# D Proofs

## D.1 Proof of Proposition 2

For any $\nu > 0$ and $\zeta \in L^2([-\nu, \nu]^D)$ (resp. $L^\infty([-\nu, \nu]^D)$), write $\|\zeta\|_{2,\nu}$ (resp. $\|\zeta\|_{\infty,\nu}$) its $L^2$ (resp. $L^\infty$) norm.

Let $\rho_0 \in [1, 2)$. Let us start with some preliminary results.

From [CMG23], Section 7.1, for all $\nu > 0$, there exists $b > 0$, $\eta > 0$, $c_M > 0$ and $c_Z > 0$ such that, writing $\epsilon(u) = b/\log\log(1/u)$, the following properties hold for any $\rho' \in [1, \rho_0]$.

* For all $\phi \in \Upsilon_{\rho',S}$ and for all $\zeta \in L^2([-\nu, \nu]^D)$ such that $\phi + \zeta \in \Upsilon_{\rho',S}$ and $\|\zeta\|_{2,\nu} \le \eta$,

$$ M(\phi + \zeta; \nu|\phi) \ge c_{\nu}^{4} \|\zeta\|_{2,\nu}^{2+2\epsilon}(\|\zeta\|_{2,\nu}). \quad (13) $$

* For all $n \ge 1$, writing $Z_n(t, \phi) = \sqrt{n}(\tilde{\phi}_n(t) - \phi(t)\Phi_{\epsilon(1)}(t_1)\Phi_{\epsilon(2)}(t_2))$, one has for all $\phi \in \Upsilon_{\rho',S}$ and $h \in L^2([-\nu, \nu]^D)$ such that $\phi + \zeta \in \Upsilon_{\rho',S}$,

$$ |M_n(\phi+\zeta)-M(\phi+\zeta; \nu_{\text{est}}|\phi)-(M_n(\phi)-M(\phi; \nu_{\text{est}}|\phi)|) \le c_M \frac{\|Z_n(\cdot, \phi)\|_{\infty, \nu_{\text{est}}}}{\sqrt{n}} \|\zeta\|_{2, \nu_{\text{est}}}^{1-\epsilon}(\|\zeta\|_{2, \nu_{\text{est}}}) \quad (14) $$

* For all $s \in [1, n]$,

$$ \mathbb{P}(\|Z_n(\cdot, \Phi_X)\|_{\infty, \nu_{\text{est}}} \ge c_Z \sqrt{s}) \le e^{-s}. \quad (15) $$

Moreover, from Lemma H.3 of [GLCL22b], there exists a constant $c_T > 0$ such that for all $\rho' \in [1, \rho_0]$, $m \ge \rho'D$ and $\phi \in \Upsilon_{\rho',S}$,

$$ \|\phi - T_m \phi\|_{\infty, \nu_{\text{est}}} \le c_T (S \nu_{\text{est}})^m m^{-m/\rho'+D}. $$

Let $\rho' \in [1, \rho_0]$ and assume that $m \ge 2\rho' \frac{\log n}{\log \log n}$, then this equation becomes $\|\phi - T_m \phi\|_{\infty, \nu_{\text{est}}} = O(n^{-2+o_n(1)})$, where $o_n(1)$ denotes a sequence tending to 0 when $n$ tends to infinity. In particular, there exists $n_0$ such that for all $n \ge n_0$,

$$ \sup_{\rho' \in [1, \rho_0]} \sup_{\nu \in (0, \nu_{\text{est}}]} \sup_{m \ge 2\rho' \frac{\log n}{\log \log n}} \sup_{\phi \in \Upsilon_{\rho',S}} \|\phi - T_m \phi\|_{2, \nu} \le \frac{1}{n} \quad (16) $$

and

$$ \sup_{\rho' \in [1, \rho_0]} \sup_{m \ge 2\rho' \frac{\log n}{\log \log n}} \sup_{\phi \in \Upsilon_{\rho',S}} |M_n(\phi) - M_n(T_m \phi)| \le c \|\phi - T_m \phi\|_{\infty, \nu_{\text{est}}} \le \frac{1}{n} \quad (17) $$

for some $c > 0$ that depends only on $\nu_{\text{est}}$, $\rho_0$ and $S$, using that $\sup_{\phi \in \Upsilon_{\rho_0,S}} \|\phi\|_{\infty, \nu_{\text{est}}} < +\infty$. Finally, following the proof of equation (25) of Section A.3 of [GLCL22a], for any $\nu' \ge \nu$, $m \ge 1$ and $\phi \in C_m[X]$

$$ \|\phi\|_{2, \nu'} \le m^{D/2} (4 \frac{\nu'}{\nu})^{m+D/2} \|\phi\|_{2, \nu}. \quad (18) $$

Let us now prove the proposition. Let $\rho \in [1, \rho_0]$ such that $\Phi_X \in \Upsilon_{\rho,S} \cap \mathcal{H}$. By definition, for any $m \ge 1$ and $\rho' \in [\rho, \rho_0]$, $\hat{\Phi}_{n,m,\rho'}$ is such that $\hat{\Phi}_{n,m,\rho'} \in \Upsilon_{\rho',S} \cap \mathcal{H}$ and

$$ \begin{aligned} M_n(T_m \hat{\Phi}_{n,m,\rho'}) &\le \inf_{\phi \in \Upsilon_{\rho',S} \cap \mathcal{H}} M_n(T_m \phi) + \frac{1}{n} \\ &\le \inf_{\phi \in \Upsilon_{\rho,S} \cap \mathcal{H}} M_n(T_m \phi) + \frac{1}{n} \\ &\le M_n(T_m \Phi_X) + \frac{1}{n} \end{aligned} $$

and thus, by (17),

$$ \sup_{\rho' \in [\rho, \rho_0]} \sup_{m \ge 2\rho' \frac{\log n}{\log \log n}} M_n(\hat{\Phi}_{n,m,\rho'}) \le M_n(\Phi_X) + \frac{3}{n}. \quad (19) $$
Finally, using Parseval's equality and the fact that $\mathcal{F}^{-1}[\mathcal{F}[\psi_{A,h}]] = \psi_{A,h}$,

$$ \|\hat{g}_{n,\kappa} - \bar{g}\|_{\infty} \le \|\psi_{A,h}\|_2 \|T_{m_\kappa} \hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,1/h}, $$

and use that $\|\psi_{A,h}\|_2 = \frac{I(A)}{h^{D/2}} \|u_A * u_A\|_2$ to conclude the proof. $\square$

Let us now show that with proper choices for the parameters $m_\kappa$ and $h$, $\Gamma_{n,\kappa}$ tends to 0 with high probability.

Let $\kappa_0 \in (1/2, 1]$, $\nu \in (0, \nu_{est}]$, $c(\nu) > 0$, $E > 0$, $S > 0$ and $C > 0$. By Proposition 2, taking $s = \log n$ and $\delta$ small enough (to be chosen later), there exists $c$ and $n_0$ such that for all $n \ge n_0$,

$$ \sup_{\kappa \in [\kappa_0, 1]} \sup_{G \in \mathrm{St}_K(a, d, r_0) \cap \mathcal{L}(\kappa, S, \mathcal{H})} \mathbb{P}_{(G*\mathbb{Q})^{\otimes n}} \left( \|\hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,1/h} \le c(\log n)n^{-(1-\delta)/(1+2\alpha)} \right) \ge 1 - \frac{2}{n}. \quad (22) $$

Take these $c$ and $n_0$, and let $\kappa \in [\kappa_0, 1]$, $\mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)$ and $G \in \mathrm{St}_K(a, d, r_0) \cap \mathcal{L}(\kappa, S, \mathcal{H})$. Using inequalities analogous to (28)-(29) p.17 of [17], we get that for all integer $m$,

$$ \|T_m \hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,1/h}^2 \le 4U(h) + 4m^D(2 + 2\frac{1}{h\nu})^{2m+D} \left( 2V(\nu) + \|\hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,\nu}^2 \right), \quad (23) $$

where

$$ U(h) = ch^{-D-2m-2/\kappa} S^{2m} m^{-2\kappa m+2D} \exp(2\kappa(S/h)^{1/\kappa}) $$

$$ \text{and } V(\nu) = c(S\nu)^{2m+2/\kappa} m^{-2\kappa m+2D}. $$

Thus, applying Lemma 6 and using $h = c_h S m_\kappa^{-\kappa}$, there exists $C > 0$ (which does not depend on $\kappa$) such that on the event where (23) holds:

$$ \Gamma_{n,\kappa}^2 \le C(c_h S)^{-2D-2m_\kappa-2/\kappa} m_\kappa^{2D(\kappa+1)+2} S^{2m_\kappa} \exp(2\kappa c_h^{-1/\kappa} m_\kappa) \\ + C m_\kappa^{D(1+\kappa)} (2+2\frac{m_\kappa^\kappa}{c_h S\nu})^{2m_\kappa+D} \left( (S\nu)^{2m_\kappa+2/\kappa} m_\kappa^{-2\kappa m_\kappa+2D} + \|\hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,\nu}^2 \right). \quad (24) $$

The first term of the upper bound is upper bounded as follows.

$$ \begin{aligned} & (c_h S)^{-2D-2m_\kappa-2/\kappa} m_\kappa^{2D(\kappa+1)+2} S^{2m_\kappa} \exp(2\kappa c_h^{-1/\kappa} m_\kappa) \\ & = S^{-2D-2/\kappa} \exp \left\{ (-2D - 2m_\kappa - 2/\kappa) \log(c_h) + (2D(\kappa+1)+2) \log(m_\kappa) + 2\kappa c_h^{-1/\kappa} m_\kappa \right\} \\ & \le C \exp \left\{ (-2 \log(c_h) + 1)m_\kappa + 2(D(\kappa+1)+1) \log(m_\kappa) \right\} \end{aligned} \quad (25) $$

$$ \le C \exp \left\{ (-2 \log(c_h) + 3 + 2D(\kappa + 1))m_\kappa \right\}, \quad (26) $$

for another constant $C > 0$, where inequality (25) holds because $2\kappa c_h^{1/\kappa} > 1$ and inequality (26) holds because $\log(m_\kappa) \le m_\kappa$. The second term of the upper bound is upper bounded by

$$ \begin{aligned} & m_\kappa^{D(1+\kappa)} (2 + 2\frac{m_\kappa^\kappa}{c_h S\nu})^{2m_\kappa+D} \left( (S\nu)^{2m_\kappa+2/\kappa} m_\kappa^{-2\kappa m_\kappa+2D} + \|\hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,\nu}^2 \right) \\ & \le C m_\kappa^{D(1+2\kappa)} (2\kappa m_\kappa)^{2\kappa m_\kappa} (2\kappa)^{-2\kappa m_\kappa} (c_h S\nu)^{-2m_\kappa-D} \\ & \qquad \times \left( (S\nu)^{2m_\kappa+2/\kappa} m_\kappa^{-2\kappa m_\kappa+2D} + \|\hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,\nu}^2 \right) \\ & \le C \left( \exp \left\{ (-2 \log(c_h) + (3D+2\kappa))m_\kappa \right\} + (2\kappa m_\kappa)^{2\kappa m_\kappa} \exp \left\{ (-2 \log(c_h) + D(1+2\kappa))m_\kappa \right\} \|\hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,\nu}^2 \right) \end{aligned} $$
Therefore, by (14), for any $\nu \in (0, \nu_{est}]$, writing $\zeta_{m,\rho'} = \hat{\Phi}_{n,m,\rho'} - \Phi_X$,

$$ M(\hat{\Phi}_{n,m,\rho'}; \nu | \Phi_X) \le M(\hat{\Phi}_{n,m,\rho'}; \nu_{est} | \Phi_X) \\ \le c_M \frac{\|Z_n(\cdot, \Phi_X)\|_{\infty, \nu_{est}} \|\zeta_{m,\rho'}\|_{2, \nu_{est}}^{1-\epsilon}(\|\zeta_{m,\rho'}\|_{2, \nu_{est}})}{\sqrt{n}} + \frac{3}{n}. \quad (20) $$

Let us show that we may apply (13). Combining (19) with Lemma A.1 of [GLCL22a] shows that for any $\delta > 0$, there exist $c_\eta > 0$ and $n_0$ (which do not depend on $\rho$) such that for all $n \ge n_0$, with probability at least $1 - 4e^{-c_\eta n}$,

$$ \sup_{\rho' \in [\rho, \rho_0]} \sup_{m \ge 2\rho \frac{\log n}{\log \log n}} M(\hat{\Phi}_{n,m,\rho'}; \nu_{est} | \Phi_X) \le \delta. $$

In addition, since $\Upsilon_{\rho_0,S} \cap \mathcal{H}$ is compact in $L^2([-\nu_{est}, \nu_{est}]^D)$, $\phi \mapsto M(\phi; \nu_{est}|\Phi_X)$ is continuous on $L^2([-\nu_{est}, \nu_{est}]^D)$, and $M(\phi; \nu_{est}|\Phi_X) = 0$ implies $\phi = \Phi_X$ for all $\phi \in \mathcal{H} \cap \Upsilon_{\rho_0,S}$ by Theorem 1, there exists $\delta > 0$ such that

$$ \inf_{\phi \in \Upsilon_{\rho_0,S} \cap \mathcal{H}} \sup_{\text{s.t.}} \|\phi - \Phi_X\|_{2, \nu_{est}} \ge \eta \quad M(\phi; \nu_{est} | \Phi_X) > \delta. $$

Therefore, there exist $c_\eta > 0$ and $n_0$ (which do not depend on $\rho$) such that for all $n \ge n_0$, with probability at least $1 - 4e^{-c_\eta n}$,

$$ \sup_{\rho' \in [\rho, \rho_0]} \sup_{m \ge 2\rho \frac{\log n}{\log \log n}} \|\zeta_{m,\rho'}\|_{2,\nu_{est}} \le \eta, \quad (21) $$

which is what we need to apply (13).

