},$
(7)

since
$\lambda^{d}(\mathbb{D}_{d}(0,\nu))=\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}\nu^{d}$.
Let $J_{d/2}$ be the Bessel function of order $(d/2)$. We collect in Section 8
results on Bessel functions that will be useful in our analysis. Using
identity (I) in in Section 8 we get that for all $x\in\mathbb{R}$,

$\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{2^{2k}k!\Gamma(\frac{d}{2}+k+1)}=2^{d/2}\frac{J_{d/2}(x)}{x^{d/2}},$
(8)

so that using (7), (8), Cauchy-Schwarz inequality and the fact that
$\mathbb{D}_{d}(0,\nu)\subset B_{\nu}^{d}$ we obtain

$2^{d}\left|\frac{J_{d/2}(\nu\widehat{R})}{(\nu\widehat{R})^{d/2}}-\frac{J_{d/2}(\nu
R^{\star})}{(\nu
R^{\star})^{d/2}}\right|^{2}\leq\frac{1}{(\sqrt{\pi}\nu)^{d}\Gamma(\frac{d}{2}+1)}\int_{B_{\nu}^{d}}\left({\Psi}_{\widehat{f},\widehat{R}}(x)-\Psi_{f^{\star},R^{\star}}(x)\right)^{2}dx.$
(9)

Let $H$ be the function defined by

$\forall x\neq
0,\;H(x)=\frac{J_{d/2}(x)}{x^{d/2}},\;H(0)=\frac{1}{2^{d/2}\Gamma(\frac{d}{2}+1)}.$

Then using (8), $H$ has infinitely many derivatives so that there exists
$\widetilde{R}\in(R^{\star},\widehat{R})$ such that

$H(\nu\widehat{R})-H(\nu
R^{\star})=\nu(\widehat{R}-R^{\star})H^{\prime}(\nu\widetilde{R}).$

Computation of the derivative and (IV) in Section 8 gives

$H^{\prime}(x)=\frac{x(J_{d/2-1}(x)-J_{d/2+1}(x))-dJ_{d/2}(x)}{2x^{d/2+1}},$

and using (V) in Section 8 we get

$H^{\prime}(x)=-\frac{J_{d/2+1}(x)}{x^{d/2}}.$

Using lemma 3 in Section 8, we get that

$J_{d/2+1}(\nu\widetilde{R})\geq\frac{(\nu\widetilde{R})^{d/2+1}}{2^{d/2+1}\Gamma(\frac{d}{2}+2)}\left(1-\frac{(\nu\widetilde{R})^{2}}{2d+8}\right).$

Since $\widetilde{R}\in(R_{min},R_{max})$, we deduce that for any
$\nu\in(0,1/R_{max})$,

$\displaystyle|H^{\prime}(\nu\widetilde{R})|$
$\displaystyle=\left|\frac{J_{d/2}(\nu\widetilde{R})}{(\nu\widetilde{R})^{d/2}}\right|\geq\frac{\nu\widetilde{R}}{2^{d/2+1}\Gamma(\frac{d}{2}+2)}\left|1-\frac{(\nu\widetilde{R})^{2}}{2d+8}\right|$
$\displaystyle\geq\frac{\nu
R_{min}}{2^{d/2+1}\Gamma(\frac{d}{2}+2)}\left(1-\frac{(\nu
R_{max})^{2}}{2d+8}\right)>0,$

so that

$|\widehat{R}-R^{\star}|\leq\frac{2^{d/2+1}\Gamma(\frac{d}{2}+2)}{\nu^{2}R_{min}(1-\frac{(\nu
R_{max})^{2}}{2d+8})}|H(\nu\widehat{R})-H(\nu R^{\star})|.$

Using (9) we get

$|\widehat{R}-R^{\star}|^{2}\leq\frac{4\Gamma(\frac{d}{2}+2)^{2}}{\nu^{4}(\sqrt{\pi}\nu)^{d}R_{min}^{2}(1-\frac{(\nu
R_{max})^{2}}{2d+8})^{2}}\int_{B_{\nu}^{d}}\left({\Psi}_{\widehat{f},\widehat{R}}(x)-\Psi_{f^{\star},R^{\star}}(x)\right)^{2}dx.$

The end of the proof follows from Corollary 1. ∎

###  3.2 The estimator of the density and of the center

In this section, we consider the case of circular signals, that is $d=2$. In
this case, we can rewrite model (3) using one dimensional angles
$U_{i}\in[0,1]$, as

$X_{i}=C^{\star}+R^{\star}\begin{pmatrix}\cos(2\pi U_{i})\\\ \sin(2\pi
U_{i})\end{pmatrix}.$ (10)

We shall focus on the following regularity classes. For any $L>0$,
$\beta>\frac{1}{2}$ and $\gamma>0$, set

$W_{\beta}(L)=\\{f\in\mathbb{L}^{2}([0,1]):\sum_{k=-\infty}^{\infty}|f_{k}|^{2}|k|^{2\beta}\leq
L^{2}\\},$
$A_{\gamma}(L)=\\{f\in\mathbb{L}^{2}([0,1]):\sum_{k=-\infty}^{\infty}|f_{k}|^{2}e^{2\gamma
k}\leq L^{2}\\},$

where for any function $f\in\mathbb{L}^{2}([0,1])$,
$(f_{k})_{k\in{\mathbb{Z}}}$ is the sequence of Fourier coefficients of $f$:

$f_{k}=\int_{0}^{1}f(\theta)e^{2i\pi k\theta}d\theta,\ \ k\in\mathbb{Z}.$

We fix $\cal F$ as a compact subset of ${\mathbb{L}}^{2}[(0,1)^{d-1}]$ such
that for all $f\in{\cal F}$, $\int_{(0,1)^{d-1}}f(u)du=1$, and containing as
subsets all $W_{\beta}(L)$ and $A_{\gamma}(L)$ for all $\beta>\frac{1}{2}$,
$\gamma>0$, and $L\leq L_{max}$ chosen. We shall now define an estimator of
$f^{\star}$ using truncated Fourier expansions of $\widehat{f}$ defined in
Section 2.3. For $N>0$ an integer to be chosen, we define $T_{N}\widehat{f}$
the trigonometric polynomial estimator of $f^{\star}$:

$\forall x\in(0,1),\ \ T_{N}\widehat{f}(x)=\sum_{|k|\leq
N}\widehat{f}_{k}e^{-2i\pi kx}.$

For any $\nu>$, $c_{\nu}>0$, $E>0$, define ${\cal Q}(\nu,c({\nu}),E)$ the set
of distribution $\mathbb{Q}=\mathbb{Q}_{1}\otimes\mathbb{Q}_{2}$ on
$\mathbb{R}^{2}$ such that $c_{\nu}\geq c(\nu)$ and
$\int_{\mathbb{R}^{2}}\|x\|^{2}d\mathbb{Q}(x)\leq E$. Define now the maximum
risk of the estimator for any class of densities $\cal C$ and any class of
noise distribution $\cal Q$ as follows.

$R\left[T_{N}\widehat{f};{\cal C};R_{\text{min}};R_{\text{max}};{\cal
Q}\right]=\sup_{f\in{\cal
C},R\in[R_{\text{min}};R_{\text{max}}],{\mathbb{Q}}\in{\cal
Q},C\in\mathbb{R}^{2}}\mathbb{E}_{C,R,f,{\mathbb{Q}}}\left(\int_{0}^{1}(T_{N}\widehat{f}(x)-f(x))^{2}dx\right).$

The following theorem shows that a good choice of $N$ leads to minimax
adaptive estimation rate over the regularity classes $W_{\beta}(L)$ and
controlled maximum risk over the regularity classes $A_{\gamma}(L)$.

######  Theorem 4.

For $\alpha\in(0,1/2)$, set

$N=\left\lfloor\alpha\frac{\log n}{\log\log n}\right\rfloor.$

Then for all $L>0$, $\beta>1/2$, (resp. $\gamma>0$), for all
$\nu\in(0,\nu_{\text{est}}]$, $c(\nu)>0$, $E>0$,

$R\left[T_{N}\widehat{f};W_{\beta}(L);R_{\text{min}};R_{\text{max}};{\cal
Q}(\nu,c({\nu}),E)\right]\leq
L^{2}\alpha^{-2\beta}\left(\frac{lnln(n)}{ln(n)}\right)^{2\beta}(1+o(1))$ (11)

as $n$ tends to infinity, and

$R\left[T_{N}\widehat{f};A_{\gamma}(L);R_{\text{min}};R_{\text{max}};{\cal
Q}(\nu,c({\nu}),E)\right]\leq
exp\left(-2\gamma\left(\alpha\frac{ln(n)}{lnln(n)}\right)\right)(1+o(1))$ (12)

as $n$ tends to infinity.

In [10], the author studies the estimation of the exploration density for
noisy circular data on the unit circle (known radius) and with known noise
distribution. Comparison of (11) with Theorem 1 in [10] shows that our
estimator is rate minimax adaptive to unknown radius, unknown noise
distribution and unknown regularity over classes $W_{\beta}(L)$ for the
signal, with a constant deteriorated by a factor at most $2^{2\beta}$.
Comparing (12) with Theorem 2 in [10] shows that a loss in the upper bound for
the rate of convergence of the maximum risk of our estimator in case of
unknown radius and unknown noise distribution on classes $A_{\gamma}(L)$ for
the signal.