Fix now $\nu \in (0, \nu_{est}]$, $c(\nu) > 0$ and $E > 0$ such that $\mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)$. In particular, $c_\nu \ge c(\nu) > 0$. Then, by (13) and (20),

$$ \|\zeta_{m,\rho'}\|_{2,\nu}^{2+2\epsilon}(\|\zeta_{m,\rho'}\|_{2,\nu}) \le \frac{2}{c(\nu)^4} \max \left( c_M \frac{\|Z_n(\cdot, \Phi_X)\|_{\infty,\nu_{est}}}{\sqrt{n}} \|\zeta_{m,\rho'}\|_{2,\nu_{est}}^{1-\epsilon}(\|\zeta_{m,\rho'}\|_{2,\nu_{est}}), \frac{3}{n} \right). \quad (22) $$

By (16) and (18), assuming $m \in [2\rho' \frac{\log n}{\log \log n}, C \frac{\log n}{\log \log n}]$ in the following series of inequalities (for some fixed $C > 2\rho'$),

$$ \begin{align} \|\zeta_{m,\rho'}\|_{2,\nu_{est}} &\le 2 \max(\|T_m \zeta_{m,\rho'}\|_{2,\nu_{est}}, \frac{3}{n}) \tag{16} \\ &\le 2 \max \left( \|T_m \zeta_{m,\rho'}\|_{2,\nu} m^{\frac{D}{2}} (4^{\frac{\nu_{est}}{\nu}})^{m+\frac{D}{2}}, \frac{3}{n} \right) \tag{18} \\ &\le 4 \max \left( \|\zeta_{m,\rho'}\|_{2,\nu} m^{\frac{D}{2}} (4^{\frac{\nu_{est}}{\nu}})^{m+\frac{D}{2}}, \frac{3m^{\frac{D}{2}} (4^{\frac{\nu_{est}}{\nu}})^{m+\frac{D}{2}}}{n} \right) \tag{16} \\ &\le n^{\epsilon(1/n)} \max \left( \|\zeta_{m,\rho'}\|_{2,\nu}, \frac{1}{n} \right), \tag{23} \end{align} $$

up to increasing the constant $b$ in the definition of $u \mapsto \epsilon(u)$, which can be done without loss of generality. Together with (22) and (15), one gets for all $s \in [1, c_\eta n]$ (assuming $c_\eta \le 1$ without loss of generality), with probability at least $1 - 4e^{-c_\eta n} - e^{-s} \ge 1 - 5e^{-s}$ (on the event where $\|Z(\cdot, \Phi_X)\|_{\infty, \nu_{est}} \le c_Z \sqrt{s}$ and (21) holds) that for all $\rho' \in [\rho, \rho_0]$ and $m \in [2\rho' \frac{\log n}{\log \log n}, C \frac{\log n}{\log \log n}]$,

$$ \|\zeta_{m,\rho'}\|_{2,\nu}^{2+2\epsilon}(\|\zeta_{m,\rho'}\|_{2,\nu}) \le c \max \left( \sqrt{\frac{s}{n}} n^{\epsilon(1/n)} \left( \|\zeta_{m,\rho'}\|_{2,\nu} \vee \frac{1}{n} \right)^{1-\epsilon}(\|\zeta_{m,\rho'}\|_{2,\nu_{est}}), \frac{1}{n} \right) $$

for some constant $c > 0$ that does not depend on $\rho$, $\rho'$ or $m$. Since $\epsilon$ is increasing, recalling
for another constant $C > 0$. Putting all together, we get that for yet another constant $C > 0$,

$$ \Gamma_{n,\kappa}^2 \le C \max \left( \exp \left\{ (-2 \log(c_h) + 3 + 2D(\kappa+1))m_\kappa \right\}, \exp \left\{ (-2 \log(c_h) + (3D+2\kappa))m_\kappa \right\}, \right. \\ \left. (2\kappa m_\kappa)^{2\kappa m_\kappa} \exp \left\{ (-2 \log(c_h) + D(1+2\kappa))m_\kappa \right\} \|\hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,\nu}^2 \right). $$

Choosing $c_h \ge \exp\{2D+2\}$ and $m_\kappa \le \frac{1}{6\kappa} \frac{\log n}{\log \log n}$, it follows that

$$ \Gamma_{n,\kappa}^2 \le C e^{-m_\kappa} \left[ 1 \vee n^{1/6} \|\hat{\Phi}_{n,\kappa} - \Phi_X\|_{2,\nu}^2 \right]. \quad (27) $$

By (22), taking $\delta < 1 - \frac{1+2\alpha}{6}$ (which is possible since $\alpha < 1$), we obtain that on the event of (22), $\Gamma_{n,\kappa}^2 \le C e^{-m_\kappa} \to 0$ (up to changing $C$), which concludes the proof.

## D.5 Proof of Theorem 4

Fix $\kappa_0 \in (1/2, 1]$, $S > 0$, $a > 0$, $d \le D$, $\nu \in (0, \nu_{est}]$, $c(\nu) > 0$, $E > 0$. Using the end of the proof of Theorem 3, there exist $n_0$ and $c'$ such that for all $\kappa \in [\kappa_0, 1]$, all $G \in St_K(a, d, r_0) \cap \mathcal{L}(\kappa, S, \mathcal{H})$ and all $\mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)$, with $(G * \mathbb{Q})^{\otimes n}$-probability at least $1 - \frac{2}{n}$,

$$ \frac{\log(n)^\kappa}{\log(\log(n))^{\kappa + \frac{A+1}{A}}} H_K(\mathcal{M}_G, \hat{\mathcal{M}}_\kappa) \le c'. \quad (28) $$

Let us now choose $c_\sigma = c'$ and consider the event where (20) holds. By the triangular inequality, for any $\kappa \in [\kappa_0, 1]$,

$$ H_K(\mathcal{M}_G, \hat{\mathcal{M}}_{\hat{\kappa}_n}) \le H_K(\mathcal{M}_G, \hat{\mathcal{M}}_\kappa) + H_K(\hat{\mathcal{M}}_\kappa, \mathcal{M}_{\hat{\kappa}_n}) \\ \le \sigma_n(\kappa) + H_K(\hat{\mathcal{M}}_\kappa, \mathcal{M}_{\hat{\kappa}_n}). $$

Now, using the definition of $B_n(\cdot)$, if $\kappa \le \hat{\kappa}_n$, then

$$ H_K(\hat{\mathcal{M}}_\kappa, \mathcal{M}_{\hat{\kappa}_n}) \le B_n(\hat{\kappa}_n) + \sigma_n(\kappa) $$

while if $\kappa \ge \hat{\kappa}_n$, then

$$ H_K(\hat{\mathcal{M}}_\kappa, \mathcal{M}_{\hat{\kappa}_n}) \le B_n(\kappa) + \sigma_n(\hat{\kappa}_n) $$

so that in all cases,

$$ H_K(\hat{\mathcal{M}}_\kappa, \mathcal{M}_{\hat{\kappa}_n}) \le B_n(\hat{\kappa}_n) + \sigma_n(\kappa) + B_n(\kappa) + \sigma_n(\hat{\kappa}_n) \\ \le 2B_n(\kappa) + 2\sigma_n(\kappa) $$

using the definition of $\hat{\kappa}_n$, and therefore

$$ H_K(\mathcal{M}_G, \hat{\mathcal{M}}_{\hat{\kappa}_n}) \le 2B_n(\kappa) + 3\sigma_n(\kappa). $$

By the triangular inequality and the definition of $B_n(\cdot)$,

$$ \begin{aligned} B_n(\kappa) &\le 0 \vee \sup_{\kappa' \in [\kappa_0, \kappa]} \{H_K(\hat{\mathcal{M}}_\kappa, \mathcal{M}_G) + H_K(\mathcal{M}_G, \hat{\mathcal{M}}_{\kappa'}) - \sigma_n(\kappa')\} \\ &\le H_K(\hat{\mathcal{M}}_\kappa, \mathcal{M}_G) + 0 \vee \sup_{\kappa' \in [\kappa_0, \kappa]} \{H_K(\mathcal{M}_G, \hat{\mathcal{M}}_{\kappa'}) - \sigma_n(\kappa')\} \\ &\le \sigma_n(\kappa). \end{aligned} $$

Thus, for all $\kappa \in [\kappa_0, 1]$, all $G \in St_K(a, d, r_0) \cap \mathcal{L}(\kappa, S, \mathcal{H})$ and all $\mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)$, with $(G * \mathbb{Q})^{\otimes n}$-probability at least $1 - \frac{2}{n}$,

$$ H_K(\mathcal{M}_G, \hat{\mathcal{M}}_{\hat{\kappa}_n}) \le 5\sigma_n(\kappa), $$

and using the fact that $H_K(\mathcal{M}_G, \hat{\mathcal{M}}_\kappa) \le \sup_{x,x' \in \mathcal{K}} d(x, x')$ on the event of probability at most $2/n$ where this does not hold, Theorem 4 follows.
that $\|\zeta_{m,\rho'}\|_{2,\nu_{est}} \le \eta$ on the event considered, by (23),

$$ \epsilon(\|\zeta_{m,\rho'}\|_{2,\nu_{est}}) \le \begin{cases} \max(\epsilon(\|\zeta_{m,\rho'}\|_{2,\nu}n^{\epsilon(1/n)}), \epsilon(n^{-1+\epsilon(1/n)})) & \text{if } \|\zeta_{m,\rho'}\|_{2,\nu} \le n^{-2\epsilon(1/n)}, \\ \epsilon(\eta) & \text{always,} \end{cases} \\ \le \begin{cases} 2\epsilon(1/n) & \text{if } \|\zeta_{m,\rho'}\|_{2,\nu} \le n^{-2\epsilon(1/n)}, \\ \epsilon(\eta) & \text{always,} \end{cases} \\ \le [\epsilon(\eta) \text{ or } 2\epsilon(1/n)] $$

for $n$ large enough (depending on $b$), up to decreasing $\eta$, where for compactness of notations, $[A \text{ or } B]$ means $\min(A, B)$ if $\|\zeta_{m,\rho'}\|_{2,\nu} \le n^{-2\epsilon(1/n)}$ and $A$ otherwise in the following. Gathering the two previous equations shows that either

$$ \|\zeta_{m,\rho'}\|_{2,\nu}^{1+3[\epsilon(\eta) \text{ or } 2\epsilon(1/n)]} \le c\sqrt{\frac{s}{n^{1-2\epsilon(1/n)}}} $$

or

$$ \|\zeta_{m,\rho'}\|_{2,\nu}^{2+2[\epsilon(\eta) \text{ or } 2\epsilon(1/n)]} \le \frac{c}{n}. $$

Therefore, assuming $3\epsilon(\eta) \le 1$ without loss of generality, $\|\zeta_{m,\rho'}\|_{2,\nu} \le n^{-\epsilon(1/n)}$ as soon as $s \le n^{1-10\epsilon(1/n)}/c^2$ and thus, up to changing the constant $c$, for $n$ large enough and for all $s \in [1, n^{1-10\epsilon(1/n)}/c^2]$, with probability at least $1 - 4e^{-c_{\eta}n} - e^{-s}$, for all $\rho' \in [\rho, \rho_0]$ and $m \in [2\rho' \frac{\log n}{\log \log n}, C \frac{\log n}{\log \log n}]$,

$$ \|\zeta_{m,\rho'}\|_{2,\nu}^2 \le c \left( \frac{s}{n^{1-2\epsilon(1/n)}} \right)^{1-6\epsilon(1/n)}. $$

Finally, note that $4e^{-c_{\eta}n}e^{n^{1-10\epsilon(1/n)}} \to 0$, so that the probability that the last equation holds is larger than $1 - 2e^{-s}$ for $n$ large enough, which concludes the proof for the version with $\tilde{\Phi}_{n,m,\rho'}$. The version for $T_m \tilde{\Phi}_{n,m,\rho'}$ follows from this and (16).

## D.2 Proof of Lemma 1

Let $y \in \mathcal{M}_G \cap \mathcal{K}$. By property (III) of $\psi_A$,

$$ \begin{align*} \bar{g}(y) &= \frac{1}{h^D} \int \psi_A \left( \frac{\|y-u\|}{h} \right) dG(u) \\ &\ge \frac{1}{h^D} \int_{\|u-y\|_2 \le c_A h} \psi_A \left( \frac{\|y-u\|}{h} \right) dG(u) \\ &\ge \frac{1}{h^D} d_A G(B(y, c_A h)) \\ &\ge \frac{1}{h^D} d_A a(c_A h)^d. \end{align*} $$

## D.3 Proof of Lemma 2

Recall the definition of $\bar{g}$: for all $y \in \mathbb{R}^D$,

$$ \bar{g}(y) = \frac{1}{h^D} \int \psi_A \left( \frac{\|y-u\|}{h} \right) dG(u). $$

Let $C_1 > 0$ and $\epsilon > 0$. By Property (V) of $\psi_A$ and (6), there exists $T > 0$ (depending on $A$ and $C_1$) such that for any $t \ge T$, $\psi_A(t) \le C_1 \exp(-\beta_A t^{A/(A+1)})$. Take $y \in \mathbb{R}^D$ such that $d(y, \mathcal{M}_G) > (\frac{\epsilon}{\beta_A})^{\frac{A+1}{A}} h \log(\frac{1}{h})^{\frac{A+1}{A}}$, then for all $u \in \mathcal{M}_G$, $\frac{\|y-u\|}{h} \ge (\beta_A^{-1} \log(\frac{1}{h^\epsilon}))^{\frac{A+1}{A}}$, therefore there exists $h_0 > 0$ depending only on $\epsilon$, $D$, $A$ and $T$ (thus $C_1$ such that $h \le h_0$ implies
## D.6 Proof of Theorem 5

As usual for the two-points method, the idea is to find two distributions having support as far as possible in $H_K$-distance, and a noise such that the joint distributions of the observations have total variation distance upper bounded by some $C < 1$. We shall consider the noise as in [17], with independent identically distributed coordinates having density $q$ defined as

$$q: x \in \mathbb{R} \mapsto c_q \frac{1 + \cos(cx)}{(\pi^2 - (cx)^2)^2}$$

for some $c > 0$, where $c_q$ is such that $q$ is a probability density, and with characteristic function

$$\mathcal{F}[q]: t \mapsto c_q \left[ \left(1 - \left|\frac{t}{c}\right|\right) \cos\left(\pi \frac{t}{c}\right) + \frac{1}{\pi} \sin\left(\pi \left|\frac{t}{c}\right|\right) \right] \mathbf{1}_{-c \le t \le c}.$$

Let us now define the two distributions to apply the two-points method. For any $\kappa \in (1/2, 1]$, we first choose a density function $f_\kappa$ according to the following Lemma.