###### Proof.

For any $f\in{\mathcal{F}}$,

$\int_{0}^{1}\left(T_{N}\widehat{f}(x)-f(x)\right)^{2}dx=\sum_{|k|\leq
N}|f_{k}-\widehat{f}_{k}|^{2}+\sum_{|k|>N}|f_{k}|^{2},$

so that for any $R\in(R_{\text{min}};R_{\text{max}})$, ${\mathbb{Q}}\in{\cal
Q}(\nu,c({\nu}),E)$, $C\in\mathbb{R}^{2}$,

$\mathbb{E}_{C,R,f,{\mathbb{Q}}}\left(\int_{0}^{1}(T_{N}\widehat{f}(x)-f(x))^{2}dx\right)=\mathbb{E}_{C,R,f,{\mathbb{Q}}}\left[\sum_{|k|\leq
N}|f_{k}-\widehat{f}_{k}|^{2}\right]+\sum_{|k|>N}|f_{k}|^{2}.$

The first term on the right hand side will be shown to be negligible with
respect to the second term thanks to the following proposition, for which a
detailed proof can be found in Section 7.4

######  Proposition 3.

Assume $f^{\star}\in{\cal F}$ and
$R^{\star}\in(R_{\text{min}},R_{\text{max}})$. For all
$\nu\in(0,\nu_{\text{est}}]$ such that $c_{\nu}^{\star}>0$, for all
$\delta>0$, there exists a constant $c>0$ depending on $\delta$, $\nu$,
$c_{\nu}^{\star}$, $d$, $R^{\star}$, $R_{\text{min}}$, $R_{\text{max}}$, and
$E(\|Y\|^{2})$ such that for all $x\geq 1$, $n\geq(1\vee xc_{1})/c_{2}$, with
probability at least $1-e^{-x}$,

$\sum_{|k|\leq N}|f_{k}^{\star}-\widehat{f}_{k}|^{2}\leq c\left(\frac{2}{\nu
R_{\text{min}}}\right)^{2N}(N+1)(N!)^{2}\left(\frac{x}{n^{1-\delta}}\vee\frac{x^{2}}{n^{2-2\delta}}\right).$
(13)

Choose $\delta$ small enough so that $2\alpha<1-\delta$. Then for large enough
$n$, for a constant $c>0$,

$\mathbb{E}_{C,R,f,{\mathbb{Q}}}\left[\sum_{|k|\leq
N}|f_{k}-\hat{f}_{k}^{n}|^{2}\right]\leq c\ (\nu
R_{\text{min}})^{-2N}(N+1)2^{2N}(N!)^{2}n^{-1+\delta},$

and using the fact that, $\forall N\geq 1$, $N!\leq eN^{N+\frac{1}{2}}e^{-N}$,
we finally have,

$\mathbb{E}_{C,R,f,{\mathbb{Q}}}\left[\sum_{|k|\leq
N}|f_{k}-\hat{f}_{k}^{n}|^{2}\right]\leq e^{2}c\ (\nu
R_{\text{min}})^{-2N}(N+1)2^{2N}N^{2N+1}e^{-2N}n^{-1+\delta}.$ (14)

The term at the right hand side of (14) is at most of order

$\exp\Bigg{\\{}(2\alpha+\delta-1)\ ln(n)\Bigg{[}1+o(1)\Bigg{]}\Bigg{\\}}.$

Now,

$\sup_{f\in W_{\beta}(L)}\sum_{|k|>N}|f_{k}|^{2}\leq L^{2}N^{-2\beta},$ (15)

and

$\sup_{f\in A_{\gamma}(L)}\sum_{|k|>N}|f_{k}|^{2}\leq L^{2}e^{-2\gamma N}.$
(16)

Equation (11) follows from (14) and (15), and equation (12) follows from (14)
and (16). ∎

######  Theorem 5.

Assume $f^{\star}\in{\cal F}$ and
$R^{\star}\in(R_{\text{min}},R_{\text{max}})$. Assume also that
$\varepsilon_{1}$ has finite variance. Then for any $\delta>0$,

$\left(\widehat{C}-C^{\star}\right)=O_{\mathbb{P}_{C^{\star},R^{\star},f^{\star},\mathbb{Q}^{\star}}}\left(n^{-1/2+\delta}\right).$

Notice that we can not get exponential deviations for the empirical mean of
the observations when nothing more is assumed about the noise apart having
finite variance. Since the estimator of the center involves the empirical mean
of the observations, we only prove tightness of
$n^{1/2-\delta}\left(\widehat{C}-C^{\star}\right)$.

###### Proof.

Notice that

$f^{\star}_{1}=\int_{0}^{1}\cos(2\pi u)f^{\star}(u)du+i\int_{0}^{1}\sin(2\pi
u)f^{\star}(u)du$

so that

$C^{\star}=\mathbb{E}[Y]-R^{\star}\begin{pmatrix}\text{Re}(f^{\star}_{1})\\\
\text{Im}(f^{\star}_{1})\end{pmatrix},$

and in the same way

$\widehat{C}=\frac{1}{n}\sum_{l=1}^{n}Y_{l}-\widehat{R}\begin{pmatrix}\text{Re}(\widehat{f}_{1})\\\
\text{Im}(\widehat{f}_{1})\end{pmatrix}.$

Thus, using the triangle inequality, and the fact that $\widehat{R}\leq
R_{\text{max}}$ and $|f^{\star}_{1}|\leq 1$, we get

$\|\widehat{C}-C^{\star}\|_{2}\leq\left\|\frac{1}{n}\sum_{l=1}^{n}Y_{l}-\mathbb{E}[Y]\right\|_{2}+|\widehat{R}-R^{\star}|+R_{\text{max}}\left\|\begin{pmatrix}\text{Re}(f^{\star}_{1})-\text{Re}(\widehat{f}_{1})\\\
\text{Im}(f^{\star}_{1})-\text{Im}(\widehat{f}_{1})\end{pmatrix}\right\|_{2}.$

The theorem follows from Theorem 3, Proposition 3 and the central limit
theorem.

∎

##  4 When the exploration density is known

In this section, we assume that $f^{\star}$ is known. By exchanging the role
of the signal and of the noise, we can look at model (1) as a semi-parametric
deconvolution problem in which the noise has known distribution (up to
centering and radius) on a sphere. But we are able to estimate the radius and
the center without solving the semi-parametric deconvolution problem, that is
without estimating $\mathbb{Q}$. We estimate the radius using the contrast
function $M_{n}(f^{\star},R)$. Since this function is continuous, we can
define

$\widetilde{R}={\text{
Argmin}}\;\\{M_{n}(f^{\star},R),\;R\in[R_{\text{min}};R_{\text{max}}]\\}.$

If moreover $\mathbb{Q}^{\star}$ has finite first moment and is a centered
distribution, then the estimator of the center is defined as

$\widetilde{C}=\frac{1}{n}\sum_{i=1}^{n}Y_{i}-\widetilde{R}\int_{(0,1)^{d-1}}S(u)f^{\star}(u)du.$

Theorem 6 states that
$\sqrt{n}(\widetilde{R}-R^{\star},\widetilde{C}-C^{\star})$ converges in
distribution as $n$ tends to infinity to some centered Gaussian distribution.
It will be a consequence of the lemma stated below. In the following, for
$R\in[R_{\text{min}},R_{\text{max}}]$, we omit $f^{\star}$ as an argument of
$M$ and $M_{n}$. We write $M^{\prime}$, $M^{\prime}_{n}$ their derivatives
with respect to $R$ and $M^{\prime\prime}$, $M^{\prime\prime}_{n}$ their
second derivatives with respect to $R$.

######  Lemma 2.

The following results hold true under the assumptions of Theorem 6.

  1. (1)

$\widetilde{R}$ is a consistent estimator of $R^{\star}$.

  2. (2)

There exists a matrix $V$ such that
$\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^{n}Y_{i}-E(Y_{1}),M^{\prime}_{n}(R^{\star})\right)$
converges in distribution to a centered Gaussian distribution with variance V.

  3. (3)

$M^{\prime\prime}(R^{\star})\neq 0$ and for any random variable
$R_{n}\in[R_{min},R_{max}]$ converging in probability to $R^{\star}$, one has

$M^{\prime\prime}_{n}(R_{n})=M^{\prime\prime}(R^{\star})+o_{\mathbb{P}_{C^{\star},R^{\star},f^{\star},\mathbb{Q}^{\star}}}(1).$

The proof of Lemma 2 is given in Section 7.5.
Define the $(d+1)\times(d+1)$ matrix

$\Sigma=\begin{pmatrix}0&-\frac{1}{M^{\prime\prime}(R^{\star})}\\\
1&\frac{E(S(U))}{M^{\prime\prime}(R^{\star})}\end{pmatrix}V\begin{pmatrix}0&1\\\
-\frac{1}{M^{\prime\prime}(R^{\star})}&\frac{E(S(U))^{T}}{M^{\prime\prime}(R^{\star})}\end{pmatrix}$

######  Theorem 6.

Assume that $\mathbb{Q}^{\star}$ has finite second moment and is a centered
distribution. Then $\sqrt{n}(\widetilde{R}-R^{\star},\widetilde{C}-C^{\star})$
converges in distribution to a centered Gaussian distribution with variance
$\Sigma$.

The proof of Theorem 6 is detailed in Section 7.6.

##  5 Simulations

The aim of this section is to illustrate our method with examples for which
the noise is not bounded. We choose $d=2$ and we consider the model (10) with
$R^{\star}=3$, $C^{\star}=0$ and $U,\varepsilon$ generated as follows.

  1. (1)

$U\sim\text{Unif}(0,1)$ and $\varepsilon\sim\mathcal{N}(0,(0.12)^{2}I)$,
figure 1

  2. (2)

$U\sim\text{Unif}(0,1)$ and for $i\in\\{1,2\\}$
$\varepsilon^{(i)}\sim\frac{1}{2}\delta_{(-1)}+\frac{1}{2}\mathcal{E}\text{xp}\left\\{\frac{1}{0.12}\right\\}$,
figure 2

  3. (3)

$U\sim\text{Unif}(0,1)$ and $\varepsilon\sim\mathcal{N}(0,I)$, figure 3

  4. (4)

$U\sim f_{U}:x\in(0,1)\mapsto\frac{\exp\left\\{\cos(2\pi
x)\right\\}}{\int_{0}^{1}\exp\left\\{\cos(2\pi u)\right\\}du}$ and
$\varepsilon\sim\mathcal{N}(\begin{pmatrix}-1.6\\\
2.5\end{pmatrix},\begin{pmatrix}(0.2)^{2}&0\\\ 0&(0.57)^{2}\end{pmatrix})$,
figure 4

For each case, we generate $n$ observed points for $n\in V$ with

$V=\\{10^{2},2\cdot 10^{2},3\cdot 10^{2},4\cdot 10^{2},5\cdot
10^{2},10^{3},2\cdot 10^{3},3\cdot 10^{3},5\cdot 10^{3},10^{4},5\cdot
10^{4},\\\ 7.5\cdot 10^{4},10^{5},3\cdot 10^{5},5\cdot 10^{5},8\cdot
10^{5},10^{6}\\}.$

We estimate the radius of the circle in the case where the exploration density
is known, and unknown.
In practice, when we want to estimate the radius, the choice of $N$ and $\nu$
does not significantly change the results thus the simulations are done with
$N=\left\lfloor{\frac{\text{log}(n)}{\text{log(log}(n))}}\right\rfloor$ and
$\nu=0.5$.
For each figure, there are $4$ plots,

  1. Top left

Scatter plot of the $10^{6}$ observed points.

  2. Top right

Scatter plot of the $10^{6}$ observed points + the support of the signal $X$.

  3. Bottom left

Plot of $(\text{log}|\widehat{R}-R^{\star}|,\text{log}(n))_{n\in V}$ \+ the
linear regression, when the density $f^{\star}$ is known and unknown.

  4. Bottom right

To avoid that the step size in $V$ is not constant and to better visualize the
graph, we choose $W\subset V$ and we plot $(\widehat{R},n)_{n\in W}$, when the
density $f^{\star}$ is known and unknown.