**Lemma 7.** For any $\kappa \in (1/2, 1)$, $p \ge 1$, there exists a continuous density function $f_\kappa: \mathbb{R} \to \mathbb{R}$ in $L_p(\mathbb{R})$, positive everywhere, and positive constants $A, B$ such that for all $u \in \mathbb{R}$,

$$|\mathcal{F}[f_\kappa](u)| \le A \exp(-B|u|^{\frac{1}{\kappa}}) \quad \text{and} \quad |\mathcal{F}[f_\kappa]'(u)| \le A \exp(-B|u|^{\frac{1}{\kappa}}).$$

For any $\delta \in (0, 1)$, there exists a continuous compactly supported density function $f_1: [-1, 1] \to \mathbb{R}$ positive everywhere such that

$$|\mathcal{F}[f_1](u)| \le A \exp(-B|u|^{\delta}) \quad \text{and} \quad |\mathcal{F}[f_1]'(u)| \le A \exp(-B|u|^{\delta}).$$

The proof of Lemma 7 is detailed in Section D.7.

Then, inspired by [20], for all $\gamma \in (0, 1]$, define $\tilde{g}_\gamma: \mathbb{R} \to \mathbb{R}$ and $g_\gamma: \mathbb{R} \to \mathbb{R}^{D-d}$ for all $x \in \mathbb{R}$ as

$$\tilde{g}_{\gamma}(x) = \cos\left(\frac{x}{\gamma}\right) \quad \text{and} \quad g_{\gamma}(x) = (\tilde{g}_{\gamma}(x), 0, \dots, 0)^{\top}.$$

Let $M_0(\gamma) = \{(u, \gamma g_\gamma(u)): u \in \mathbb{R}\}$, $M_1(\gamma) = \{(u, -\gamma g_\gamma(u)): u \in \mathbb{R}\}$, and let $J \in \mathbb{R}^{D \times D}$ be the matrix

$$J = \begin{pmatrix} 1 & 0 & 0 & \dots & 0 \\ 1 & 1/2 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \end{pmatrix}.$$

For any $\kappa \in (1/2, 1]$, let $U(\kappa)$ be the random variable in $\mathbb{R}$ having density $f_\kappa$ defined in Lemma 7 and let $S_0(\kappa) = (U(\kappa), \gamma g_\gamma(U(\kappa)))$, $S_1(\kappa) = (U(\kappa), -\gamma g_\gamma(U(\kappa)))$. For $i \in \{0, 1\}$, we shall denote $T_i(\kappa)$ the distribution of $S_i(\kappa)$. Finally we define $X_i(\kappa) = JS_i(\kappa)$, $i = 0, 1$ and $G_i(\kappa)$ the distribution of $X_i(\kappa)$. When $\kappa < 1$, the supports of $S_i(\kappa)$ and $X_i(\kappa)$ do not depend on $\kappa$. They are illustrated in Figure 4. The transformation $J$ is such that the the support of $X_i$ is the graph of a bijective function, which makes it easy to satisfy (Adep) through for instance assumptions (H1), (H2) and Theorem 2 from Section 2.3.

**Lemma 8.** For any $i \in \{0, 1\}$ and $\kappa \in (1/2, 1]$, $X_i(\kappa)$ satisfies $A(\kappa)$.

The proof of Lemma 8 is detailed in Section D.8.

**Lemma 9.** There exists $a_0 > 0$ such that for $i \in \{0, 1\}$, for any $d \ge 1$, $r_0 < 1$ and $a \le a_0$, $G_i(\kappa) \in St_K(a, d, r_0)$.

The proof of Lemma 9 is detailed in Section D.9.

**Lemma 10.** For any $i \in \{0, 1\}$ and $\kappa \in (1/2, 1]$, $\Phi_{X_i(\kappa)}$ satisfy (Adep)
$\frac{\|y-u\|}{h} \ge T$ and thus

$$
\begin{align*}
\psi_A(\frac{\|y-u\|}{h}) &\le C_1 \exp \{-\beta_A(\frac{\|y-u\|}{h})^{A/(A+1)}\} \\
&\le C_1 \exp \{-\log(\frac{1}{h^\epsilon})\} \\
&= C_1 h^\epsilon,
\end{align*}
$$

and finally $\bar{g}(y) \le C_1(\frac{1}{h})^{D-\epsilon}$ since $G$ is a probability distribution. Lemma 2 follows by taking $\epsilon = D$.

D.4 Proof of Lemma 3

For $y \in \mathbb{R}^D$,

$$
\hat{g}_{n,\kappa}(y) - \bar{g}(y) = \left(\frac{1}{2\pi}\right)^D \int e^{-it^\top y} \mathcal{F}[\psi_A](th) (T_{m_\kappa} \hat{\Phi}_{n,1/\kappa}(t) - \Phi_X(t)) dt.
$$

Since $\mathcal{F}[\psi_A](th)$ is 0 for $\|t\|_2 > 1/h$,

$$
\begin{align}
\hat{g}_{n,\kappa}(y) - \bar{g}(y) &= \left(\frac{1}{2\pi}\right)^D \int e^{-it^\top y} \mathcal{F}[\psi_A](th) (T_{m_\kappa} \hat{\Phi}_{n,1/\kappa}(t) - \Phi_X(t)) 1_{\|t\|_2 \le 1/h} dt \nonumber \\
&= \mathcal{F}^{-1}[\mathcal{F}[\psi_h]\{(T_{m_\kappa} \hat{\Phi}_{n,1/\kappa} - \Phi_X)1_{\|t\|_2 \le 1/h}\}](y) \nonumber \\
&= \mathcal{F}^{-1}[\mathcal{F}[\psi_{A,h}]] * \mathcal{F}^{-1}[(T_{m_\kappa} \hat{\Phi}_{n,1/\kappa} - \Phi_X)1_{\|t\|_2 \le 1/h}](y). \tag{24}
\end{align}
$$

By Young's convolution inequality,

$$
\|\hat{g}_{n,\kappa} - \bar{g}\|_{\infty} \le \|\mathcal{F}^{-1}[\mathcal{F}[\psi_{A,h}]]\|_{2} \|\mathcal{F}^{-1}[(T_{m_\kappa} \hat{\Phi}_{n,1/\kappa} - \Phi_X)1_{\|t\|_2 \le 1/h}]\|_{2}.
$$

Finally, using Parseval's equality and the fact that $\mathcal{F}^{-1}[\mathcal{F}[\psi_{A,h}]] = \psi_{A,h}$,

$$
\| \hat{g}_{n, \kappa} - \bar{g} \|_{\infty} \le \| \psi_{A, h} \|_{2} \| T_{m_{\kappa}} \hat{\Phi}_{n, 1 / \kappa} - \Phi_{X} \|_{2, 1 / h},
$$

and use (7) to conclude the proof.

D.5 Proof of Theorem 2

Let $\kappa_0 \in (1/2, 1]$, $\nu \in (0, \nu_{est}]$, $c(\nu) > 0$, $E > 0$, $S > 0$ and $C > 0$. Let $\kappa \in [\kappa_0, 1]$,
$\mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)$ and $G \in \text{St}_{\mathcal{K}}(a, d, r_0) \cap \mathcal{L}(\kappa, S, \mathcal{H})$.

Using inequalities analogous to (28)-(29) p.17 of [GLCL22a], we get that for all $\kappa' \in [\kappa_0, \kappa]$ and all integer $m$,

$$
\| T_m \hat{\Phi}_{n,1/\kappa'} - \Phi_X \|_{2,1/h}^2 \le 4U(h) + 4m^D \left(2 + 2\frac{1}{h\nu}\right)^{2m+D} \left( 2V(\nu) + \|\hat{\Phi}_{n,1/\kappa'} - \Phi_X\|_{2,\nu}^2 \right), \quad (25)
$$

where

$$
\begin{align*}
U(h) &= c h^{-D-2m-2/\kappa'} S^{2m} m^{-2\kappa'm+2D} \exp(2\kappa'(S/h)^{1/\kappa'}) \\
\text{and } V(\nu) &= c(S\nu)^{2m+2/\kappa'} m^{-2\kappa'm+2D}.
\end{align*}
$$

Thus, applying Lemma 3 and using $h = c_h S m_{\kappa'}^{-\kappa}$, there exists $C > 0$ such that on the event
where (25) holds:

$$
\begin{align}
\Gamma_{n, \kappa'}^2 \le C(c_h S)^{-2D-2m_{\kappa'}-2/\kappa'} m_{\kappa'}^{2D(\kappa'+1)+2} S^{2m_{\kappa'}} \exp(2\kappa' c_h^{-1/\kappa'} m_{\kappa'}) \nonumber \\
+ C m_{\kappa'}^{D(1+\kappa')} (2+2\frac{m_{\kappa'}^\kappa}{c_h S \nu})^{2m_{\kappa'}+D} \left( (S\nu)^{2m_{\kappa'}+2/\kappa'} m_{\kappa'}^{-2\kappa' m_{\kappa'}+2D} + \|\hat{\Phi}_{n,1/\kappa'} - \Phi_X\|_{2,\nu}^2 \right). \tag{26}
\end{align}
$$
<figure><img src="image_5.png" /><figcaption>Figure 4: Left: support of $S_0$ and $S_1$ when $\gamma = 1$. Right: support of $X_0$ and $X_1$ when $\gamma = 1$.</figcaption></figure>

The proof of Lemma 10 is detailed in Section D.10.

We finally set $\mathcal{H}_{\kappa}^{*} = \{\Phi_{X_0(\kappa)}, \Phi_{X_1(\kappa)}\}$. As a finite set, it is closed in $L_2([-\nu, \nu]^D)$.

Let us comment on dimensionality. The distributions used here are distributions with support of dimension 1. This is not an issue since the $d$ in the definition of $\mathrm{St}_K(a, d, r_0)$ is an upper bound on the dimension of the support. We could also have used supports with dimension $d$ by adding to $X_i(\kappa)$ an independent uniform distribution on a Euclidean ball of a linear space of dimension $d$.

The support of $G_i(\kappa)$ is $JM_i(\gamma)$, and the following lemma follows easily from the fact that for $i \in \{0, 1\}$, $JM_i(\gamma) = \gamma JM_i(1)$.

**Lemma 11.** For any $\gamma > 0$,

$$H_K(JM_0(\gamma), JM_1(\gamma)) = \gamma H_K(JM_0(1), JM_1(1)). \quad (29)$$

To obtain the lower bound, the parameter $\gamma$ will be chosen as large as possible while making sure that the joint distributions of the observations have total variation distance smaller than some $C < 1$.

In the following, we will write $A, B, C$ (with upper case letters) positive constants that can change from line to line. As in [17] and [20], we use the upper bound:

$$\|(G_0(\kappa) * Q)^{\otimes n} - (G_1(\kappa) * Q)^{\otimes n}\|_{TV} \le 1 - (1 - \|(G_0(\kappa) * Q) - (G_1(\kappa) * Q)\|_{TV})^n,$$

where $\| \cdot - \cdot \|_{TV}$ denotes the total variation distance. Using Le Cam's two-points method, the minimax rate will be lower bounded by $H(JM_0(\gamma), JM_1(\gamma))$, that is $\gamma$, (see Lemma 11) provided that there exists a constant $C > 0$ such that $\|(G_0(\kappa) * Q)^{\otimes n} - (G_1(\kappa) * Q)^{\otimes n}\|_{TV} \le C < 1$, so that we only need to find $C > 0$ such that

$$\int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le \frac{C}{n}.$$

Since $Q$ has a density $q$ over $\mathbb{R}^D$, $G_0(\kappa) * Q$ and $G_1(\kappa) * Q$ also have a density over $\mathbb{R}^D$. We first prove that for $i \in \{0, 1\}$,

$$\int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^2 \left| \frac{d(G_i(\kappa) * Q)}{dx}(x) \right|^2 dx < +\infty. \quad (30)$$

Indeed,

$$\int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^2 \left| \frac{d(G_i(\kappa) * Q)}{dx}(x) \right|^2 dx \le \left\| \frac{d(G_i(\kappa) * Q)}{dx} \right\|_{\infty} \int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^2 d(G_i(\kappa) * Q)(x).$$
The first term of the upper bound is upper bounded as follows.

$$
\begin{align}
& (c_h S)^{-2D-2m_{\kappa'}-2/\kappa'} m_{\kappa'}^{2D(\kappa'+1)+2} S^{2m_{\kappa'}} \exp(2\kappa' c_h^{-1/\kappa'} m_{\kappa'}) \nonumber \\
&= S^{-2D-2/\kappa'} \exp \{(-2D - 2m_{\kappa'} - 2/\kappa') \log(c_h) + (2D(\kappa' + 1) + 2) \log(m_{\kappa'}) + 2\kappa' c_h^{-1/\kappa'} m_{\kappa'} \} \nonumber \\
&\le C \exp \{(-2\log(c_h) + 1)m_{\kappa'} + 2(D(\kappa' + 1) + 1) \log(m_{\kappa'}) \} \tag{27} \\
&\le C \exp \{(-2\log(c_h) + 3 + 2D(\kappa' + 1))m_{\kappa'} \}, \tag{28}
\end{align}
$$

for another constant $C > 0$, where inequality (27) holds because $2\kappa c_h^{1/\kappa} > 1$ and inequality (28) holds because $\log(m_\kappa) \le m_\kappa$. The second term of the upper bound is upper bounded by

$$
\begin{align*}
& m_{\kappa'}^{D(1+\kappa')} (2 + 2 \frac{m_{\kappa'}^{\kappa'}}{c_h S \nu})^{2m_{\kappa'} + D} \left( (S\nu)^{2m_{\kappa'} + 2/\kappa'} m_{\kappa'}^{-2\kappa' m_{\kappa'} + 2D} + \|\hat{\Phi}_{n,1/\kappa'} - \Phi_X\|_{2,\nu}^2 \right) \\
&\le C m_{\kappa'}^{D(1+2\kappa')} (2\kappa' m_{\kappa'})^{2\kappa' m_{\kappa'}} (2\kappa')^{-2\kappa' m_{\kappa'}} (c_h S \nu)^{-2m_{\kappa'} - D} \\
&\qquad \times \left( (S\nu)^{2m_{\kappa'} + 2/\kappa'} m_{\kappa'}^{-2\kappa' m_{\kappa'} + 2D} + \|\hat{\Phi}_{n,1/\kappa'} - \Phi_X\|_{2,\nu}^2 \right) \\
&\le C \left( \exp \left\{ (-2\log(c_h) + (3D + 2\kappa')) m_{\kappa'} \right\} \right. \\
&\qquad \left. + (2\kappa' m_{\kappa'})^{2\kappa' m_{\kappa'}} \exp \left\{ (-2\log(c_h) + D(1 + 2\kappa')) m_{\kappa'} \right\} \|\hat{\Phi}_{n,1/\kappa'} - \Phi_X\|_{2,\nu}^2 \right)
\end{align*}
$$

for another constant $C > 0$. Putting all together, we get that for yet another constant $C > 0$,

$$
\Gamma_{n,\kappa'}^2 \le C \max \left( \exp \left\{ (-2\log(c_h) + 3 + 2D(\kappa' + 1))m_{\kappa'} \right\}, \exp \left\{ (-2\log(c_h) + (3D + 2\kappa'))m_{\kappa'} \right\}, \right. \\
\left. (2\kappa'm_{\kappa'})^{2\kappa'm_{\kappa'}} \exp \left\{ (-2\log(c_h) + D(1 + 2\kappa'))m_{\kappa'} \right\} \|\hat{\Phi}_{n,1/\kappa'} - \Phi_X\|_{2,\nu}^2 \right).
$$

Choosing $c_h \ge \exp\{2D+2\}$ and $m_{\kappa'} = \frac{1}{4\kappa'} \frac{\log n}{\log \log n}$ for some $\gamma \in (0,1)$, it follows that

$$
\Gamma_{n,\kappa'}^2 \le C e^{-m_{\kappa'}} \left[ 1 \vee n^{1/2} \| \hat{\Phi}_{n,1/\kappa'} - \Phi_X \|_{2,\nu}^2 \right]. \quad (29)
$$

By Proposition 2, taking $s = \log n$ and $\delta, \delta''$ such that $(1 - \delta)(1 - \delta'') > 1/2$, we obtain that with probability at least $1 - 2/n$, for all $\kappa' \le \kappa$, $\Gamma_{n,\kappa'}^2 \le Ce^{-m_{\kappa'}} \to 0$. Note that we could also take $s = n^{1/2-\delta'''}$ for any $\delta'' > 0$ and still have $\Gamma_{n,\kappa'}^2 \le Ce^{-m_{\kappa'}}$ with probability at least $1 - 2e^{-s}$, up to changing the constant $C$, by picking $\delta$ and $\delta''$ small enough in Proposition 2. Now, by Lemma 1, for any $h \le (r_0/c_A) \wedge 1$,

$$
\begin{align*}
\inf_{y \in \mathcal{M}_G \cap \mathcal{K}} \hat{g}_{n, \kappa'}(y) &\ge \inf_{y \in \mathcal{M}_G \cap \mathcal{K}} \bar{g}(y) - \Gamma_{n, \kappa'} \\
&\ge c_A^d d_A a \left(\frac{1}{h}\right)^{D-d} - \Gamma_{n, \kappa'} \\
&\ge \frac{c_A^d d_A a}{2} \left(\frac{1}{h}\right)^{D-d}
\end{align*}
$$

as soon as $\Gamma_{n,\kappa'} \le \frac{c_A^d d_A a}{2}$, and this lower bound is strictly larger than $\lambda_{n,\kappa}$ for any $d$. This implies that on the event where $\Gamma_{n,\kappa'} \le \frac{c_A^d d_A a}{2}$, $M_G \cap \mathcal{K} \subset \tilde{M}_{\kappa'} \cap \mathcal{K}$. Next,

$$
\sup_{y \in \mathcal{K}, d(y, \mathcal{M}_G) \ge h} \hat{g}_{n, \kappa'}(y) \le \sup_{y \in \mathcal{K}, d(y, \mathcal{M}_G) \ge h} \bar{g}(y) + \Gamma_{n, \kappa'}.
$$
First, $\left\| \frac{d(G_i(\kappa) * Q)}{dx} \right\|_{\infty} \le \|q\|_{\infty}^D < \infty$. Moreover, for $k \in \{1, \dots, D\}$, writing $X_i(\kappa)^{[k]}$ and $\varepsilon^{[k]}$ for the k-th coordinate of $X_i(\kappa)$ and $\varepsilon$,

$$ \int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^2 |d(G_i(\kappa) * Q)(x)| = \mathbb{E}[\prod_{k=1}^{D} (X_i(\kappa)^{[k]} + \varepsilon^{[k]})^2] \\ = \mathbb{E}[(X_i(\kappa)^{[1]} + \varepsilon^{[1]})^2 (X_i(\kappa)^{[2]} + \varepsilon^{[2]})^2] \prod_{k=3}^{D} \mathbb{E}[(\varepsilon^{[k]})^2]. \quad (31) $$

We have that $(X_i(\kappa)^{[2]} + \varepsilon^{[2]})^2 \le a^2(X_i(\kappa)^{[1]})^2 + 2\gamma X_i(\kappa)^{[1]} + 2X_i(\kappa)^{[1]}\varepsilon^{[2]} + (1+\gamma)(\varepsilon^{[2]})^2 + \gamma^2$, using (31) and the fact that $\varepsilon^{[2]}$ is independent of all other variables and that, for $k \in \{1, 2\}$, $X_i^{[1]}$ is independent of $\varepsilon^{[k]}$, we finally get that $\int \prod_{j=1}^{D} x_j^2 |d(G_i(\kappa) * Q)(x)|$ is upper bounded by product and sum of expectation of $((\varepsilon^{[j]})^2)_{j \in \{1, \dots, D\}}$, $(X_i(\kappa)^{[1]})^2$, $(X_i(\kappa)^{[1]})^3$ and $(X_i(\kappa)^{[1]})^4$ which are all finite thanks to Lemma 7.

By the Cauchy-Schwarz inequality,

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \\ \le \pi^{D/2} \left( \int \prod_{j=1}^{D} (1+x_j^2) \left| \frac{d((G_0(\kappa) - G_1(\kappa)) * Q)}{dx}(x) \right|^2 dx \right)^{1/2} . \quad (32) $$

By Parseval's identity, for all $\eta \in \{0, 1\}^D$,

$$ \int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^{2\eta_j} \left| \frac{d((G_0(\kappa) - G_1(\kappa)) * Q)}{dx}(x) \right|^2 dx \\ = \int_{\mathbb{R}^D} \left| \left( \prod_{j=1}^{D} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[G_0(\kappa)] - \mathcal{F}[G_1(\kappa)])(t) \mathcal{F}[Q](t) \right|^2 dt \\ = \int_{[-c, c]^D} \left| \left( \prod_{j=1}^{D} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[G_0(\kappa)] - \mathcal{F}[G_1(\kappa)])(t) \mathcal{F}[Q](t) \right|^2 dt, $$

since $\mathcal{F}[Q]$ and for $\eta \in \{0, 1\}^D$, $\partial^\eta \mathcal{F}[Q]$ are supported on $[-c, c]^D$. Moreover, they are bounded functions, so that there exists a constant $C$ (depending only on $D$) such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \\ \le C \sum_{\eta \in \{0,1\}^D} \int_{[-c,c]^D} \left| \left( \prod_{j=1}^{D} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[G_0(\kappa)] - \mathcal{F}[G_1(\kappa)])(t) \right|^2 dt \\ = \sum_{\eta \in \{0,1\}^D} \int_{[-c,c]^D} \left| \left( \prod_{j=1}^{D} \partial_{t_j}^{\eta_j} \right) (t \mapsto \mathcal{F}[S_0] - \mathcal{F}[S_1])(A_a^\top t) \right|^2 dt. $$

Using the change of variable $u = A^\top t$, and noticing that $\{A^\top t ; t \in [-c, c]^D\} \subset [-2c, 2c]^D$, there exists a constant $C > 0$ depending on $D$ and $a$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \\ \le C \sum_{\eta \in \{0,1\}^D} \int_{[-2c,2c]^D} \left| \left( \prod_{j=1}^{D} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[S_0] - \mathcal{F}[S_1])(u) \right|^2 du. $$
Choosing $C_1 = \frac{c_A^d d_A a}{16}$ and applying Lemma 2 we get that, on the event where $\Gamma_{n, \kappa'} \le \frac{c_A^d d_A a}{16}$,

$$ \sup_{y \in \mathcal{K}, d(y, \mathcal{M}_G) \ge h \left[ \frac{D}{\beta_A} \log\left(\frac{1}{h}\right) \right]^{\frac{A+1}{A}}} \hat{g}(y) \le 2C_1 $$

for $n$ large enough, and this upper bound is strictly less than $\lambda_{n, \kappa'}$ for any $d$. This implies that

$$ \left\{ y : y \in \mathcal{K}, d(y, \mathcal{M}_G) > h \left[ \frac{D}{\beta_A} \log \left( \frac{1}{h} \right) \right]^{\frac{A+1}{A}} \right\} \cap \hat{\mathcal{M}}_{\kappa'} = \emptyset. $$

We may now take $h$ as in the statement of the Theorem. As a result, we have proved that: for all $\kappa_0 \in (1/2, 1]$, $S > 0$, $a > 0$ $d \le D$, $\nu \in (0, \nu_{est}]$, $c(\nu) > 0$ and $E > 0$, there exist $c' > 0$ and $n_0$ such that for all $n \ge n_0$, for all $\kappa \in [\kappa_0, 1]$, $G \in \text{St}_\mathcal{K}(a, d, r_0) \cap \mathcal{L}(\kappa, S, \mathcal{H})$ and $\mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)$, with $(G * \mathbb{Q})^{\otimes n}$-probability at least $1 - \frac{2}{n}$,

$$ \sup_{\kappa' \in [\kappa_0, \kappa]} \frac{\log(n)^{\kappa'}}{\log(\log(n))^{\kappa' + \frac{A+1}{A}}} H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\kappa'}) \le c'. \quad (30) $$

Using the fact that $H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\kappa})$ is uniformly upper bounded, the theorem follows.

## D.6 Proof of Theorem 3

Fix $\kappa_0 \in (1/2, 1]$, $S > 0$, $a > 0$ $d \le D$, $\nu \in (0, \nu_{est}]$, $c(\nu) > 0$ and $E > 0$. Using the end of the proof of Theorem 2, there exist $n_0$ and $c'$ such that for all $\kappa \in [\kappa_0, 1]$, all $G \in \text{St}_\mathcal{K}(a, d, r_0) \cap \mathcal{L}(\kappa, S, \mathcal{H})$ and all $\mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)$, with $(G * \mathbb{Q})^{\otimes n}$-probability at least $1 - \frac{2}{n}$, (30) holds. Let us now choose $c_\sigma = c'$ and consider the event where (30) holds. By the triangular inequality, for any $\kappa \in [\kappa_0, 1]$,

$$ \begin{aligned} H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\kappa_n}) &\le H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\kappa}) + H_{\mathcal{K}}(\hat{\mathcal{M}}_{\kappa}, \hat{\mathcal{M}}_{\kappa_n}) \\ &\le \sigma_n(\kappa) + H_{\mathcal{K}}(\hat{\mathcal{M}}_{\kappa}, \hat{\mathcal{M}}_{\kappa_n}). \end{aligned} $$

Now, using the definition of $B_n(\cdot)$, if $\kappa \le \hat{\kappa}_n$, then

$$ H_{\mathcal{K}}(\hat{\mathcal{M}}_{\kappa}, \hat{\mathcal{M}}_{\hat{\kappa}_n}) \le B_n(\hat{\kappa}_n) + \sigma_n(\kappa) $$

while if $\kappa \ge \hat{\kappa}_n$, then

$$ H_{\mathcal{K}}(\hat{\mathcal{M}}_{\kappa}, \hat{\mathcal{M}}_{\hat{\kappa}_n}) \le B_n(\kappa) + \sigma_n(\kappa) $$

so that in all cases,

$$ \begin{aligned} H_{\mathcal{K}}(\hat{\mathcal{M}}_{\kappa}, \hat{\mathcal{M}}_{\hat{\kappa}_n}) &\le B_n(\hat{\kappa}_n) + \sigma_n(\kappa) + B_n(\kappa) + \sigma_n(\hat{\kappa}_n) \\ &\le 2B_n(\kappa) + 2\sigma_n(\kappa) \end{aligned} $$

using the definition of $\hat{\kappa}_n$, and therefore

$$ H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\hat{\kappa}_n}) \le 2B_n(\kappa) + 3\sigma_n(\kappa). $$

By the triangular inequality and the definition of $B_n(\cdot)$,

$$ \begin{aligned} B_n(\kappa) &\le 0 \vee \sup_{\kappa' \in [\kappa_0, \kappa]} \left\{ H_{\mathcal{K}}(\hat{\mathcal{M}}_{\kappa}, \mathcal{M}_G) + H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\kappa'}) - \sigma_n(\kappa') \right\} \\ &\le H_{\mathcal{K}}(\hat{\mathcal{M}}_{\kappa}, \mathcal{M}_G) + 0 \vee \sup_{\kappa' \in [\kappa_0, \kappa]} \left\{ H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\kappa'}) - \sigma_n(\kappa') \right\} \\ &\le \sigma_n(\kappa). \end{aligned} $$