Figure 1:  $U\sim\text{Unif}(0,1)$ ,
$\varepsilon\sim\mathcal{N}(0,(0.12)^{2}I)$ and $W=\\{3\cdot 10^{2},4\cdot
10^{2},5\cdot 10^{2},10^{3},2\cdot 10^{3},3\cdot 10^{3},5\cdot
10^{3},10^{4},5\cdot 10^{4},7.5\cdot 10^{4},10^{5}\\}$ Figure 2:
$U\sim\text{Unif}(0,1)$ , for $i\in\\{1,2\\}$
$\varepsilon^{(i)}\sim\frac{1}{2}\delta_{(-1)}+\frac{1}{2}\mathcal{E}\text{xp}\left\\{\frac{1}{0.12}\right\\}$
and $W=\\{3\cdot 10^{2},4\cdot 10^{2},5\cdot 10^{2},10^{3},2\cdot
10^{3},3\cdot 10^{3},5\cdot 10^{3},10^{4},5\cdot 10^{4},7.5\cdot
10^{4},10^{5},3\cdot 10^{5},5\cdot 10^{5},8\cdot 10^{5},10^{6}\\}$ Figure 3:
$U\sim\text{Unif}(0,1)$ , $\varepsilon\sim\mathcal{N}(0,I)$ and $W=\\{3\cdot
10^{2},4\cdot 10^{2},5\cdot 10^{2},10^{3},2\cdot 10^{3},3\cdot 10^{3},5\cdot
10^{3},10^{4},5\cdot 10^{4},7.5\cdot 10^{4},10^{5},3\cdot 10^{5},5\cdot
10^{5}\\}$ Figure 4:  $U\sim f_{U}:x\in(0,1)\mapsto\frac{\exp\left\\{\cos(2\pi
x)\right\\}}{\int_{0}^{1}\exp\left\\{\cos(2\pi u)\right\\}du}$ ,
$\varepsilon\sim\mathcal{N}(\begin{pmatrix}-1.6\\\
2.5\end{pmatrix},\begin{pmatrix}(0.2)^{2}&0\\\ 0&(0.57)^{2}\end{pmatrix})$ and
$W=\\{3\cdot 10^{2},4\cdot 10^{2},5\cdot 10^{2},10^{3},2\cdot 10^{3},3\cdot
10^{3},5\cdot 10^{3},10^{4},5\cdot 10^{4},7.5\cdot 10^{4},10^{5},3\cdot
10^{5},5\cdot 10^{5}\\}$

The graph of $log|R^{\star}-\widehat{R}|$ of figure 1, 2 and 3 drive us to
reasonably conjecture that the rate of convergence of
$|R^{\star}-\widehat{R}|$ is the same when the density $f^{\star}$ is known
and unknown.
We use Monte-Carlo to estimate $M_{n}$ and the package optimize.minimize in
Python to minimize $M_{n}$, that is, there is possibly a numerical bias that
can explain the fluctuations on the values of $\widehat{R}$ as we can see in
the figure 5 and figure 6. (left histogram) The histograms are computed with
Monte-Carlo replications of $50$ values of $\widehat{R}$ for $n=10.000$ in the
case of figure 1 and figure 2.

Figure 5:  $U\sim\text{Unif}(0,1)$ and
$\varepsilon\sim\mathcal{N}(0,(0.12)^{2}I)$ Figure 6:  $U\sim\text{Unif}(0,1)$
and for $i\in\\{1,2\\}$
$\varepsilon^{(i)}\sim\frac{1}{2}\delta_{(-1)}+\frac{1}{2}\mathcal{E}\text{xp}\left\\{\frac{1}{0.12}\right\\}$

Finally, for each $n\in V$ in the case of figure 1, we computed $30$ values of
$\widehat{R}$ denoted by $(\widehat{R}^{n}_{k})_{1\leq k\leq 30}$ when the
density $f^{\star}$ is known and unknown, we give the following table which
gives the emperical mean squarred error.

$n$ |  $f^{\star}$ known |  $f^{\star}$ unknown
---|---|---
$\frac{1}{30}\sum_{k=1}^{30}|R^{\star}-\widetilde{R}^{n}_{k}|^{2}$ | $\frac{1}{30}\sum_{k=1}^{30}|R^{\star}-\widehat{R}^{n}_{k}|^{2}$
$10^{2}$ | $7.09\cdot 10^{-2}$ | $2.86\cdot 10^{-3}$
$2\cdot 10^{2}$ | $3.87\cdot 10^{-2}$ | $1.02\cdot 10^{-3}$
$3\cdot 10^{2}$ | $1.87\cdot 10^{-2}$ | $5.72\cdot 10^{-4}$
$4\cdot 10^{2}$ | $1.5\cdot 10^{-2}$ | $6.15\cdot 10^{-4}$
$5\cdot 10^{2}$ | $7.28\cdot 10^{-3}$ | $4.24\cdot 10^{-4}$
$10^{3}$ | $1.27\cdot 10^{-3}$ | $2.07\cdot 10^{-4}$
$2\cdot 10^{3}$ | $3.64\cdot 10^{-4}$ | $9.97\cdot 10^{-5}$
$3\cdot 10^{3}$ | $1.39\cdot 10^{-4}$ | $8.73\cdot 10^{-5}$
$5\cdot 10^{3}$ | $1.31\cdot 10^{-4}$ | $5.29\cdot 10^{-5}$
$10^{4}$ | $6.99\cdot 10^{-5}$ | $6.63\cdot 10^{-6}$
$5\cdot 10^{4}$ | $1.01\cdot 10^{-5}$ | $4.96\cdot 10^{-7}$
$7.5\cdot 10^{4}$ | $7.75\cdot 10^{-6}$ | $2.72\cdot 10^{-7}$
$10^{5}$ | $7.63\cdot 10^{-6}$ | $1.96\cdot 10^{-7}$
$3\cdot 10^{5}$ | $2.47\cdot 10^{-6}$ | $6.03\cdot 10^{-8}$
$5\cdot 10^{5}$ | $1.88\cdot 10^{-6}$ | $3.27\cdot 10p^{-8}$
$8\cdot 10^{5}$ | $1.37\cdot 10^{-6}$ | $2.32\cdot 10^{-8}$
$10^{6}$ | $1.02\cdot 10^{-6}$ | $1.78\cdot 10^{-8}$

##  6 Discussion

In this paper, we proved that deconvolution of spherical data is possible
without any knowledge of the distribution of the noise, and that the radius of
the sphere can be recovered at nearly parametric rate. The question whether
the rate $1/\sqrt{n}$ can be attained is still open. To get the almost
parametric rate following the proposed analysis would require first to be able
to strengthen the lower bound of $M$ in (17). But in [7], getting a lower
bound for $M$ requires delicate arguments involving a technical truncation
from which it is not possible to derive a quadratic lower bound. If ever such
a lower bound can be proved, new ideas have to be developed. Also, we were
able to prove the identifiability theorem for all possible densities on a
circle, but in higher dimensions the proof holds only for densities that are
positive near the origin. Extending the result to hold for any density for any
$d$ would be nice.
We also proved, for noisy data on a circle that the exploration density can be
recovered at the same minimax convergence rate on Sobolev regularity classes
as when the noise distribution is known. The analysis we propose here does not
extend to $d>2$, and the question of the convergence rate for $d>2$ remains
unsolved.
More generally, deconvolution of data coming from observations supported on a
lower dimensional manifold and corrupted by additive noise has been
investigated earlier for known noise in [9], see also [4]. The extension of
the methodology proposed here to analyze those settings and to deal with
unknown noise distribution will be developed in a further work. Understanding
how to deal with noisy observations in topological data analysis is a
challenging question, see for instance [1] and [2], and our solution for
additive noise having independent components can be understood as a
contribution in this perspective.

##  7 Proofs

###  7.1 Proof of Proposition 1

We shall denote $\|\cdot\|_{2,\nu}$ the ${\mathbb{L}}^{2}(B_{\nu}^{m})$-norm
and $\|\cdot\|_{\infty,\nu}$ the ${\mathbb{L}}^{\infty}(B_{\nu}^{m})$-norm,
where the dimension $m$ may be $d$, $d_{1}$ or $d_{2}$ and is clear from the
context.
Following the proof of Proposition A.2 in [7] and the proof of Proposition 24
in Appendix B.5 in [8], we easily get that there exist positive constants $b$,
$\eta_{1}<1$ and $\eta_{2}<1$ depending only on $\nu$, $S$, $d$, $\rho$ such
that for all $\Phi\in\Upsilon_{\rho,S}$,

$\|h\|_{2,\nu}\leq\eta_{1}\Longrightarrow M(\Phi+h;\nu|\Phi)\geq
c_{\nu}^{4}\|h\|_{2,\nu}^{2+2\epsilon(\|h\|_{2,\nu})},$ (17)

with, for any $u\in(0,1)$,

$\epsilon(u)=\frac{b}{\log\log\frac{1}{u}},$

and such that for any $t_{1}\in\mathbb{R}^{d_{1}}$ and any
$t_{2}\in\mathbb{R}^{d_{2}}$,

$\displaystyle\|h\|_{2,\nu}\leq\eta_{2}$ $\displaystyle\Longrightarrow$
$\displaystyle\|h(\cdot,0)\|_{2,\nu}^{2}\leq\|h\|_{2,\nu}^{2-2\epsilon(\|h\|_{2,\nu})}$
(18) $\displaystyle{\text{ and
}}\|h(0,\cdot)\|_{2,\nu}^{2}\leq\|h\|_{2,\nu}^{2-2\epsilon(\|h\|_{2,\nu})}.$

We now fix $\eta=\eta_{1}\wedge\eta_{2}$. Let $Z_{n}$ be the random process
defined, for all
$t=(t_{1},t_{2})\in\mathbb{R}^{d_{1}}\times\mathbb{R}^{d_{2}}$, by

$Z_{n}(t)=\sqrt{n}\left(\tilde{\phi}_{n}(t)-\Phi(t)\Phi_{\varepsilon^{(1)}}(t_{1})\Phi_{\varepsilon^{(2)}}(t_{2})\right)$

Using explicit computation, straightforward upper bounds and (18) we easily
get that there exists a constant $C$ that depends only on $\nu_{\text{est}}$,
$\rho$ and $S$ such that if $h\in\Upsilon_{\rho,S}$ is such that
$\|h\|_{2,\nu_{\text{est}}}\leq\eta$, then