Thus, for all $\kappa \in [\kappa_0, 1]$, all $G \in \text{St}_\mathcal{K}(a, d, r_0) \cap \mathcal{L}(\kappa, S, \mathcal{H})$ and all $\mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)$, with $(G * \mathbb{Q})^{\otimes n}$-probability at least $1 - \frac{2}{n}$,

$$ H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\hat{\kappa}_n}) \le 5\sigma_n(\kappa), $$

and using the fact that $H_{\mathcal{K}}(\mathcal{M}_G, \hat{\mathcal{M}}_{\hat{\kappa}}) \le \sup_{x, x' \in \mathcal{K}} d(x, x')$ on the event of probability at most $2/n$ where this does not hold, Theorem 3 follows.
For all $t = (t_1, \dots, t_D) \in \mathbb{R}^D$, for $i \in \{0, 1\}$, $\mathcal{F}[T_i](t) = \mathcal{F}[\tilde{T}_i](t_1, t_2)$, where $\tilde{T}_i$ is the distribution of the 2 first coordinates of $S_i(\kappa)$ under $T_i$. There exists a constant $C > 0$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \\ \le C \sum_{\eta \in \{0,1\}^2} \int_{[-2c, 2c]^2} \left| \left( \prod_{j=1}^2 \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right|^2 dt. \quad (33) $$

Following the same approach as [20], we get that for all $t = (t_1, t_2) \in \mathbb{R}^2$,

$$ \begin{aligned} (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) &= \int_{\mathbb{R}} \{e^{it_1 u + i\gamma t_2 \tilde{g}_{\gamma}(u)} - e^{it_1 u - i\gamma t_2 \tilde{g}_{\gamma}(u)}\} f_{\kappa}(u) du \\ &= 2i \int_{\mathbb{R}} e^{it_1 u} \sin(t_2 \gamma \tilde{g}_{\gamma}(u)) f_{\kappa}(u) du \\ &= 2i \int_{\mathbb{R}} e^{it_1 u} \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} \tilde{g}_{\gamma}^{2k+1}(u) f_{\kappa}(u) du. \end{aligned} $$

Since $\sum_{k=0}^{\infty} \int_{\mathbb{R}} \frac{|t_2|^{2k+1} \gamma^{2k+1}}{(2k+1)!} |\tilde{g}_{\gamma}^{2k+1}(u)| f_{\kappa}(u) du$ is finite, we can switch integral and sum thanks to Fubini Theorem, so that

$$ \begin{aligned} (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) &= 2i \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} \int_{\mathbb{R}} e^{it_1 u} \tilde{g}_{\gamma}^{2k+1}(u) f_{\kappa}(u) du \\ &= 2i \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} m_k(t_1), \end{aligned} $$

with for all $u \in \mathbb{R}$,

$$ m_k(u) = \mathcal{F}[\tilde{g}^{2k+1} f_{\kappa}](u) = \underbrace{(\mathcal{F}[\tilde{g}] * \mathcal{F}[\tilde{g}] * \dots * \mathcal{F}[\tilde{g}])}_{2k+1 \text{ times}} * \mathcal{F}[f_{\kappa}](u). \quad (34) $$

Since

$$ \mathcal{F}[x \mapsto \cos(\frac{x}{\gamma})] = \frac{1}{2}\delta_{-\frac{1}{\gamma}} + \frac{1}{2}\delta_{\frac{1}{\gamma}}, $$

for all $u \in \mathbb{R}$,

$$ \begin{aligned} \underbrace{(\mathcal{F}[\tilde{g}] * \mathcal{F}[\tilde{g}] * \dots * \mathcal{F}[\tilde{g}])}_{2k+1 \text{ times}}(u) &= \underbrace{\mathcal{F}[\cos(\frac{\cdot}{\gamma})] * \dots * \mathcal{F}[\cos(\frac{\cdot}{\gamma})]}_{2k+1 \text{ times}} \\ &= \left(\frac{1}{2}\right)^{2k+1} \sum_{j=1}^{2k+1} \binom{2k+1}{j} \delta_{a_j}, \end{aligned} $$

where $a_j = (2j - 2k - 1)/\gamma$. By (34),

$$ m_k(u) = \left(\frac{1}{2}\right)^{2k+1} \sum_{j=0}^{2k+1} \binom{2k+1}{j} \mathcal{F}[f_{\kappa}](u - a_j). $$

Therefore,

$$ \sup_{|t| \le c} |m_k(t)| \le \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_{\kappa}] \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right| $$

and

$$ \sup_{|t| \le c} |m'_k(t)| \le \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_{\kappa}]'(t - \frac{2j - 2k - 1}{\gamma}) \right|. $$
## D.7 Proof of Lemma 4

**Case** $\kappa \neq 1$ This case is based on [Mor34]. In the following, we will note constants that can change with upper case A, B and C. In [Mor34], Theorem 2, the author defines for any positive constants $\mu > 0$, $q > 1$ and $a > 0$ a function $\zeta_{q,\mu,a}$, such that for $x \in \mathbb{R}$,

$$ \zeta_{q,\mu,a}(x) = -i \int_C z^{\mu} \exp(z^q - qax^2z) dz, $$

where $C$ is a curve in the complex plane so that the maximum of $|z^{\mu} \exp(z^q - qax^2z)|$ for $z \in C$ is attained on the positive real line. The author shows that $\zeta_{q,\mu,a}$ and $\zeta_{q,\mu,a}^2$ are integrable functions.

The author uses the saddle-point integration method to show that there exist $A > 0$ and $B > 0$ which depend on $q, \mu$ and $a$ such that

$$ |\mathcal{F}[\zeta_{q,\mu,a}](t)| \le A \exp(-Bx^{\frac{2q}{q+1}}). \quad (31) $$

Finally, for $\kappa \in (1/2, 1)$, fix $\mu > 0$, $a > 0$, and define

$$ f_{\kappa} = c_{f_{\kappa}} \mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2 * u_1, $$

where $u_1: x \in \mathbb{R} \mapsto \exp(-\frac{1}{1-4x^2})1|_{(-1/2,1/2)}(x)$ and $c_{f_\kappa}$ is a constant that ensures that $f_\kappa$ is a density.

Let us first prove that there exist $A > 0$ and $B > 0$ positive constants such that $|\mathcal{F}[\mathrm{Re}[\zeta_{q,\mu,a}]^2](t)| \le A \exp(-B|x|^{1/\kappa})$.

$$ \begin{aligned} |\mathcal{F}[\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2](t)| &= |\mathcal{F}[\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]] * \mathcal{F}[\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]](t)| \\ &\le A \int_{\mathbb{R}} \exp(-B|x-y|^{1/\kappa} - B|y|^{1/\kappa}) dy \\ &= \int_{|y-x| \ge |x|/2} \exp(-B|x-y|^{1/\kappa} - B|y|^{1/\kappa}) dy \\ &\quad + \int_{|y-x| < |x|/2} \exp(-B|x-y|^{1/\kappa} - B|y|^{1/\kappa}) dy \\ &\le A \exp(-B|x|^{1/\kappa}). \end{aligned} \quad (32) $$

Finally, for all $t \in \mathbb{R}$, using that $|\mathcal{F}[u_1](t)| \le 1$,

$$ \begin{aligned} |\mathcal{F}[f_{\kappa}](t)| &= |\mathcal{F}[\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2](t)| |\mathcal{F}[u_1](t)| \\ &\le c_{f_{\kappa}} A \exp(-B|x|^{\frac{1}{\kappa}}). \end{aligned} $$

For $x \in \mathbb{R}$, $\mathcal{F}[f_\kappa(x)]' = \mathcal{F}[x \mapsto xf_\kappa(x)]$ and

$$ xf_{\kappa}(x) = c_{f_{\kappa}} v * \mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2(x) + c_{f_{\kappa}} u_1 * \tilde{\zeta}(x), $$

where $v: x \in \mathbb{R} \mapsto xu_1(x)$ and $\tilde{\zeta}: x \in \mathbb{R} \mapsto x\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2(x)$.

Following the same proof as Theorem 2 of [Mor34], there exists $A > 0$ and $B > 0$ such that for all $t \in \mathbb{R}$, $|\mathcal{F}[x \mapsto x\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}](t)]| \le A \exp(-B|t|^{1/\kappa})$, so that, following the proof of (32), $|\mathcal{F}[\tilde{\zeta}]|(t) \le A \exp(-B|t|^{1/\kappa})$. Hence, there exists $A > 0$ and $B > 0$ such that $|\mathcal{F}[f_\kappa]'(t)| \le A \exp(-B|t|^{1/\kappa})$.

Finally, note that $f_\kappa$ is continuous as a convolution of an integrable function with a smooth function, and that for all $x \in \mathbb{R}$, $f_\kappa(x) > 0$ since $\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]$ and $u$ are not the null function almost everywhere.
Assume first that $\kappa \in (1/2, 1)$

For $\gamma$ that satisfies $\gamma \le \frac{1}{2c}$, by Lemma 7, there exist two constants $A, B$ independent of $\gamma$ and $k$ such that

$$ \sup_{|t|\le c, 0\le j\le 2k+1} \left| \mathcal{F}[f_\kappa] \left( t - \frac{2j-2k-1}{\gamma} \right) \right| \le A \exp(-B\gamma^{-\frac{1}{\kappa}}) $$

and

$$ \sup_{|t|\le c, 0\le j\le 2k+1} \left| \mathcal{F}[f_\kappa]' \left( t - \frac{2j-2k-1}{\gamma} \right) \right| \le A \exp(-B\gamma^{-\frac{1}{\kappa}}). $$

Thus,

$$ \sup_{|t|\le c} |m_k(t)| \le A \exp(-B\gamma^{-\frac{1}{\kappa}}), \quad (35) $$

and

$$ \sup_{|t|\le c} |m'_k(t)| \le A \exp(-B\gamma^{-\frac{1}{\kappa}}). \quad (36) $$

For all $\eta \in \{0, 1\}^2$, and $t \in [-c, c]^2$,

$$ \begin{aligned} \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) &= \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \left[ 2i \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} m_k(t_1) \right] \\ &= 2i\eta_2 \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k} \gamma^{2k+1}}{(2k)!} \partial_{t_1}^{\eta_1} m_k(t_1) + 2i(1-\eta_2) \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} \partial_{t_1}^{\eta_1} m_k(t_1), \end{aligned} $$

so that

$$ \left| \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right| \le 2 \sum_{k=0}^{\infty} \frac{|t_2|^{2k} \gamma^{2k+1}}{(2k)!} |\partial_{t_1}^{\eta_1} m_k(t_1)| + 2 \sum_{k=0}^{\infty} \frac{|t_2|^{2k+1} \gamma^{2k+1}}{(2k+1)!} |\partial_{t_1}^{\eta_1} m_k(t_1)|. $$

By (35) and (36), there exists a constant $C > 0$ which depends only on $D$ and $A$ such that

$$ \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{S}_0] - \mathcal{F}[\tilde{S}_1])(t) \le C \exp(-B\gamma^{-\frac{1}{\kappa}}) \sup_{|t_2|\le c} \left( \gamma \cosh(|t_2|\gamma) + \sinh(|t_2|\gamma) \right). $$

For $\gamma$ small enough, there exists a constant $C_1 > 0$ which depends only on $D$, $A > 0$ and $C > 0$ such that

$$ \left| \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right| \le C \exp(-B\gamma^{-\frac{1}{\kappa}}). $$

Finally, using (33), there exist constants $C > 0$ and $B > 0$ which depend only on $D$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le C \exp(-B\gamma^{-\frac{1}{\kappa}}). $$

Taking $\gamma = c_\gamma(\log n)^{-\kappa}$ with $c_\gamma \le B_1^\kappa$ shows that there exists $C > 0$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le \frac{C}{n}. $$

Let us now consider the case $\kappa = 1$
**Case** $\kappa = 1$ Let $\delta \in (0,1)$ and define $f_1 : x \in \mathbb{R} \mapsto c_{f_1}(u_{\frac{1}{1-\delta}} * u_{\frac{1}{1-\delta}})(x)$, where $c_{f_1}$ is a constant that ensures that $f_1$ is a probability density.

There exist $A > 0$ and $B > 0$ such that $\mathcal{F}[f_1](x) \le A \exp(-B|x|^{\delta})$, see Lemma in [Tla22]. Moreover, $\mathcal{F}[f_1]'(x) = 2c_{f_1}\mathcal{F}[u_{\frac{1}{1-\delta}}](x)\mathcal{F}[u_{\frac{1}{1-\delta}}]'(x) \le A\|x\| \mapsto xu_{\frac{1}{1-\delta}}(x)\|_1 \exp(-B|x|^{\delta})$.

Finally, note that $f_1$ is continuous and does not vanish on its support.

## D.8 Proof of Lemma 5

First, by Lemma 4, for any $\kappa \in (1/2, 1]$, $U(\kappa)$ satisfies $A(1/\kappa)$.

Let $i \in \{0, 1\}$. For any $\lambda = (\lambda_1, \dots, \lambda_D) \in \mathbb{R}^D$,

$$ 
\begin{aligned}
\mathbb{E}[\exp(\lambda^\top X_i(\kappa))] &= \mathbb{E} \left[ \exp \left( \left( \lambda_1 + \frac{1}{2}\lambda_2 \right) U(\kappa) + (-1)^i \gamma \lambda_2 \frac{1}{2} \cos \left( \frac{U(\kappa)}{\gamma} \right) \right) \right] \\
&\le e^{\gamma \frac{1}{2} |\lambda_2|} \mathbb{E} \left[ \exp \left( \left( \lambda_1 + \frac{1}{2}\lambda_2 \right) U(\kappa) \right) \right].
\end{aligned} 
\quad (33) $$

Since $U(\kappa)$ satisfies $A(1/\kappa)$, there exist positive constants $A > 0$ and $B > 0$ such that for all $\lambda = (\lambda_1, \dots, \lambda_D) \in \mathbb{R}^D$, $\mathbb{E}[\exp(\lambda^\top X_i(\kappa))] \le A \exp(B|\lambda|^{\frac{1}{\kappa}})$. Applying this in (33),

$$ 
\begin{aligned}
\mathbb{E}[\exp(\lambda^\top X_i(\kappa))] &\le A \exp \left( \gamma \frac{1}{2} |\lambda_2| + B \left| \left( \lambda_1 + \frac{1}{2}\lambda_2 \right) \right|^{\frac{1}{\kappa}} \right) \\
&\le A' \exp(B'|\lambda|^{\frac{1}{\kappa}})
\end{aligned} 
$$

for some other constants $A'$ and $B'$ since $1 \le \kappa$, so that $X_i(\kappa)$ satisfies $A(1/\kappa)$.