$\displaystyle\left|\left(M_{n}(\Phi+h)-M(\Phi+h,\nu_{\text{est}}|\Phi)\right)\right.$
$\displaystyle-$
$\displaystyle\left.\left(M_{n}(\Phi)-M(\Phi,\nu_{\text{est}}|\Phi)\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\left[\frac{\|Z_{n}\|_{\infty,\nu_{\text{est}}}}{\sqrt{n}}+\frac{\|Z_{n}\|^{2}_{\infty,\nu_{\text{est}}}}{n}\right]\cdot\left[\|h\|_{2,\nu_{\text{est}}}^{1-\epsilon(\|h\|_{2,\nu_{\text{est}}})}+\|h\|_{2,\nu_{\text{est}}}^{2-2\epsilon(\|h\|_{2,\nu_{\text{est}}})}\right]$
$\displaystyle\leq$ $\displaystyle
3C\frac{\|Z_{n}\|_{\infty,\nu_{\text{est}}}}{\sqrt{n}}\cdot\|h\|_{2,\nu_{\text{est}}}^{1-\epsilon(\|h\|_{2,\nu_{\text{est}}})}$
(20)

since $\|Z_{n}\|_{\infty,\nu_{\text{est}}}\leq 2\sqrt{n}$ and
$\|h\|_{2,\nu_{\text{est}}}\leq 1$. We shall use the following deviation
inequality which is proved in Appendix G of [8]. There exist a numerical
constant $c$ and a constant $C$ that depends only on $d$, $\nu_{\text{est}}$
and $E(\|Y_{1}\|^{2})$ such that for all $n\geq 1$ and $x>0$, with probability
at least $1-4e^{-x}$,

$\|Z_{n}\|_{\infty,\nu_{\text{est}}}\leq C+c\sqrt{x}+c\frac{x}{\sqrt{n}}.$
(21)

The end of the proof follows from a classical peeling device. Let
$\widehat{h}=\widehat{\phi}_{n}-\Phi$. For any $a>0$, let $j_{0}$ the integer
such that $2^{j_{0}}\leq a<2^{j_{0}+1}$, and
$J_{n}(a)=\\{j\in\mathbb{N}\;:j_{0}\leq j,\;2^{j+1}\leq r_{n}\eta\\}$. Then

$P\left(r_{n}\|\widehat{h}\|_{2,\nu}\geq a\right)\leq
P\left(\|\widehat{h}\|_{2,\nu}\geq\eta/2\right)+\sum_{j\in
J_{n}(a)}P\left(2^{j}\leq r_{n}\|\widehat{h}\|_{2,\nu}\leq 2^{j+1}\right).$

Now, if $2^{j}\leq r_{n}\|\widehat{h}\|_{2,\nu}\leq 2^{j+1}$, then

$\inf_{h:2^{j}\leq r_{n}\|h\|_{2,\nu}\leq 2^{j+1}}M_{n}(\Phi+h)\leq
M_{n}(\Phi),$

which implies that

$\displaystyle\sup_{h:2^{j}\leq r_{n}\|h\|_{2,\nu}\leq
2^{j+1}}\left|\left(M_{n}(\Phi+h)-M(\Phi+h,\nu_{\text{est}}|\Phi)\right)\right.$
$\displaystyle-$
$\displaystyle\left.\left(M_{n}(\Phi)-M(\Phi,\nu_{\text{est}}|\Phi)\right)\right|$
$\displaystyle\geq$ $\displaystyle\inf_{h:2^{j}\leq r_{n}\|h\|_{2,\nu}\leq
2^{j+1},\|h\|_{2,\nu}\leq\eta(\delta)}M(\Phi+h,\nu_{\text{est}}|\Phi)$
$\displaystyle\geq$ $\displaystyle\inf_{h:2^{j}\leq r_{n}\|h\|_{2,\nu}\leq
2^{j+1},\|h\|_{2,\nu}\leq\eta(\delta)}M(\Phi+h,\nu|\Phi)$

where the last inequality follows from the fact that since
$\nu\leq\nu_{\text{est}}$, $M(\cdot,\nu_{\text{est}}|\Phi)\geq
M(\cdot,\nu|\Phi)$. Using (17) and (7.1) we then get that for $j\in J_{n}(a)$,

$P\left(2^{j}\leq r_{n}\|\widehat{h}\|_{2,\nu}\leq 2^{j+1}\right)\leq
P\left(c_{\nu}^{4}\left(\frac{2^{j}}{r_{n}}\right)^{2+2\epsilon(2^{j}/r_{n})}\leq
3C\frac{\|Z_{n}\|_{\infty,\nu_{\text{est}}}}{\sqrt{n}}\left(\frac{2^{j+1}}{r_{n}}\right)^{1-\epsilon(2^{j+1}/r_{n})}\right)$

so that

$P\left(2^{j}\leq r_{n}\|\widehat{h}\|_{2,\nu}\leq 2^{j+1}\right)\leq
P\left(\|Z_{n}\|_{\infty,\nu_{\text{est}}}\geq\frac{c_{\nu}^{4}}{3C}2^{j}\sqrt{n}\left(\frac{1}{r_{n}}\right)^{1+3\epsilon(2^{j}/r_{n})}\right).$

Now, for $j\in J_{n}(a)$, $(2^{j}/r_{n})\leq\eta$, and for
$c(\eta)=3/\log\log(1/\eta)$ we have $3\epsilon(2^{j}/r_{n})\leq c(\eta)$. We
then get for $j\in J_{n}(a)$,

$P\left(2^{j}\leq r_{n}\|\widehat{h}\|_{2,\nu}\leq 2^{j+1}\right)\leq
P\left(\|Z_{n}\|_{\infty,\nu_{\text{est}}}\geq\frac{c_{\nu}^{4}}{3C}2^{j}\sqrt{n}\left(\frac{1}{r_{n}}\right)^{1+c(\eta)}\right).$

We now take $r_{n}$ such that $r_{n}^{1+c(\eta)}=\sqrt{n}$, that is
$r_{n}=n^{1/2(1+c(\eta))}$, and we get

$P\left(r_{n}\|\widehat{h}\|_{2,\nu}\geq a\right)\leq
P\left(\|\widehat{h}\|_{2,\nu}\geq\eta/2\right)+\sum_{j\geq
j_{0}}P\left(\|Z_{n}\|_{\infty,\nu_{\text{est}}}\geq C2^{j}\right)$ (22)

for some constant $C$. To deal with the first term in the upper bound we use
(25) in [7]. The second term is a series which is upper bounded using (21) and
Proposition 1 follows.

###  7.2 Proof of Lemma 1

To begin with, for any $z_{0}\in\mathbb{C}$ and $z\in\mathbb{C}^{d-1}$,

$E\left[\exp\left(iz_{0}\widetilde{X}^{(1)}+iz^{T}\widetilde{X}^{(2)}\right)\right]=E\left[E\left[\exp\left(iz_{0}\widetilde{X}^{(1)}\right)|\widetilde{X}^{(2)}\right]\exp\left(iz^{T}\widetilde{X}^{(2)}\right)\right],$

and usual arguments for multivariate analytic functions on $\mathbb{C}^{d-1}$
prove that $z\mapsto
E\left[\exp\left(iz_{0}\widetilde{X}^{(1)}+iz^{T}\widetilde{X}^{(2)}\right)\right]$
is the nul function if and only if
$E\left[\exp\left(iz_{0}\widetilde{X}^{(1)}\right)|\widetilde{X}^{(2)}\right]$
is zero $P_{\widetilde{X}^{(2)}}$-a.s. In the same way, for any
$z_{0}\in\mathbb{C}^{d-1}$, $z\mapsto
E\left[\exp\left(iz\widetilde{X}^{(1)}+iz_{0}^{T}\widetilde{X}^{(2)}\right)\right]$
is the nul function if and only if
$E\left[\exp\left(iz_{0}^{T}\widetilde{X}^{(2)}\right)|\widetilde{X}^{(1)}\right]$
is zero $P_{\widetilde{X}^{(1)}}$-a.s. Also, the value of the center
$C^{\star}$ can only change the function
$E\left[\exp\left(z^{T}(\widetilde{X}^{(1)},\widetilde{X}^{(2)})\right)\right]$,
$z\in\mathbb{C}^{d}$, by a factor $\exp(z^{T}C^{\star})$ which is non zero, so
that we may assume $C^{\star}=0$ to prove Lemma 1.
In the following, we write for all $u\in(0,1),\widetilde{S}^{(1)}(u)=cos(2\pi
u)$ and for all $u\in(0,1)^{d-1}$ ,

$\widetilde{S}^{(2)}(u)=\begin{pmatrix}\sin(2\pi u^{(1)})\cos(\pi u^{(2)})\\\
\sin(2\pi u^{(1)})\sin(\pi u^{(2)})\cos(\pi u^{(3)})\\\ \vdots\\\ \sin(2\pi
u^{(1)})\sin(\pi u^{(2)})\cdots\sin(\pi u^{(d-2)})\cos(\pi u^{(d-1)})\\\
\sin(2\pi u^{(1)})\sin(\pi u^{(2)})\cdots\sin(\pi u^{(d-2)})\sin(\pi
u^{(d-1)})\end{pmatrix}.$

We first prove that for any $z_{0}\in\mathbb{C}$,
$E\left[\exp\left(iz_{0}\widetilde{X}^{(1)}\right)|\widetilde{X}^{(2)}\right]$
is not $P_{\widetilde{X}^{(2)}}$-a.s. zero.
Since $f^{\star}$ is not identically zero, there exists a closed interval
$[\alpha,\beta]$ subset of one of the four following intervals :
$(0,\frac{1}{4})$, $(\frac{1}{4},\frac{1}{2})$, $(\frac{1}{2},\frac{3}{4})$,
$(\frac{3}{4},1)$, a vector $a=(a^{(i)})_{1\leq i\leq
d-1}\in(\alpha,\beta)\times(0,1)^{d-2}$ (if $d=2$, $a$ is a real number in
$(\alpha,\beta)$) and $\lambda>0$ such that if we define

$I_{1}=\\{u=(u^{(i)})_{i\in\\{1,\ldots,d-1\\}}\in(\alpha,\beta)\times(0,1)^{d-2}\;:\;\left\|\widetilde{S}^{(2)}(u)-\widetilde{S}^{(2)}(a)\right\|^{2}_{2}<\lambda^{2}\\},$

then the restriction of $f^{\star}$ to $I_{1}$, $f^{\star}|_{I_{1}}$, is not
the null function.
We choose $\lambda$ small enough such that, if we define
$A\subset(-1,1)^{d-1}$ as $A=\widetilde{S}^{(2)}(I_{1})$, we have that there
exists $I_{2}\subset(0,1)^{d-1}$ such that $I_{1}\cap I_{2}=\emptyset$ and
$(\widetilde{S}^{(2)})^{-1}(A)=I_{1}\cup I_{2}$.
We define, for $i,j\in\\{1,2\\}$, the $\mathcal{C}^{1}$ diffeomorphisms
$\eta_{i,j}:I_{i}\longrightarrow I_{j}$, $u\mapsto(S^{(2)})^{-1}(S^{(2)}(u))$,
such that $\eta_{i,i}=Id|_{I_{i}}$ and
$\eta_{i,j}\circ\eta_{j,i}=Id|_{I_{i}}$.
Note that we can explicitly calculate $\eta_{i,j}(x)$ for the different
possible inclusions of $[\alpha,\beta]$:

  1. 1.

for $[\alpha,\beta]\subset(0,\frac{1}{4})$ or
$[\alpha,\beta]\subset(\frac{1}{4},\frac{1}{2})$, we have
$\eta_{1,2}(u^{(1)},u^{(2)},\ldots,u^{(d-1)})=(\frac{1}{2}-u^{(1)},u^{(2)},\ldots,u^{(d-1)})$,

  2. 2.

and for $[\alpha,\beta]\subset(\frac{1}{2},\frac{3}{4})$ or
$[\alpha,\beta]\subset(\frac{3}{4},1)$ , we have
$\eta_{1,2}(u^{(1)},u^{(2)},\ldots,u^{(d-1)})=(\frac{3}{2}-u^{(1)},u^{(2)},\ldots,u^{(d-1)})$.

We now compute
$E[\exp(iz_{0}\widetilde{X}^{(1)})|\widetilde{X}^{(2)}]1_{\widetilde{X}^{(2)}\in
A}$. For any measurable bounded function $\omega$ on
$(-R^{\star},R^{\star})^{d-1}$, we have

$\displaystyle
E[\exp(iz_{0}\widetilde{X}^{(1)})\omega(\widetilde{X}^{(2)})1_{\widetilde{X}^{(2)}\in
A}]$ $\displaystyle=$
$\displaystyle\int_{(\widetilde{S}^{(2)})^{-1}(A)}\omega(R^{\star}\widetilde{S}^{(2)}(u))\exp(iz_{0}R^{\star}\cos(2\pi
u^{(1)}))f^{\star}(u)du$ $\displaystyle=$
$\displaystyle\int_{I_{1}}\omega(R^{\star}\widetilde{S}^{(2)}(u))\exp(iz_{0}R^{\star}\cos(2\pi
u^{(1)}))f^{\star}(u)du$
$\displaystyle+\int_{I_{2}}\omega(R^{\star}\widetilde{S}^{(2)}(u))\exp(iz_{0}R^{\star}\cos(2\pi
u^{(1)}))f^{\star}(u)du.$

Define the change of variables $u=\eta_{1,2}(v)$ in the second integral. Using
the explicit definition of $\eta_{1,2}$ which is differentiable with Jacobian
equal to $1$ we get

$\displaystyle
E[\exp(iz_{0}\widetilde{X}^{(1)})\omega(\widetilde{X}^{(2)})1_{\widetilde{X}^{(2)}\in
A}]$
$\displaystyle=\int_{I_{1}}\omega(R^{\star}\widetilde{S}^{(2)}(u))\\{\exp(iR^{\star}z_{0}\cos(2\pi
u^{(1)}))f^{\star}(u)+\exp(iR^{\star}z_{0}\cos(2\pi\eta_{1,2}(u)^{(1)}))f^{\star}(\eta_{1,2}(u))\\}du$
$\displaystyle=\int_{I_{1}}\omega(R^{\star}\widetilde{S}^{(2)}(u))\frac{\exp(iR^{\star}z_{0}\cos(2\pi
u^{(1)}))f^{\star}(u)+\exp(iR^{\star}z_{0}\cos(2\pi\eta_{1,2}(u)^{(1)}))f^{\star}(\eta_{1,2}(u))}{f^{\star}(u)+f^{\star}(\eta_{1,2}(u))}f^{\star}(u)du$
$\displaystyle+\int_{I_{1}}\omega(R^{\star}\widetilde{S}^{(2)}(u))\frac{\exp(iR^{\star}z_{0}\cos(2\pi
u^{(1)}))f^{\star}(u)+\exp(iR^{\star}z_{0}\cos(2\pi\eta_{1,2}(u)^{(1)}))f^{\star}(\eta_{1,2}(u))}{f^{\star}(u)+f^{\star}(\eta_{1,2}(u))}f^{\star}(\eta_{1,2}(u))du.$

Thus if we define $\nu_{1}:A\longrightarrow I_{1}$ such that for all $u\in
I_{1}$, $\nu_{1}(\widetilde{S}^{(2)}(u))=u$, and $\nu_{2}:A\longrightarrow
I_{2}$ such that for all $u\in I_{2}$, $\nu_{2}(\widetilde{S}^{(2)}(u))=u$, we
get

$\displaystyle
E\left[\exp(iz_{0}\widetilde{X}^{(1)})|\widetilde{X}^{(2)}\right]1_{\widetilde{X}^{(2)}\in
A}$
$\displaystyle=\frac{\exp(iR^{\star}z_{0}\cos(2\pi\nu_{1}(\widetilde{X}^{(2)}))^{(1)})f^{\star}(\nu_{1}(\widetilde{X}^{(2)}))+\exp(-iR^{\star}z_{0}\cos(2\pi\nu_{2}(\widetilde{X}^{(2)}))^{(1)})f^{\star}(\nu_{2}(\widetilde{X}^{(2)}))}{f^{\star}(\nu_{1}(\widetilde{X}^{(2)}))+f^{\star}(\nu_{2}(\widetilde{X}^{(2)}))}1_{\widetilde{X}^{(2)}\in
A}.$

Finally,
$E\left[\exp(iz_{0}\widetilde{X}^{(1)})|\widetilde{X}^{(2)}\right]1_{\widetilde{X}^{(2)}\in
A}$ is nul $\mathbb{P}_{\widetilde{X}^{(2)}}$-a.s if and only if for
$f^{\star}du$ almost all $u\in I_{1}$,

$\exp(iR^{\star}z_{0}\cos(2\pi
u^{(1)}))f^{\star}(u)+\exp(iR^{\star}z_{0}\cos(2\pi\eta_{1,2}(u)^{(1)}))f^{\star}(\eta_{1,2}(u))=0,$

that is for $f^{\star}du$ almost all $u\in I_{1}$,

$\exp\left(iR^{\star}z_{0}\left(\cos(2\pi
u^{(1)})-\cos(2\pi\eta_{1,2}(u)^{(1)})\right)\right)=-\frac{f^{\star}(\eta_{1,2}(u))}{f^{\star}(u)}.$

Since for almost all $u\in I_{1}$, $f^{\star}(u)\neq 0$, this would imply in
particular that for almost all $u\in I_{1}$

$R^{\star}\text{Re}(z_{0})\left(\cos(2\pi
u^{(1)})-\cos(2\pi\eta_{1,2}(u)^{(1)})\right)=\pi\;\;[\text{mod}2\pi],$ (23)

which gives a contradiction.

Now, let’s prove that for any $z_{0}\in\mathbb{C}^{d-1}$,
$E\left[\exp\left(iz_{0}^{T}\widetilde{X}^{(2)}\right)|\widetilde{X}^{(1)}\right]$
is not $P_{\widetilde{X}^{(1)}}$-a.s. zero.
Since $f^{\star}$ is not identically zero, there exists a closed interval
$[\alpha,\beta]$ in one of the four following intervals : $(0,\frac{1}{4})$,
$(\frac{1}{4},\frac{1}{2})$, $(\frac{1}{2},\frac{3}{4})$, $(\frac{3}{4},1)$,
such that $f|_{(\alpha,\beta)\times(0,1)^{d-2}}$ is not the null function.
Define $J_{1}=(\alpha,\beta)$, $B=\widetilde{S}^{(1)}(J_{1})$, and $J_{2}$
such that $J_{2}\cap J_{1}=\emptyset$ and
$(\widetilde{S}^{(1)})^{-1}(B)=J_{1}\cup J_{2}$. We define the
$\mathcal{C}^{1}$ diffeomorphism $\sigma_{1,2}:J_{1}\longrightarrow J_{2}$,
$u\mapsto(\widetilde{S}^{(1)})^{-1}(\widetilde{S}^{(1)}(u))$, such that
$\sigma_{1,1}=Id|_{I_{1}}$ and $\sigma_{1,2}\circ\sigma_{2,1}=Id|_{I_{1}}$.
Note that we can explicitly calculate $\sigma_{1,2}(u)$, indeed, for $u\in
J_{1}$, we have $\sigma_{1,2}(u)=1-u$. The reason of choosing $J_{1}$ in one
of these four intervals is to have the decomposition of
$(\widetilde{S}^{(1)})^{-1}(B)$ in exactly 2 disjoint open sets on which
$\widetilde{S}^{(1)}$ is one to one.
For any bounded and measurable function $\omega$ on $(-R^{\star},R^{\star})$,
we have

$\displaystyle
E\left[\exp\left(iz_{0}^{T}\widetilde{X}^{(2)}\right)\omega(\widetilde{X}^{(1)})1_{\widetilde{X}^{(1)}\in
B}\right]$ $\displaystyle=$
$\displaystyle\int_{J_{1}\times(0,1)^{d-2}}\omega(R^{\star}\cos(2\pi
u^{(1)}))\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(u))f^{\star}(u)du$
$\displaystyle+\int_{J_{2}\times(0,1)^{d-2}}\omega(R^{\star}\cos(2\pi
u^{(1)}))\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(u))f^{\star}(u)du.$

Define the following change of variables for the second integral,
$u=\overline{\sigma}(v)=(\sigma_{1,2}\times Id_{{(0,1)}^{d-2}})(v)$, which has
Jacobien equal to 1. Then

$\displaystyle
E\left[\exp\left(iz_{0}^{T}\widetilde{X}^{(2)}\right)\omega(\widetilde{X}^{(1)})1_{\widetilde{X}^{(1)}\in
B}\right]$
$\displaystyle=\int_{J_{1}\times(0,1)^{d-2}}\omega(R^{\star}\cos(2\pi
u^{(1)}))\\{\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(u))f^{\star}(u)+\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(\overline{\sigma}(u)))f^{\star}(\overline{\sigma}(u))\\}du$
$\displaystyle=\int_{J_{1}\times(0,1)^{d-2}}\omega(R^{\star}\cos(2\pi
u^{(1)}))\frac{\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(u))f^{\star}(u)+\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(\overline{\sigma}(u)))f^{\star}(\overline{\sigma}(u))}{f^{\star}(u)+f^{\star}(\overline{\sigma}(u))}f^{\star}(u)du$
$\displaystyle+\int_{J_{1}\times(0,1)^{d-2}}\omega(R^{\star}\cos(2\pi
u^{(1)}))\frac{\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(u))f^{\star}(u)+\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(\overline{\sigma}(u)))f^{\star}(\overline{\sigma}(u))}{f^{\star}(u)+f^{\star}(\overline{\sigma}(u))}f^{\star}(\overline{\sigma}(u))du.$

We now define $\tau_{1}:B\longrightarrow J_{1}$ such that for all $u\in
J_{1}$, $\tau_{1}(\widetilde{S}^{(1)}(u))=u$, and $\tau_{2}:B\longrightarrow
J_{2}$ such that for all $u\in J_{2}$, $\tau_{2}(\widetilde{S}^{(1)}(u))=u$.
First when $d=2$, we get