## D.9 Proof of Lemma 6

The proof is done in five steps.

1. We show that $\gamma g_{\gamma}$ is 1-lipschitz.

2. For $i \in \{0, 1\}$ and $\kappa \in (\frac{1}{2}, 1]$, we compute the density $p_i$ of $T_i(\kappa)$ with respect to the 1-dimensional Hausdorff measure $\mu_H$ and we show that for any compact set $K$, there exists $b(\kappa, K) > 0$ such that, for all $u \in M_i(\gamma) \cap K$, $|p_i(u)| \ge b(\kappa, K)$.

3. We show that for $i \in \{0, 1\}$, $\mu_H(\cdot \cap M_i(\gamma))$ is in $\text{St}_K(2, d, r_0)$.

4. We deduce that for $i \in \{0, 1\}$ and $d \ge 1$, $T_i$ is in $\text{St}_K(2b(\kappa, K), d, r_0)$.

5. Finally, we show that for $i \in \{0, 1\}$, $d \ge 1$ and a small enough, $G_i(\kappa) \in \text{St}_K(a, d, r_0)$.

**Proof of 1** For all $x \in \mathbb{R}$, $|\gamma \tilde{g}'_{\gamma}(x)| = |\sin(\frac{x}{\gamma})| \le 1$, which implies that $\gamma g_{\gamma}$ is 1-Lipschitz.

**Proof of 2** Let us first compute the density $p_i$ of $T_i(\kappa)$ with respect to $\mu_H$. For $i \in \{0, 1\}$, denote $\zeta_i : x \in \mathbb{R} \mapsto (x, (-1)^i \gamma g_{\gamma}(x))$. Let $\mathcal{B}$ be an open subset of $\mathbb{R}^D$. For any $\kappa \in (\frac{1}{2}, 1]$,

$$ T_i(\kappa)(\mathcal{B}) = \mathbb{P}[\zeta_i(U(\kappa)) \in \mathcal{B}] = \mathbb{P}[U(\kappa) \in \zeta_i^{-1}(\mathcal{B})] = \int_{\zeta_i^{-1}(\mathcal{B})} f_{\kappa}(u)du. $$

Let Jac $\zeta_i : u \in \mathbb{R} \mapsto \sqrt{1 + \gamma^2 \tilde{g}_{\gamma}(u)^2}$ be the Jacobian of $\zeta_i$. By the Area Formula (see equation (2.47) in [AFP00]),

$$ s_i(\kappa)(\mathcal{B}) = \int_{\zeta_i^{-1}(\mathcal{B})} \frac{f_{\kappa}(u)}{\text{Jac} \zeta_i(u)} \text{Jac} \zeta_i(u)du = \int_{\mathcal{B} \cap M_i(\gamma)} \frac{f_{\kappa}(\pi^{(1)}(u))}{\text{Jac} \zeta_i(\pi^{(1)}(u))} d\mu_H(u). $$

We then have that for all $x \in \mathbb{R}^D$,
For $\gamma$ that satisfies $\gamma \le \frac{1}{2c}$, by Lemma 7, for all $\delta \in (0, 1)$, there exist two constants $A > 0$, $B > 0$ independent of $\gamma$ and $k$ such that

$$ \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_1] \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right| \le A \exp(-B\gamma^{-\delta}), $$

and

$$ \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_1]' \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right| \le A \exp(-B\gamma^{-\delta}). $$

Thus, there exist constants $A > 0$ and $B > 0$ independent of $\gamma$ and $k$ such that

$$ \sup_{|t| \le c} |m_k(t)| \le A \exp(-B\gamma^{-\delta}), \qquad (37) $$

and

$$ \sup_{|t| \le c} |m_k'(t)| \le A \exp(-B\gamma^{-\delta}). \qquad (38) $$

Doing the same computation as in the case $\kappa \in (1/2, 1)$ shows that for all $\eta \in \{0, 1\}^2$ and $t \in [-c, c]^2$,

$$ \left| \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right| \le 2 \sum_{k=0}^{\infty} \frac{|t_2|^{2k} \gamma^{2k+1}}{(2k)!} |\partial_{t_1}^{\eta_1} m_k(t_1)| + 2 \sum_{k=0}^{\infty} \frac{|t_2|^{2k+1} \gamma^{2k+1}}{(2k+1)!} |\partial_{t_1}^{\eta_1} m_k(t_1)|. $$

By (37) and (38), there exist constants $C > 0$ and $B > 0$ which depend only on $D$ such that

$$ \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \le C \exp(-B\gamma^{-\delta}) \sup_{|t_2| \le c} \left( \gamma \cosh(|t_2|\gamma) + \sinh(|t_2|\gamma) \right). $$

For $\gamma$ small enough, there exists a constant $C > 0$ which depends only on $D$ such that

$$ \left| \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right| \le C \exp(-B\gamma^{-\delta}). $$

Finally, using (33), there exists a constant $C > 0$ which depends only on $D$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le C \exp(-B\gamma^{-\delta}). $$

Taking $\gamma = c_\gamma(\log n)^{-\frac{1}{\delta}}$ with $c_\gamma \le B^{\frac{1}{\delta+1}}$ shows that there exists $C > 0$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le \frac{C}{n}. $$

## D.7 Proof of Lemma 7

Case $\kappa \ne 1$.

This case is based on [23]. In the following, we will note constants that can change with upper case A, B and C. In [23], Theorem 2, the author defines for any positive constants $\mu > 0$, $q > 1$ and $a > 0$ a function $\zeta_{q,\mu,a}$, such that for $x \in \mathbb{R}$,

$$ \zeta_{q,\mu,a}(x) = -i \int_C z^\mu \exp(z^q - qax^2z) dz, $$
$$p_i(x) = \frac{f_{\kappa}(\pi^{(1)}(x))}{\mathrm{Jac}\,\zeta_i(\pi^{(1)}(x))} 1|_{M_i(\gamma)}(x).$$

Since $f_κ$ is continuous and does not vanish on its support, for any compact set $K$, $M_i(\gamma) \cap K$ is a compact subset of the support of $f_κ$. Thus, since $f_κ$ is continuous and does not vanish on its support, for any compact set $K$, there exists $c(\kappa, K) > 0$ such that for all $u \in M_i(\gamma) \cap K$, $f_κ(u) \ge c(\kappa, K)$. Moreover, for $i \in \{0, 1\}$, $\mathrm{Jac}\,\zeta_i(u) \le \sqrt{2}$. Therefore, for all $x \in M_i(\gamma) \cap K$, $|p_i(x)| \ge \frac{c(\kappa, K)}{\sqrt{2}}$.

**Proof of 3** Recall that the 1-dimensional Hausdorff measure $\mu_H$ is defined as the limit $\lim_{\eta \to 0} \mu_H^\eta$, where for any set $Z$

$$\mu_H^\eta(Z) = \inf \left\{ \sum_{i \in \mathbb{N}} \mathrm{Diam}(A_i) : X \subset \bigcup_i A_i \text{ and } \forall i, \mathrm{Diam}(A_i) \le \eta \right\}.$$

For any $z \in M_i(\gamma)$, there exists $x_0 \in \mathbb{R}$ such that $z = (x_0, (-1)^i \gamma g_\gamma(x_0))$ and, for any $r > 0$,

$$B(z, r) \cap M_i(\gamma) \supset \{(x, (-1)^i \gamma g_\gamma(x)), x \in B(x_0, r)\}$$

since $|x - x_0| \le r$ implies $\|\gamma g_\gamma(x) - \gamma g_\gamma(x_0)\|_\infty \le r$.

Let $(A_i)_{i \in \mathbb{N}}$ be a covering of $\{(x, (-1)^i \gamma g_\gamma(x)), x \in B(x_0, r)\}$, and $B_i = \pi^{(1)}(A_i)$, then $B_i$ is a covering of $B(x_0, r)$. For all $\eta > 0$,

$$\mu_H^\eta(\{(x, (-1)^i \gamma g_\gamma(x)), x \in B(x_0, r)\}) \ge \mu_H^\eta(B(x_0, r)),$$

thus $\mu_H(B(z, r) \cap M_i(\gamma)) \ge \mu_H(B(x_0, r)) = 2r$. If $r_0 \le 1$, then for any $r \le r_0$,

$$\mu_H(B(z, r) \cap M_i(\gamma)) \ge 2r^d,$$

which proves 3.

**Proof of 4** Let $x_i \in M_i(\gamma) \cap K$ and $r_0 < 1$. Then for all $r \le r_0$,

$$T_i(B(x_i, r) \cap M_i(\gamma)) = \int_{B(x_i, r) \cap M_i(\gamma)} p_i(u) d\mu_H(u) \ge b(\kappa, K) \mu_H(B(x_i, r) \cap M_i(\gamma)) \ge 2b(\kappa, K) r^d.$$

**Proof of 5** For $i \in \{0, 1\}$, let $x_i \in JM_i(\gamma) \cap K$, $r_0 < 1$, and take $\tilde{K}$ such that $J^{-1}K \subset \tilde{K}$. For all $r \le r_0$,

$$\begin{aligned} G_i(\kappa)(B(x_i, r)) &= \mathbb{P}[JS_i(\kappa) \in B(x_i, r)] \\ &\ge \mathbb{P} \left[ S_i(\kappa) \in B \left( J^{-1}x_i, \frac{r}{\|J\|_{\mathrm{op}}} \right) \right] \\ &\ge \frac{2b(\kappa, J^{-1}K)}{\|J\|_{\mathrm{op}}} r \\ &\ge \frac{2b(\kappa, \tilde{K})}{\|J\|_{\mathrm{op}}} r, \end{aligned}$$

so that for some $a_0$ and all $a \le a_0$, $G_i(\kappa)(B(x_i, r)) \ge ar^d$.

### D.10 Proof of Lemma 7

Let us write $m_{i,\gamma}(x) = (x + (-1)^{i\frac{\gamma}{2}} \cos(\frac{x}{\gamma}), 0, \dots, 0)$, so that $X_i(\kappa) = (U(\kappa), m_{i,\gamma}(U(\kappa)))$. For $i \in \{0, 1\}$, let $w_{i,\kappa,\gamma}$ be the density of the first coordinate of $m_{i,\gamma}(U(\kappa))$, then

$$M_{\kappa} = \sup_{x \in \mathbb{R}, \gamma \in [0,1], i \in \{0,1\}} \{w_{i,\kappa,\gamma}(x) \lor f_{\kappa}(x)\}$$
where $\mathcal{C}$ is a curve in the complex plane so that the maximum of $|z^\mu \exp(z^q - qax^2z)|$ for $z \in \mathcal{C}$ is attained on the positive real line. The author shows that $\zeta_{q,\mu,a}$ and $\zeta_{q,\mu,a}^2$ are integrable functions.

The author uses the saddle-point integration method to show that there exist $A > 0$ and $B > 0$ which depend on $q, \mu$ and $a$ such that

$$|\mathcal{F}[\zeta_{q,\mu,a}](t)| \le A \exp(-Bx^{\frac{2q}{q+1}}). \quad (39)$$

Finally, for $\kappa \in (1/2, 1)$, fix $\mu > 0$, $a > 0$, and define

$$f_{\kappa} = c_{f_{\kappa}} \mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2 * u_1,$$

where $u_1: x \in \mathbb{R} \mapsto \exp(-\frac{1}{1-4x^2})1|_{(-1/2,1/2)}(x)$ and $c_{f_\kappa}$ is a constant that ensures that $f_\kappa$ is a density.

Let us first prove that there exist $A$ and $B$ positive constants such that $|\mathcal{F}[\mathrm{Re}[\zeta_{q,\mu,a}]^2](t)| \le A \exp(-B|t|^{1/\kappa})$.

$$ \begin{aligned} |\mathcal{F}[\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2](t)| &= |\mathcal{F}[\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]] * \mathcal{F}[\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]](t)| \\ &\le A \int_{\mathbb{R}} \exp(-B|x-y|^{1/\kappa} - B|y|^{1/\kappa})dy \\ &= \int_{|y-x| \ge |x|/2} \exp(-B|x-y|^{1/\kappa} - B|y|^{1/\kappa})dy \\ &\quad + \int_{|y-x| < |x|/2} \exp(-B|x-y|^{1/\kappa} - B|y|^{1/\kappa})dy \\ &\le A \exp(-B|x|^{1/\kappa}). \end{aligned} \quad (40) $$

Finally, for all $t \in \mathbb{R}$, using that $|\mathcal{F}[u_1](t)| \le 1$,

$$ \begin{aligned} |\mathcal{F}[f_{\kappa}](t)| &= |\mathcal{F}[\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2](t)| |\mathcal{F}[u_1](t)| \\ &\le c_{f_{\kappa}} A \exp(-B|x|^{\frac{1}{\kappa}}). \end{aligned} $$

For $x \in \mathbb{R}$, $\mathcal{F}[f_\kappa(x)]' = \mathcal{F}[x \mapsto xf_\kappa(x)]$ and

$$xf_{\kappa}(x) = c_{f_{\kappa}} v * \mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2(x) + c_{f_{\kappa}} u_1 * \tilde{\zeta}(x),$$

where $v: x \in \mathbb{R} \mapsto xu_1(x)$ and $\tilde{\zeta}: x \in \mathbb{R} \mapsto x\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]^2(x)$.

Following the same proof as Theorem 2 of [23], there exists $A > 0$ and $B > 0$ such that for all $t \in \mathbb{R}$, $|\mathcal{F}[x \mapsto x\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}](t)]| \le A \exp(-B|t|^{1/\kappa})$, so that, following the proof of (40), $|\mathcal{F}[\tilde{\zeta}]|(t) \le A \exp(-B|t|^{1/\kappa})$. Hence, there exists $A > 0$ and $B > 0$ such that $|\mathcal{F}[f_\kappa]'(t)| \le A \exp(-B|t|^{1/\kappa})$.

Finally, note that $f_\kappa$ is continuous as a convolution of an integrable function with a smooth function, and that for all $x \in \mathbb{R}$, $f_\kappa(x) > 0$ since $\mathrm{Re}[\zeta_{\frac{1}{2\kappa-1}, \mu, a}]$ and $u$ are not the null function almost everywhere.

Case $\kappa = 1$.

Let $\delta \in (0, 1)$ and define $f_1: x \in \mathbb{R} \mapsto c_{f_1}(u_{\frac{1}{1-\delta}} * u_{\frac{1}{1-\delta}})(x)$, where $c_{f_1}$ is a constant that ensures that $f_1$ is a probability density.

There exist $A > 0$ and $B > 0$ such that $\mathcal{F}[f_1](x) \le A \exp(-B|x|^{\delta})$, see Lemma in [26]. Moreover, $\mathcal{F}[f_1]'(x) = 2c_{f_1}\mathcal{F}[u_{\frac{1}{1-\delta}}](x)\mathcal{F}[u_{\frac{1}{1-\delta}}]'(x) \le A\|x\mapsto xu_{\frac{1}{1-\delta}}(x)\|_1 \exp(-B|x|^{\delta})$.

Finally, note that $f_1$ is continuous and does not vanish on its support.
is an upper bound of the density of $X_i(\kappa)^{(1)}$ and of the first coordinate of $X_i(\kappa)^{(2)}$ with respect to the Lebesgue measure. Let us show that $M_\kappa$ is finite. First, note that $m_{i,\gamma}$ is one-to-one from $\mathbb{R}$ to $\mathbb{R} \times \{0\}^{D-2}$ and $m_{i,\gamma}^{-1}$ is Lipschitz with Lipschitz constant upper bounded by 1/2. One can easily check that for all $x \in \mathbb{R}$,

$$w_{i,\kappa,\gamma}(x) = f_{\kappa}((m_{i,\gamma})_1(x, 0, \dots, 0)) \frac{1}{(m_{i,\gamma})'_1(m_{i,\gamma}^{-1}(x, 0, \dots, 0))},$$

where $(m_{i,\gamma})_1(x)$ is the first coordinate of $m_{i,\gamma}(x)$. Since $(m_{i,\gamma})'_1$ is lower bounded by 1/2, $M_\kappa \le \sup_{x \in \mathbb{R}} f_\kappa(x)$, which is finite.

For any $\Delta > 0$, define the sets:

$$A_{\Delta}^{(1)} = [-\Delta, \Delta] \quad \text{and} \quad B_{\Delta}^{(2)} = \bar{B}\left(0, \left(\frac{1}{2} + 2\right)\Delta\right) \cap \left(\mathbb{R} \times \{0\}^{D-2}\right),$$

$$A_{\Delta}^{(2)} = [-\Delta, \Delta] \times \{0\}^{D-2} \quad \text{and} \quad B_{\Delta}^{(1)} = \bar{B}(0, \Delta) \cap \mathbb{R}.$$

Define $c_{\Delta,\kappa} = \mathbb{P}[U(\kappa) \in A_{\Delta}^{(1)}] \wedge \inf_{\gamma \in [0,1], i \in \{0,1\}} \mathbb{P}[m_{i,\gamma}(U(\kappa)) \in A_{\Delta}^{(2)}]$, and let us prove that $c_{\Delta,\kappa} > 0$.

First, $\mathbb{P}[U(\kappa) \in A_{\Delta}^{(1)}] > 0$ since the density of $U(\kappa)$ is positive everywhere on its support. Then, for $i \in \{0,1\}$,

$$\begin{aligned}
\mathbb{P}[m_{i,\gamma}(U(\kappa)) \in A_{\Delta}^{(2)}] &= \mathbb{P}\left(U(\kappa) + (-1)^i \frac{1}{2}\gamma \cos\left(\frac{U(\kappa)}{\gamma}\right) \in [-\Delta, \Delta]\right) \\
&\ge \mathbb{P}\left(U(\kappa) \in [-\Delta/2, \Delta/2], (-1)^i \frac{1}{2}\gamma \cos\left(\frac{U(\kappa)}{\gamma}\right) \in [-\Delta/2, \Delta/2]\right) \\
&\ge \mathbb{P}\left(U(\kappa) \in [-\Delta/2, \Delta/2], \cos\left(\frac{U(\kappa)}{\gamma}\right) \in [-\Delta/\gamma, \Delta/\gamma]\right) \\
&\ge \mathbb{P}\left(U(\kappa) \in \left[-\frac{\Delta}{2}, \frac{\Delta}{2}\right] \cap [\arccos(\Delta), \pi - \arccos(\Delta)]\right),
\end{aligned}$$

which is positive.

For any $\Delta > 0$ define $B_{\Delta,i,\gamma}^{(1)} = m_{i,\gamma}^{-1}(A_{\Delta}^{(2)})$. Then

$$\begin{aligned}
\mathrm{Diam}(B_{\Delta,i,\gamma}^{(1)}) &= \sup_{x,y \in B_{\Delta,i,\gamma}^{(1)}} |x - y| \\
&= \sup_{x,y \in A_{\Delta}^{(2)}} |m_{i,\gamma}^{-1}(x) - m_{i,\gamma}^{-1}(y)| \\
&\le \frac{1}{2} \sup_{x,y \in A_{\Delta}^{(2)}} \|x - y\| \le \Delta.
\end{aligned}$$

Thus, $B_{\Delta,i,\gamma}^{(1)} \subset B_{\Delta}^{(1)}$, and

$$\mathbb{P}[(X_i(\kappa))^{(1)} \in B_{\Delta,i}^{(1)} | (X_i(\kappa))^{(2)} \in A_{\Delta}^{(2)}] \ge \mathbb{P}[(X_i(\kappa))^{(1)} \in B_{\Delta,i,\gamma}^{(1)} | (X_i(\kappa))^{(2)} \in A_{\Delta}^{(2)}] = 1.$$

Similarly, define $B_{\Delta,i,\gamma}^{(2)} = m_{i,\gamma}(A_{\Delta}^{(1)}) = \{(x + (-1)^i \gamma \frac{1}{2} \cos(\frac{x}{\gamma}), 0, \dots, 0), x \in [-\Delta, \Delta]\}$, then

$$\begin{aligned}
\mathrm{Diam}(B_{\Delta,i,\gamma}^{(2)}) &= \sup_{x,y \in A_{\Delta}^{(1)}} \left| x + (-1)^i \gamma \frac{1}{2} \cos\left(\frac{x}{\gamma}\right) - y - (-1)^i \gamma \frac{1}{2} \cos\left(\frac{y}{\gamma}\right) \right| \\
&\le 2\Delta + \gamma \frac{1}{2} \left| \cos\left(\frac{x}{\gamma}\right) - \cos\left(\frac{y}{\gamma}\right) \right| \\
&\le \left(\frac{1}{2} + 2\right) \Delta.
\end{aligned}$$
## D.8 Proof of Lemma 8

First, by Lemma 7, for any $\kappa \in (1/2, 1]$, $U(\kappa)$ satisfies $A(\kappa)$.

Let $i \in \{0, 1\}$. For any $\lambda = (\lambda_1, \dots, \lambda_D) \in \mathbb{R}^D$,

$$ 
\begin{aligned}
\mathbb{E}[\exp(\lambda^\top X_i(\kappa))] &= \mathbb{E} \left[ \exp \left( \left( \lambda_1 + \frac{1}{2}\lambda_2 \right) U(\kappa) + (-1)^i \gamma \lambda_2 \frac{1}{2} \cos \left( \frac{U(\kappa)}{\gamma} \right) \right) \right] \\
&\le e^{\gamma \frac{1}{2} |\lambda_2|} \mathbb{E} \left[ \exp \left( \left( \lambda_1 + \frac{1}{2}\lambda_2 \right) U(\kappa) \right) \right].
\end{aligned} 
\quad (41) $$

Since $U(\kappa)$ satisfies $A(\kappa)$, there exist positive constants $A > 0$ and $B > 0$ such that for all $\lambda = (\lambda_1, \dots, \lambda_D) \in \mathbb{R}^D$, $\mathbb{E}[\exp(\lambda^\top X_i(\kappa))] \le A \exp(B|\lambda|^{\frac{1}{\kappa}})$. Applying this in (41),

$$ 
\begin{aligned}
\mathbb{E}[\exp(\lambda^\top X_i(\kappa))] &\le A \exp \left( \gamma \frac{1}{2} |\lambda_2| + B \left| \left( \lambda_1 + \frac{1}{2}\lambda_2 \right) \right|^{\frac{1}{\kappa}} \right) \\
&\le A' \exp(B' |\lambda|^{\frac{1}{\kappa}})
\end{aligned} 
$$

for some other constants $A'$ and $B'$ since $1 \le \kappa$, so that $X_i(\kappa)$ satisfies $A(\kappa)$.

## D.9 Proof of Lemma 9

The proof is done in five steps.

1. We show that $\gamma g_\gamma$ is 1-lipschitz.

2. For $i \in \{0, 1\}$ and $\kappa \in (\frac{1}{2}, 1]$, we compute the density $p_i$ of $T_i(\kappa)$ with respect to the 1-dimensional Hausdorff measure $\mu_H$ and we show that for any compact set $K$, there exists $b(\kappa, K) > 0$ such that, for all $u \in M_i(\gamma) \cap K$, $|p_i(u)| \ge b(\kappa, K)$.

3. We show that for $i \in \{0, 1\}$, $\mu_H(\cdot \cap M_i(\gamma))$ is in $\text{St}_K(2, d, r_0)$.

4. We deduce that for $i \in \{0, 1\}$ and $d \ge 1$, $T_i$ is in $\text{St}_K(2b(\kappa, K), d, r_0)$.

5. Finally, we show that for $i \in \{0, 1\}$, $d \ge 1$ and $a$ small enough, $G_i(\kappa) \in \text{St}_K(a, d, r_0)$.

*Proof of 1* For all $x \in \mathbb{R}$, $|\gamma \tilde{g}'_\gamma(x)| = |\sin(\frac{x}{\gamma})| \le 1$, which implies that $\gamma g_\gamma$ is 1-Lipschitz.

*Proof of 2* Let us first compute the density $p_i$ of $T_i(\kappa)$ with respect to $\mu_H$. For $i \in \{0, 1\}$, denote $\zeta_i : x \in \mathbb{R} \mapsto (x, (-1)^i \gamma g_\gamma(x))$. Let $\mathcal{B}$ be an open subset of $\mathbb{R}^D$. For any $\kappa \in (\frac{1}{2}, 1]$,

$$ T_i(\kappa)(\mathcal{B}) = \mathbb{P}[\zeta_i(U(\kappa)) \in \mathcal{B}] = \mathbb{P}[U(\kappa) \in \zeta_i^{-1}(\mathcal{B})] = \int_{\zeta_i^{-1}(\mathcal{B})} f_{\kappa}(u) du. $$

Let Jac $\zeta_i : u \in \mathbb{R} \mapsto \sqrt{1 + \gamma^2 \tilde{g}_\gamma(u)^2}$ be the Jacobian of $\zeta_i$. By the Area Formula (see equation (2.47) in [4]),

$$ s_i(\kappa)(\mathcal{B}) = \int_{\zeta_i^{-1}(\mathcal{B})} \frac{f_{\kappa}(u)}{\text{Jac} \zeta_i(u)} \text{Jac} \zeta_i(u) du = \int_{\mathcal{B} \cap M_i(\gamma)} \frac{f_{\kappa}(\pi^{(1)}(u))}{\text{Jac} \zeta_i(\pi^{(1)}(u))} d\mu_H(u). $$

We then have that for all $x \in \mathbb{R}^D$,

$$ p_i(x) = \frac{f_{\kappa}(\pi^{(1)}(x))}{\text{Jac} \zeta_i(\pi^{(1)}(x))} 1_{M_i(\gamma)}(x). $$

Since $f_\kappa$ is continuous and does not vanish on its support, for any compact set $K$, $M_i(\gamma) \cap K$ is a compact subset of the support of $f_\kappa$. Thus, since $f_\kappa$ is continuous and does not vanish on its support, for any compact set $K$, there exists $c(\kappa, K) > 0$ such that for all $u \in M_i(\gamma) \cap K$, $f_\kappa(u) \ge c(\kappa, K)$. Moreover, for $i \in \{0, 1\}$, Jac $\zeta_i(u) \le \sqrt{2}$. Therefore, for all $x \in M_i(\gamma) \cap K$, $|p_i(x)| \ge \frac{c(\kappa, K)}{\sqrt{2}}$.
Thus, $B_{\Delta,i,\gamma}^{(2)} \subset B_{\Delta,i}^{(2)}$, and

$$ \mathbb{P}[(X_i(\kappa))^{(2)} \in B_{\Delta,i}^{(2)} | (X_i(\kappa))^{(1)} \in A_{\Delta}^{(1)}] \ge \mathbb{P}[(X_i(\kappa))^{(2)} \in B_{\Delta,i,\gamma}^{(2)} | (X_i(\kappa))^{(1)} \in A_{\Delta}^{(1)}] = 1. $$

Thus, the distribution of $X_i(\kappa)$ satisfies (H1) and (H2). Lemma 7 follows from Lemma 5 and Theorem 7.