$\displaystyle
E\left[\exp\left(iz_{0}^{T}\widetilde{X}^{(2)}\right)|\widetilde{X}^{(1)}\right]1_{\widetilde{X}^{(1)}\in
B}$
$\displaystyle=\frac{\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}({\tau}_{1}(\frac{\widetilde{X}^{(1)}}{R^{\star}})))f^{\star}({\tau}_{1}(\frac{\widetilde{X}^{(1)}}{R^{\star}}))+\exp(-iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}({\tau}_{2}(\frac{\widetilde{X}^{(1)}}{R^{\star}})))f^{\star}({\tau}_{2}(\frac{\widetilde{X}^{(1)}}{R^{\star}}))}{f^{\star}({\tau}_{1}(\frac{\widetilde{X}^{(1)}}{R^{\star}}))+f^{\star}({\tau}_{2}(\frac{\widetilde{X}^{(1)}}{R^{\star}}))}1_{\widetilde{X}^{(1)}\in
B}$

and
$E\left[\exp\left(iz_{0}^{T}\widetilde{X}^{(2)}\right)|\widetilde{X}^{(1)}\right]$
can not be nul $\mathbb{P}_{\widetilde{X}^{(1)}}$-a.s. since it would require
that for almost all $(u,v)\in J_{1}\times(0,1)^{d-2}$,

$\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(u,v))f^{\star}(u,v)+\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(1-u,v))f^{\star}(1-u,v)=0.$

Then when $d>2$, we can choose $J_{1}=(0,\gamma]$

$\displaystyle
E\left[\exp\left(iz_{0}^{T}\widetilde{X}^{(2)}\right)|\widetilde{X}^{(1)}\right]1_{\widetilde{X}^{(1)}\in
B}$
$\displaystyle=\left(\int_{(0,1)^{d-2}}\frac{\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}({\tau}_{1}(\frac{\widetilde{X}^{(1)}}{R^{\star}}),v))f^{\star}({\tau}_{1}(\frac{\widetilde{X}^{(1)}}{R^{\star}}),u)+\exp(-iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}({\tau}_{2}(\frac{\widetilde{X}^{(1)}}{R^{\star}}),v))f^{\star}({\tau}_{2}(\frac{\widetilde{X}^{(1)}}{R^{\star}}),u)}{f^{\star}({\tau}_{1}(\frac{\widetilde{X}^{(1)}}{R^{\star}}),u)+f^{\star}({\tau}_{2}(\frac{\widetilde{X}^{(1)}}{R^{\star}}),u)}du\right)1_{\widetilde{X}^{(1)}\in
B}$

and
$E\left[\exp\left(iz_{0}^{T}\widetilde{X}^{(2)}\right)|\widetilde{X}^{(1)}\right]1_{\widetilde{X}^{(1)}\in
B}$ is nul $\mathbb{P}_{\widetilde{X}^{(1)}}$-a.s. if and only if for all
$u\in J_{1}$,

$\int_{(0,1)^{d-2}}\frac{\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(u,v))f^{\star}(u,v)+\exp(iR^{\star}z_{0}^{\intercal}\widetilde{S}^{(2)}(1-u,v))f^{\star}(1-u,v)}{f^{\star}(u,v)+f^{\star}(1-u,v)}dv=0.$

In particular, this implies that for all $u\in J_{1}$,

$\int_{(0,1)^{d-2}}\cos({\text{Re}}(z_{0})^{\intercal}\widetilde{S}^{(2)}(u,v))\frac{[f^{\star}(u,v)\exp(-R^{\star}{\text{Im}}(z_{0})^{\intercal}\widetilde{S}^{(2)}(u,v))+f^{\star}(1-u,v)\exp(R^{\star}{\text{Im}}(z_{0})^{\intercal}\widetilde{S}^{(2)}(u,v))]}{f^{\star}(u,v)+f^{\star}(1-u,v)}dv=0.$

But for small enough $u\in J_{1}$,
$\cos({\text{Re}}(z_{0})^{\intercal}\widetilde{S}^{(2)}(u,v))$ stays positive
for all $v\in(0,1)^{d-2}$ which gives a contradiction.

###  7.3 Proof of Proposition 2

For $f,f^{\star}\in\mathcal{F}$ and
$R,R^{\star}\in[R_{\text{min}},R_{\text{max}}]$, denote $\theta=(f,R)$,
$\theta^{\star}=(f^{\star},R^{\star})$, and define the distance $d$ by
$d(\theta,\theta^{\star})=\left(\int_{(0,1)^{d-1}}(f(u)-f^{\star}(u))^{2}du\right)^{1/2}+|R-R^{\star}|$.
First, using Lemma A.1 in [7] we get

$\sup_{\theta\in\mathcal{F}\times[R_{\text{min}},R_{\text{max}}]}\left|M_{n}(\theta)-M(\theta)\right|=o_{\mathbb{P}_{C^{\star},R^{\star},f^{\star},\mathbb{Q}^{\star}}}(1).$
(24)

Then, using the continuity of $M$ with respect to the distance $d$ and the
compacity of $\mathcal{F}\times[R_{\text{min}},R_{\text{max}}]$, using Theorem
2 we get that for any $\delta>0$,

$\inf_{\theta\in\mathcal{F}\times[R_{\text{min}},R_{\text{max}}],\
d(\theta,\theta^{\star})>\delta}M(\theta)>M(\theta^{\star})=0.$ (25)

Consistency of $\widehat{f}$ and $\widehat{R}$ follows from (24), (25) and
Theorem 5.7 in [15]. Consistency of $\widehat{C}$ is then a consequence of the
continuity theorem and the law of large numbers.

###  7.4 Proof of proposition 3

The functions $\Psi_{f,R}$ on $\mathbb{R}^{2}$ can be written as functions
$\Phi_{f,R}$ on $[0,+\infty[\times[0,1)$ using polar representation. For any
$r\geq 0$ and $\theta\in[0,1)$, define

$\Phi_{f,R}(r,\theta)=\Psi_{f,R}(r\cos(2\pi\theta),r\sin(2\pi\theta)).$

For all $r\geq 0$, let $(\lambda_{p}(r))_{p\in\mathbb{Z}}$ be the sequence of
Fourier coefficients of $\Phi_{f,R}(r,\cdot)$,

$\lambda_{p}(r)=\int_{0}^{1}\Phi_{f,R}(r,u)e^{2i\pi pu}du,\ \ p\in\mathbb{Z}.$

Using (III) in Section 8, we get that for all $r\geq 0$,

$\displaystyle\Phi_{f,R}(r,\theta)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}f(u)\text{exp}(irRcos(2\pi u-2\pi\theta))du$
$\displaystyle=$
$\displaystyle\sum_{p\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}i^{p}J_{p}(rR)f_{k}(\int_{0}^{1}e^{-2i\pi
u(k-p)}du)e^{-2i\pi p\theta}$ $\displaystyle=$
$\displaystyle\sum_{p\in\mathbb{Z}}i^{p}f_{p}J_{p}(rR)e^{-2i\pi p\theta},$

so that for all $p\in\mathbb{Z}$,

$\lambda_{p}(r)=i^{p}J_{p}(rR)f_{p},$

and also

$\lambda_{p}^{\star}(r)=i^{p}J_{p}(rR^{\star})f_{p}^{\star},\ \ \
\widehat{\lambda}_{p}(r)=i^{p}J_{p}(r\widehat{R})\widehat{f}_{p},$

where $(f_{p}^{\star})_{p\in\mathbb{Z}}$ (resp.
$(\widehat{f}_{p})_{p\in\mathbb{Z}}$) are the Fourier coefficients of
$f^{\star}$(resp. $\hat{f}$) and $(\widehat{\lambda}_{p}(r))_{p\in\mathbb{Z}}$
are the Fourier coefficients of $\Phi_{\widehat{f},\widehat{R}}(r,.)$. We
have, using Parseval’s identity,

$\displaystyle\int_{0}^{1}|\Phi_{f^{\star},R^{\star}}(r,\theta)-\Phi_{\widehat{f},\widehat{R}}(r,\theta)|^{2}d\theta$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}}|\lambda_{k}^{\star}(r)-\widehat{\lambda}_{k}(r)|^{2}$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}}|f_{k}^{\star}J_{k}(rR^{\star})-\widehat{f}_{k}J_{k}(r\widehat{R})|^{2}$
$\displaystyle\geq$ $\displaystyle\sum_{|k|\leq
N}|f_{k}^{\star}J_{k}(rR^{\star})-\widehat{f}_{k}J_{k}(r\widehat{R})|^{2}.$

We use the fact that $|a-b|^{2}\geq\frac{|a|^{2}}{2}-|b|^{2}$ for all
$a,b\in\mathbb{C}$, to get

$\int_{0}^{1}|\Phi_{f^{\star},R^{\star}}(r,\theta)-\Phi_{\widehat{f},\widehat{R}}(r,\theta)|^{2}d\theta\geq\sum_{|k|\leq
N}\frac{|f_{k}^{\star}-\widehat{f}_{k}|^{2}}{2}J_{k}(r\widehat{R})^{2}-\sum_{|k|\leq
N}|f^{\star}_{k}|^{2}|J_{k}(rR^{\star})-J_{k}(r\widehat{R})|^{2},$

so that

$\sum_{|k|\leq
N}|f_{k}^{\star}-\widehat{f}_{k}|^{2}J_{k}(r\widehat{R})^{2}\leq
2\int_{0}^{1}|\Phi_{f^{\star},R^{\star}}(r,\theta)-\Phi_{\widehat{f},\widehat{R}}(r,\theta)|^{2}d\theta+\sum_{|k|\leq
N}|f_{k}^{\star}|^{2}|J_{k}(rR^{\star})-J_{k}(r\widehat{R})|^{2}.$

Then, for all $\nu\in(0,\nu_{\text{est}}]$ such that $c_{\nu}^{\star}>0$, we
integrate from $0$ to $\nu$ and we use (IV) in Section 8 to obtain

$\sum_{|k|\leq
N}|f_{k}^{\star}-\widehat{f}_{k}|^{2}\int_{0}^{\nu}rJ_{k}(r\widehat{R})^{2}dr\leq
2||\Psi_{f^{\star},R^{\star}}-\Psi_{\widehat{f},\widehat{R}}||^{2}_{\mathbb{L}_{2}(\mathbb{D}_{2}(0,\nu))}+2\sum_{|k|\leq
N}|f_{k}^{\star}|^{2}|R^{\star}-\widehat{R}|^{2}\int_{0}^{\nu}r^{2}dr.$

Using Proposition 1, Theorem 3 and the fact that $\sum_{|k|\leq
N}|f_{k}^{\star}|^{2}\leq\int_{0}^{1}(f^{\star}(u))^{2}du$ is uniformly upper
bounded in the compact set $\cal F$, we have that there exists a constant
$c>0$ depending on $\delta$, $\nu$, $c_{\nu}^{\star}$, $d$, $R^{\star}$,
$R_{\text{min}}$, $R_{\text{max}}$, and $E(\|Y\|^{2})$ such that for all
$x\geq 1$ and for $c_{1}$ and $c_{2}$ coming from Proposition 1, for all
$n\geq(1\vee xc_{1})/c_{2}$, with probability at least $1-e^{-x}$,

$\sum_{|k|\leq
N}|f_{k}-\widehat{f}_{k}|^{2}\int_{0}^{\nu}rJ_{k}(r\widehat{R})^{2}dr\leq
c\left(\frac{x}{n^{1-\delta}}\vee\frac{x^{2}}{n^{2-2\delta}}\right).$

Using lemma 4, we finally have that with probability at least $1-e^{-x}$,

$\sum_{|k|\leq N}|f_{k}^{\star}-\widehat{f}_{k}|^{2}\leq
c\frac{32}{9\nu^{2}}(\nu\widehat{R})^{-2N}(N+1)2^{2N}(N!)^{2}\left(\frac{x}{n^{1-\delta}}\vee\frac{x^{2}}{n^{2-2\delta}}\right).$

We finally use $\widehat{R}\in[R_{\text{min}},R_{\text{max}}]$ to end the
proof.

###  7.5 Proof of lemma 2

The proof of (1) follows from the same arguments as in the proof of
Proposition 2.
For all $(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}^{d-1}$, define

$m_{n,R}(t_{1},t_{2})=\Psi_{f^{\star},R}(t_{1},t_{2})\tilde{\psi}_{n}(t_{1},0)\tilde{\psi}_{n}(0,t_{2})-\tilde{\psi}_{n}(t_{1},t_{2})\Psi_{f^{\star},R}(t_{1},0)\Psi_{f^{\star},R}(0,t_{2})$

and

$m_{R}(t_{1},t_{2})=\Psi_{f^{\star},R}(t_{1},t_{2})\Psi_{f^{\star},R^{\star}}(t_{1},0)\Psi_{f^{\star},R^{\star}}(0,t_{2})-\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\Psi_{f^{\star},R}(t_{1},0)\Psi_{f^{\star},R}(0,t_{2}),$

such that $M_{n}(R)=\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}|m_{n,R}(t_{1},t_{2})|^{2}dt_{1}dt_{2}$ and
$M(R)=\int_{B_{\nu}\times
B_{\nu}^{d-1}}|m_{R}(t_{1},t_{2})|^{2}|\Phi_{\epsilon}(t_{1},t_{2})|^{2}dt_{1}dt_{2}$.

Let us prove (2). Differentiation of $M_{n}$ gives

$M^{\prime}_{n}(R)=\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}\left(\frac{d}{dR}\\{m_{n,R}(t_{1},t_{2})\\}\overline{m_{n,R}(t_{1},t_{2})}+\frac{d}{dR}\\{\overline{m_{n,R}(t_{1},t_{2})}\\}m_{n,R}(t_{1},t_{2})\right)dt_{1}dt_{2},$

where $\overline{z}$ denotes the complex conjugate of $z$. Since
$\overline{m_{n,R}(t_{1},t_{2})}=m_{n,R}(-t_{1},-t_{2})$ we get

$M^{\prime}_{n}(R)=\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}\left(\frac{d}{dR}\\{m_{n,R}(t_{1},t_{2})\\}m_{n,R}(-t_{1},-t_{2})+\frac{d}{dR}\\{m_{n,R}(-t_{1},-t_{2})\\}m_{n,R}(t_{1},t_{2})\right)dt_{1}dt_{2}.$

Let $\textbf{Z}_{n}$ be the random process defined for
$(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}^{d-1}$ by

$\textbf{Z}_{n}(t_{1},t_{2})=\sqrt{n}\left(\tilde{\psi}_{n}(t_{1},t_{2})-\Psi_{f^{\star},R^{\star},}(t_{1},t_{2})\Phi_{\epsilon}(t_{1},t_{2})\right).$
(26)

The random process $\textbf{Z}_{n}$ converges weakly to a Gaussian process
$(\textbf{Z}(t_{1},t_{2}))_{(t_{1},t_{2})}$ in the set of complex continuous
functions endowed with the uniform norm.
Using (26),
$\tilde{\psi}_{n}(t_{1},t_{2})=\frac{1}{\sqrt{n}}\textbf{Z}_{n}(t_{1},t_{2})+\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\Phi_{\epsilon}(t_{1},t_{2})$,
so that

$\displaystyle\sqrt{n}M^{\prime}_{n}(R^{\star})$ $\displaystyle=$
$\displaystyle\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}C(t_{1},t_{2})\left\\{\Psi_{f^{\star},R^{\star}}(-t_{1},-t_{2})\left[\textbf{Z}_{n}(-t_{1},0)\Psi_{f^{\star},R^{\star}}(0,-t_{2})+\textbf{Z}_{n}(0,-t_{2})\Psi_{f^{\star},R^{\star}}(-t_{1},0)\right]\right.$
$\displaystyle\left.-\textbf{Z}_{n}(-t_{1},-t_{2})\Psi_{f^{\star},R^{\star}}(-t_{1},0)\Psi_{f^{\star},R^{\star}}(0,-t_{2})\right\\}dt_{1}dt_{2}$
$\displaystyle+\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}C(-t_{1},-t_{2})\left\\{\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\left[\textbf{Z}_{n}(t_{1},0)\Psi_{f^{\star},R^{\star}}(0,t_{2})+\textbf{Z}_{n}(0,t_{2})\Psi_{f^{\star},R^{\star}}(t_{1},0)\right]\right.$
$\displaystyle\left.-\textbf{Z}_{n}(t_{1},t_{2})\Psi_{f^{\star},R^{\star}}(t_{1},0)\Psi_{f^{\star},R^{\star}}(0,t_{2})\right\\}dt_{1}dt_{2}+O_{\mathbb{P}}(\frac{1}{\sqrt{n}})$

where all $O_{\mathbb{P}}$ (and later $o_{\mathbb{P}}$) are in
$\mathbb{P}_{C^{\star},R^{\star},f^{\star},\mathbb{Q}^{\star}}$ probability.
Now, the empirical process converges uniformly in distribution to a Gaussian
process over the set of functions
$\\{Id,\exp(it^{T}\cdot),|t|\leq\nu_{\text{est}}\\}$, so that
$\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^{n}Y_{i}-E(Y_{1}),M^{\prime}_{n}(R^{\star})\right)$
converges in distribution to $\mathcal{N}(0,V)$ as $n$ goes to infinity for
$V$ the covariance matrix of the random vector.

Let us now prove (3). Twice differentiation of $M$ gives

$\displaystyle M^{\prime\prime}(R)$ $\displaystyle=$
$\displaystyle\int_{B_{\nu}\times
B_{\nu}^{d-1}}|\Phi_{\epsilon}(t_{1},t_{2})|^{2}\left(\frac{d^{2}}{dR^{2}}m_{R}(t_{1},t_{2})m_{R}(-t_{1},-t_{2})+2\frac{d}{dR}m_{R}(t_{1},t_{2})\right.$
$\displaystyle\left.\frac{d}{dR}m_{R}(-t_{1},-t_{2})+\frac{d^{2}}{dR^{2}}m_{R}(-t_{1},-t_{2})m_{R}(t_{1},t_{2})\right)dt_{1}dt_{2}.$

But $m_{R^{\star}}(t_{1},t_{2})=0$ for all $(t_{1},t_{2})$, so that

$M^{\prime\prime}(R^{\star})=2\int_{B_{\nu}\times
B_{\nu}^{d-1}}\left|\frac{d}{dR}m_{R^{\star}}(t_{1},t_{2})\right|^{2}|\Phi_{\epsilon}(t_{1},t_{2})|^{2}dt_{1}dt_{2}.$

We shall prove $M^{\prime\prime}(R^{\star})\neq 0$ by contradiction.
If it is not the case, we have, for almost all $(t_{1},t_{2})\in B_{\nu}\times
B_{\nu}^{d-1}$,
$\frac{d}{dR}m_{R^{\star}}(t_{1},t_{2})\Phi_{\epsilon}(t_{1},t_{2})=0$. Now,
there exists $r_{\epsilon}\in(0,\nu)$ such that for all $(t_{1},t_{2})\in
B_{r_{\epsilon}}\times B_{r_{\epsilon}}^{d-1}$,
$\Phi_{\epsilon}(t_{1},t_{2})\neq 0$. Since $\frac{d}{dR}m_{R^{\star}}$ is a
continuous function on $\mathbb{C}^{d}$ we get
$\frac{d}{dR}m_{R^{\star}}(t_{1},t_{2})=0$ for all $(t_{1},t_{2})\in
B_{r_{\epsilon}}\times B_{r_{\epsilon}}^{d-1}$, that is

$\displaystyle\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\Psi_{f^{\star},R^{\star}}(t_{1},0)\Psi_{f^{\star},R^{\star}}(0,t_{2})$
$\displaystyle=$
$\displaystyle\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(t_{1},0)\Psi_{f^{\star},R^{\star}}(0,t_{2})$
(27) $\displaystyle+$
$\displaystyle\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\Psi_{f^{\star},R^{\star}}(t_{1},0)\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(0,t_{2}),$

with

$\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})=\int_{(0,1)^{d-1}}f^{\star}(u)\exp(iR^{\star}t^{\intercal}S(u))du$

and

$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})=i\int_{(0,1)^{d-1}}[t^{\intercal}S(u)]f^{\star}(u)e^{iR^{\star}t^{\intercal}S(u)}du.$

But $\Psi_{f^{\star},R^{\star}}$ and $\frac{d}{dR}\Psi_{f^{\star},R^{\star}}$
are multivariate analytic functions, so that, using Lemma C.1 in [8], we have
that (27) holds for all $(t_{1},t_{2})\in\mathbb{C}\times\mathbb{C}^{d-1}$. We
shall now investigate the set of zeros of the functions
$\Psi_{f^{\star},R^{\star}}(\cdot,0)$ and
$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(\cdot,0)$. Let $t_{1}\in\mathbb{C}$ be
such that $\Psi_{f^{\star},R^{\star}}(t_{1},0)=0$. Then by Lemma 1 it is
possible to choose $t_{2}\in\mathbb{C}^{d-1}$ such that
$\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\neq 0$, and also such that
$\Psi_{f^{\star},R^{\star}}(0,t_{2})\neq 0$ since
$\Psi_{f^{\star},R^{\star}}(0,\cdot)$ is a multivariate analytic function
having only isolated zeros. Equation (27) then leads to
$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(t_{1},0)=0$ so that the set of zeros
of the function $\Psi_{f^{\star},R^{\star}}(\cdot,0)$ is a subset of that of
the function $\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(\cdot,0)$. Then, using
Hadamard’s factorization theorem (see [14] Chapter 4, Theorem 4.1), and the
fact that $\Psi_{f^{\star},R^{\star}}(\cdot,0)$ and
$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(\cdot,0)$ have exponential growth
order $1$, we get that there exists an entire function $G$ of exponential
growth order $1$ such that for any $t_{1}\in\mathbb{C}$,

$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(t_{1},0)=\Psi_{f^{\star},R^{\star}}(t_{1},0)G(t_{1}).$

Plugging into (27) we get that for all
$(t_{1},t_{2})\in\mathbb{C}\times\mathbb{C}^{d-1}$,

$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\Psi_{f^{\star},R^{\star}}(0,t_{2})=\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})G(t_{1})\Psi_{f^{\star},R^{\star}}(0,t_{2})+\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(0,t_{2}).$

The same arguments applied for each coordinate of $t_{2}$ gives that there
exists a multivariate anlytic function $H$ such that for any
$t_{2}\in\mathbb{C}^{d-1}$,

$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(0,t_{2})=\Psi_{f^{\star},R^{\star}}(0,t_{2})H(t_{2}),$

so that for all $(t_{1},t_{2})\in\mathbb{C}\times\mathbb{C}^{d-1}$,

$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})=\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})\left(G(t_{1})+H(t_{2})\right).$
(28)

But for any $t\in\mathbb{C}^{d}$,

$\frac{d}{dR}\Psi_{f^{\star},R^{\star}}(t)=\frac{1}{R}\frac{d}{du}\Psi_{f^{\star},R^{\star}}(ut),\;u\in\mathbb{R}$

so that solving the derivative equation (28) we find that
$\Psi_{f^{\star},R^{\star}}(t_{1},t_{2})$ is a product of a function of
$t_{1}$ only by a function of $t_{2}$ only, meaning that $S^{(1)}(U)$ and
$S^{(2)}(U)$ are independent variables, which is not true and we get a
contradiction. We conclude that $M^{\prime\prime}(R^{\star})\neq 0$.

To end the proof of (3), for all $R\in[0,+\infty[$,

$\displaystyle M_{n}^{\prime\prime}(R)-M_{n}^{\prime\prime}(R^{\star})$
$\displaystyle=$ $\displaystyle\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}|\tilde{\psi}_{n}(t_{1},t_{2})|^{2}[a_{1}(t_{1},t_{2},R)-a_{1}(t_{1},t_{2},R^{\star})]dt_{1}dt_{2}$
$\displaystyle+\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}|\tilde{\psi}_{n}(t_{1},0)|^{2}|\tilde{\psi}_{n}(0,t_{2})|^{2}[a_{2}(t_{1},t_{2},R)-a_{2}(t_{1},t_{2},R^{\star})]dt_{1}dt_{2}$
$\displaystyle+\text{Re}\Bigg{\\{}\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}\tilde{\psi}_{n}(-t_{1},t_{2})\tilde{\psi}_{n}(t_{1},0)\tilde{\psi}_{n}(0,t_{2})[a_{3}(t_{1},t_{2},R)-a_{3}(t_{1},t_{2},R^{\star})]dt_{1},dt_{2}\Bigg{\\}},$

for functions $a_{1}$, $a_{2}$ and $a_{3}$ functions that are, for all
$(t_{1},t_{2})$, continuous in the variable $R$ and uniformly upper bounded
for bounded $R$. Since for all $(t_{1},t_{2})$,
$|\tilde{\psi}_{n}(t_{1},t_{2})|\leq 1$, we get that
$\left|M_{n}^{\prime\prime}(R)-M_{n}^{\prime\prime}(R^{\star})\right|$ is
upper bounded by

$\int_{B_{\nu_{\text{est}}}\times
B_{\nu_{\text{est}}}^{d-1}}\left(\left|a_{1}(t_{1},t_{2},R)-a_{1}(t_{1},t_{2},R^{\star})\right|+\left|a_{2}(t_{1},t_{2},R)-a_{2}(t_{1},t_{2},R^{\star})\right|+\left|a_{3}(t_{1},t_{2},R)-a_{3}(t_{1},t_{2},R^{\star})\right|\right)dt_{1},dt_{2}$

from which, applying the continuity theorem, we deduce that
$M_{n}^{\prime\prime}(R_{n})-M_{n}^{\prime\prime}(R^{\star})$ converges in
probability to $0$ whenever $R_{n}$ is a random variable converging in
probability to $R^{\star}$. Then, for any random variable
$R_{n}\in[R_{\text{min}},R_{\text{max}}]$ converging in probability to
$R^{\star}$, $M^{\prime\prime}_{n}(R_{n})$ converges in probability to
$M^{\prime\prime}(R^{\star})$.

###  7.6 Proof of Theorem 6

Using Taylor expansion of $M^{\prime}_{n}$ near $R^{\star}$, there exists
$R_{n}\in(\widetilde{R},R^{\star})$ such that

$M^{\prime}_{n}(\widetilde{R})=M^{\prime}_{n}(R^{\star})+(R_{n}-R^{\star})M^{\prime\prime}_{n}(R_{n}).$

Using $M^{\prime}_{n}(\widetilde{R})=0$ and Lemma 2 we get
$M^{\prime\prime}_{n}(R_{n})=M^{\prime\prime}(R^{\star})+o_{\mathbb{P}}(1)$,
so that

$\sqrt{n}(\widetilde{R}-R^{\star})=-\sqrt{n}\frac{M^{\prime}_{n}(R^{\star})}{M^{\prime\prime}(R^{\star})}(1+o_{\mathbb{P}}(1)).$

We deduce that

$\sqrt{n}\begin{pmatrix}\widetilde{R}-R^{\star}\\\
\widetilde{C}-C^{\star}\end{pmatrix}=\begin{pmatrix}0\\\
\sqrt{n}(\frac{1}{n}\sum_{l=1}^{n}Y_{l}-\mathbb{E}[Y_{1}])\end{pmatrix}-\begin{pmatrix}1\\\
-\mathbb{E}[S(U)]\end{pmatrix}\frac{\sqrt{n}M^{\prime}_{n}(R^{\star})}{M^{\prime\prime}(R^{\star})}(1+o_{\mathbb{P}}(1))$

and the conclusion follows.

##  8 Results on Bessel functions

Denote $J_{\alpha}$ the Bessel function of order $\alpha\in[0,+\infty[$.
We shall use the following results that can be found in [16].

  1. (I)

The Bessel function of order $\alpha\in[0,+\infty[$ can be represented as

$J_{\alpha}(z)=\sum_{m=0}^{\infty}(-1)^{m}\frac{(z/2)^{\alpha+2m}}{m!\Gamma(\alpha+m+1)},\
\ \ z\in\mathbb{C},$

where for all $z\in]0,+\infty[$,
$\Gamma(z)=\int_{-\infty}^{+\infty}t^{z-1}e^{-t}dt$.

  2. (II)

For $k\geq 0$ and $z\in\mathbb{C}$

$J_{-k}(z)=(-1)^{k}J_{k}(z).$

  3. (III)

For $z\in\mathbb{C}$ and $\theta\in\mathbb{R}$

$\text{exp}(izcos(\theta))=\sum_{k\in\mathbb{Z}}i^{k}J_{k}(z)e^{-ik\theta}.$

  4. (IV)

For $k\geq 0$ and $x,y>0$

$|J_{k}(x)-J_{k}(y)|\leq|x-y|.$

Indeed, since, $J_{k}\in\mathcal{C}^{1}(0,+\infty)$, for $k>0$,
$J_{k}^{\prime}(z)=\frac{1}{2}(J_{k-1}(z)-J_{k+1}(z))$,
$J_{0}^{\prime}(z)=-J_{1}(z)$ and $|J_{k}(x)|\leq 1$.

  5. (V)

For $\alpha\geq 1$ and $x>0$

$J_{\alpha+1}(x)=\frac{2\alpha}{x}J_{\alpha}(x)-J_{\alpha-1}(x).$

We prove lemmas giving useful lower bounds.

######  Lemma 3.

For all $0\leq x<1$, for all $\alpha\in[0,+\infty[$, we have

$J_{\alpha}(x)\geq\frac{x^{\alpha}}{2^{\alpha}\Gamma(\alpha+1)}(1-\frac{x^{2}}{4(\alpha+1)}).$

###### Proof.

Let $0\leq x<1$, for $\alpha\geq 0$, we have
$J_{\alpha}(x)=\sum_{m=0}^{\infty}(-1)^{m}\frac{(x/2)^{\alpha+2m}}{m!\Gamma(\alpha+m+1)}$,
so, if we expand the sum using that $\Gamma(x+1)=x\Gamma(x)$ for $x>0$ :

$J_{\alpha}(x)=\frac{x^{\alpha}}{2^{\alpha}\Gamma(\alpha+1)}(1-\frac{x^{2}}{4(\alpha+1)}+\sum_{m\geq
2}^{\infty}(-1)^{m}\frac{(x/2)^{2m}}{m!(\alpha+1)\cdots(\alpha+m)}).$

Since $0\leq x<1$, we have
$\sum_{m=2}^{\infty}(-1)^{m}\frac{(x/2)^{2m}}{m!(\alpha+1)\cdots(\alpha+m)}\geq
0$ thus ,

$J_{k}(x)\geq\frac{x^{\alpha}}{2^{\alpha}\Gamma(\alpha+1)}(1-\frac{x^{2}}{4(\alpha+1)}).$

∎

######  Lemma 4.

For all $R>0$ and $0<\nu<1/R$, for all $0\leq k\leq N$, we have :

$\int_{0}^{\nu}rJ_{k}(rR)^{2}dr\geq\frac{9\nu^{2}}{32}\frac{(\nu
R)^{2N}}{(N+1)2^{2N}(N!)^{2}}.$

###### Proof.

Let $R>0$ and $0\leq r\leq\nu<1/R$. For all $k\geq 0$, since $0\leq rR<1$, we
have from lemma 3,

$J_{k}(rR)\geq\frac{(rR)^{k}}{2^{k}k!}(1-\frac{(rR)^{2}}{4(k+1)}),$

and

$rJ_{k}(rR)^{2}\geq\frac{r(rR)^{2k}}{2^{2k}(k!)^{2}}(1-\frac{(rR^{\star})^{2}}{4(k+1)})^{2}.$

Then,

$\int_{0}^{\nu}rJ_{k}(rR)^{2}dr\geq\int_{0}^{\nu}\frac{r(rR)^{2k}}{2^{2k}(k!)^{2}}(1-\frac{(rR)^{2}}{4(k+1)})^{2}dr\geq\frac{R^{2N}}{(2N+2)2^{2N}(N!)^{2}}(1-\frac{(\nu
R)^{2}}{4})^{2}\nu^{2N+2}.$

To conclude the proof, we use that $\nu<1/R$, so that $(1-\frac{(\nu
R)^{2}}{4})^{2}\geq\frac{9}{16}$, which gives the result.

∎

### Acknowledgements

Jérémie Capitao-Miniconi would like to thank the IA Chair BisCottE
(ANR-19-CHIA-0021-01), Elisabeth Gassiat would like to thank Institut
Universitaire de France for supporting this project.

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