## D.11 Proof of Theorem 4

In the following, we will write A, B, C (with upper case letters) positive constants that can change from line to line. As in [GLCL22a] and [Gen+12], we use the upper bound:

$$ \|(G_0(\kappa) * Q)^{\otimes n} - (G_1(\kappa) * Q)^{\otimes n}\|_{TV} \le 1 - (1 - \|(G_0(\kappa) * Q) - (G_1(\kappa) * Q)\|_{TV})^n, $$

where $\| \cdot - \cdot \|_{TV}$ denotes the total variation distance. Using Le Cam's two-points method, the minimax rate will be lower bounded by $H(JM_0(\gamma), JM_1(\gamma))$, that is $\gamma$, (see Lemma 8) provided that there exists a constant $C > 0$ such that $\|(G_0(\kappa) * Q)^{\otimes n} - (G_1(\kappa) * Q)^{\otimes n}\|_{TV} \le C < 1$, so that we only need to find $C > 0$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le \frac{C}{n}. $$

Since $Q$ has a density $q$ over $\mathbb{R}^D$, $G_0(\kappa) * Q$ and $G_1(\kappa) * Q$ also have a density over $\mathbb{R}^D$. We first prove that for $i \in \{0,1\}$,

$$ \int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^2 \left| \frac{d(G_i(\kappa) * Q)}{dx}(x) \right|^2 dx < +\infty. \quad (34) $$

Indeed,

$$ \int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^2 \left| \frac{d(G_i(\kappa) * Q)}{dx}(x) \right|^2 dx \le \left\| \frac{d(G_i(\kappa) * Q)}{dx} \right\|_{\infty} \int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^2 d(G_i(\kappa) * Q)(x). $$

First, $\left\| \frac{d(G_i(\kappa) * Q)}{dx} \right\|_{\infty} \le \|q\|_{\infty}^D < \infty$. Moreover, for $k \in \{1, \dots, D\}$, writing $X_i(\kappa)^{[k]}$ and $\varepsilon^{[k]}$ for the k-th coordinate of $X_i(\kappa)$ and $\varepsilon$,

$$ \begin{aligned} \int_{\mathbb{R}^D} \prod_{j=1}^{D} x_j^2 |d(G_i(\kappa) * Q)(x)| &= \mathbb{E}[\prod_{k=1}^{D} (X_i(\kappa)^{[k]} + \varepsilon^{[k]})^2] \\ &= \mathbb{E}[(X_i(\kappa)^{[1]} + \varepsilon^{[1]})^2 (X_i(\kappa)^{[2]} + \varepsilon^{[2]})^2] \prod_{k=3}^{D} \mathbb{E}[(\varepsilon^{[k]})^2]. \end{aligned} \quad (35) $$

We have that $(X_i(\kappa)^{[2]} + \varepsilon^{[2]})^2 \le a^2(X_i(\kappa)^{[1]})^2 + 2\gamma X_i(\kappa)^{[1]} + 2X_i(\kappa)^{[1]}\varepsilon^{[2]} + (1+\gamma)(\varepsilon^{[2]})^2 + \gamma^2$, using (35) and the fact that $\varepsilon^{[2]}$ is independent of all other variables and that, for $k \in \{1, 2\}$, $X_i^{[1]}$ is independent of $\varepsilon^{[k]}$, we finally get that $\int \prod_{j=1}^{D} x_j^2 |d(G_i(\kappa) * Q)(x)|$ is upper bounded by product and sum of expectation of $((\varepsilon^{[j]})^2)_{j \in \{1, \dots, D\}}$, $(X_i(\kappa)^{[1]})^2$, $(X_i(\kappa)^{[1]})^3$ and $(X_i(\kappa)^{[1]})^4$ which are all finite thanks to Lemma 4.

By the Cauchy-Schwarz inequality,

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \\ \le \pi^{D/2} \left( \int \prod_{j=1}^{D} (1 + x_j^2) \left| \frac{d((G_0(\kappa) - G_1(\kappa)) * Q)}{dx}(x) \right|^2 dx \right)^{1/2} . \quad (36) $$
Proof of 3 Recall that the 1-dimensional Hausdorff measure $\mu_H$ is defined as the limit $\lim_{\eta \to 0} \mu_H^\eta$, where for any set $Z$

$$ \mu_H^\eta(Z) = \inf \left\{ \sum_{i \in \mathbb{N}} \mathrm{Diam}(A_i) : X \subset \bigcup_i A_i \text{ and } \forall i, \mathrm{Diam}(A_i) \le \eta \right\}. $$

For any $z \in M_i(\gamma)$, there exists $x_0 \in \mathbb{R}$ such that $z = (x_0, (-1)^i \gamma g_\gamma(x_0))$ and, for any $r > 0$,

$$ B(z, r) \cap M_i(\gamma) \supset \{(x, (-1)^i \gamma g_\gamma(x)), x \in B(x_0, r)\} $$

since $|x - x_0| \le r$ implies $\|\gamma g_\gamma(x) - \gamma g_\gamma(x_0)\|_\infty \le r$.

Let $(A_i)_{i \in \mathbb{N}}$ be a covering of $\{(x, (-1)^i \gamma g_\gamma(x)), x \in B(x_0, r)\}$, and $B_i = \pi^{(1)}(A_i)$, then $B_i$ is a covering of $B(x_0, r)$. For all $\eta > 0$,

$$ \mu_H^\eta(\{(x, (-1)^i \gamma g_\gamma(x)), x \in B(x_0, r)\}) \ge \mu_H^\eta(B(x_0, r)), $$

thus $\mu_H(B(z, r) \cap M_i(\gamma)) \ge \mu_H(B(x_0, r)) = 2r$. If $r_0 \le 1$, then for any $r \le r_0$,

$$ \mu_H(B(z, r) \cap M_i(\gamma)) \ge 2r^d, $$

which proves 3.

Proof of 4 Let $x_i \in M_i(\gamma) \cap \mathcal{K}$ and $r_0 < 1$. Then for all $r \le r_0$,

$$ \begin{aligned} T_i(B(x_i, r) \cap M_i(\gamma)) &= \int_{B(x_i, r) \cap M_i(\gamma)} p_i(u) d\mu_H(u) \\ &\ge b(\kappa, \mathcal{K}) \mu_H(B(x_i, r) \cap M_i(\gamma)) \ge 2b(\kappa, \mathcal{K}) r^d. \end{aligned} $$

Proof of 5 For $i \in \{0, 1\}$, let $x_i \in JM_i(\gamma) \cap \mathcal{K}$, $r_0 < 1$, and take $\tilde{\mathcal{K}}$ such that $J^{-1}\mathcal{K} \subset \tilde{\mathcal{K}}$. For all $r \le r_0$,

$$ \begin{aligned} G_i(\kappa)(B(x_i, r)) &= \mathbb{P}[JS_i(\kappa) \in B(x_i, r)] \\ &\ge \mathbb{P} \left[ S_i(\kappa) \in B \left( J^{-1}x_i, \frac{r}{\|J\|_{\mathrm{op}}} \right) \right] \\ &\ge \frac{2b(\kappa, J^{-1}\mathcal{K})}{\|J\|_{\mathrm{op}}} r \\ &\ge \frac{2b(\kappa, \tilde{\mathcal{K}})}{\|J\|_{\mathrm{op}}} r, \end{aligned} $$

so that for some $a_0$ and all $a \le a_0$, $G_i(\kappa)(B(x_i, r)) \ge ar^d$.

D.10 Proof of Lemma 10

Let us write $m_{i,\gamma}(x) = (x + (-1)^{i/2} \cos(x/\gamma), 0, \dots, 0)$, so that $X_i(\kappa) = (U(\kappa), m_{i,\gamma}(U(\kappa)))$. For $i \in \{0, 1\}$, let $w_{i,\kappa,\gamma}$ be the density of the first coordinate of $m_{i,\gamma}(U(\kappa))$, then

$$ M_{\kappa} = \sup_{x \in \mathbb{R}, \gamma \in [0,1], i \in \{0,1\}} \{w_{i,\kappa,\gamma}(x) \vee f_{\kappa}(x)\} $$

is an upper bound of the density of $X_i(\kappa)^{(1)}$ and of the first coordinate of $X_i(\kappa)^{(2)}$ with respect to the Lebesgue measure. Let us show that $M_\kappa$ is finite. First, note that $m_{i,\gamma}$ is one-to-one from $\mathbb{R}$ to $\mathbb{R} \times \{0\}^{D-2}$ and $m_{i,\gamma}^{-1}$ is Lipschitz with Lipschitz constant upper bounded by 1/2. One can easily check that for all $x \in \mathbb{R}$,

$$ w_{i,\kappa,\gamma}(x) = f_{\kappa}((m_{i,\gamma})_1(x, 0, \dots, 0)) \frac{1}{(m_{i,\gamma})'_1(m_{i,\gamma}^{-1}(x, 0, \dots, 0))}, $$

where $(m_{i,\gamma})_1(x)$ is the first coordinate of $m_{i,\gamma}(x)$. Since $(m_{i,\gamma})'_1$ is lower bounded by 1/2, $M_\kappa \le \sup_{x \in \mathbb{R}} f_\kappa(x)$, which is finite.
By Parseval's identity, for all $\eta \in \{0, 1\}^D$,

$$ 
\begin{aligned}
\int_{\mathbb{R}^D} \prod_{j=1}^D x_j^{2\eta_j} \left| \frac{d((G_0(\kappa) - G_1(\kappa)) * Q)}{dx}(x) \right|^2 dx &= \int_{\mathbb{R}^D} \left| \left( \prod_{j=1}^D \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[G_0(\kappa)] - \mathcal{F}[G_1(\kappa)])(t) \mathcal{F}[Q](t) \right|^2 dt \\
&= \int_{[-c, c]^D} \left| \left( \prod_{j=1}^D \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[G_0(\kappa)] - \mathcal{F}[G_1(\kappa)])(t) \mathcal{F}[Q](t) \right|^2 dt,
\end{aligned}
$$

since $\mathcal{F}[Q]$ and for $\eta \in \{0, 1\}^D$, $\partial^\eta \mathcal{F}[Q]$ are supported on $[-c, c]^D$. Moreover, they are bounded functions, so that there exists a constant $C$ (depending only on $D$) such that

$$ 
\begin{aligned}
\int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| &\le C \sum_{\eta \in \{0, 1\}^D} \int_{[-c, c]^D} \left| \left( \prod_{j=1}^D \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[G_0(\kappa)] - \mathcal{F}[G_1(\kappa)])(t) \right|^2 dt \\
&= \sum_{\eta \in \{0, 1\}^D} \int_{[-c, c]^D} \left| \left( \prod_{j=1}^D \partial_{t_j}^{\eta_j} \right) (t \mapsto \mathcal{F}[S_0] - \mathcal{F}[S_1])(A_a^\top t) \right|^2 dt.
\end{aligned}
$$

Using the change of variable $u = A^\top t$, and noticing that $\{A^\top t ; t \in [-c, c]^D\} \subset [-2c, 2c]^D$, there exists a constant $C > 0$ depending on $D$ and $a$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le C \sum_{\eta \in \{0, 1\}^D} \int_{[-2c, 2c]^D} \left| \left( \prod_{j=1}^D \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[S_0] - \mathcal{F}[S_1])(u) \right|^2 du. $$

For all $t = (t_1, \dots, t_D) \in \mathbb{R}^D$, for $i \in \{0, 1\}$, $\mathcal{F}[T_i](t) = \mathcal{F}[\tilde{T}_i](t_1, t_2)$, where $\tilde{T}_i$ is the distribution of the 2 first coordinates of $S_i(\kappa)$ under $T_i$. There exists a constant $C > 0$ such that

$$ 
\begin{aligned}
\int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| & \\
&\le C \sum_{\eta \in \{0, 1\}^2} \int_{[-2c, 2c]^2} \left| \left( \prod_{j=1}^2 \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right|^2 dt. \quad (37)
\end{aligned}
$$

Following the same approach as [Gen+12], we get that for all $t = (t_1, t_2) \in \mathbb{R}^2$,

$$ 
\begin{aligned}
(\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) &= \int_{\mathbb{R}} \{e^{it_1 u + i\gamma t_2 \tilde{g}_\gamma(u)} - e^{it_1 u - i\gamma t_2 \tilde{g}_\gamma(u)}\} f_\kappa(u) du \\
&= 2i \int_{\mathbb{R}} e^{it_1 u} \sin(t_2 \gamma \tilde{g}_\gamma(u)) f_\kappa(u) du \\
&= 2i \int_{\mathbb{R}} e^{it_1 u} \sum_{k=0}^\infty \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} \tilde{g}_\gamma^{2k+1}(u) f_\kappa(u) du.
\end{aligned}
$$

Since $\sum_{k=0}^\infty \int_{\mathbb{R}} \frac{|t_2|^{2k+1} \gamma^{2k+1}}{(2k+1)!} |\tilde{g}_\gamma^{2k+1}(u)| f_\kappa(u) du$ is finite, we can switch integral and sum thanks to Fubini Theorem, so that

$$ 
\begin{aligned}
(\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) &= 2i \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} \int_{\mathbb{R}} e^{it_1 u} \tilde{g}_\gamma^{2k+1}(u) f_\kappa(u) du \\
&= 2i \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} m_k(t_1),
\end{aligned}
$$

with for all $u \in \mathbb{R}$,

$$ m_k(u) = \mathcal{F}[\tilde{g}^{2k+1} f_\kappa](u) = \underbrace{(\mathcal{F}[\tilde{g}] * \mathcal{F}[\tilde{g}] * \dots * \mathcal{F}[\tilde{g}] * \mathcal{F}[f_\kappa])(u)}_{2k+1 \text{ times}}. \quad (38) $$
For any $\Delta > 0$, define the sets:

$$A_{\Delta}^{(1)} = [-\Delta, \Delta] \quad \text{and} \quad B_{\Delta}^{(2)} = \bar{B}\left(0, \left(\frac{1}{2} + 2\right)\Delta\right) \cap \left(\mathbb{R} \times \{0\}\right)^{D-2},$$

$$A_{\Delta}^{(2)} = [-\Delta, \Delta] \times \{0\}^{D-2} \quad \text{and} \quad B_{\Delta}^{(1)} = \bar{B}(0, \Delta) \cap \mathbb{R}.$$

Define $c_{\Delta, \kappa} = \mathbb{P}[U(\kappa) \in A_{\Delta}^{(1)}] \wedge \inf_{\gamma \in [0,1], i \in \{0,1\}} \mathbb{P}[m_{i,\gamma}(U(\kappa)) \in A_{\Delta}^{(2)}]$, and let us prove that $c_{\Delta, \kappa} > 0$.

First, $\mathbb{P}[U(\kappa) \in A_{\Delta}^{(1)}] > 0$ since the density of $U(\kappa)$ is positive everywhere on its support. Then, for $i \in \{0, 1\}$,

$$\begin{aligned}
\mathbb{P}[m_{i,\gamma}(U(\kappa)) \in A_{\Delta}^{(2)}] &= \mathbb{P}\left(U(\kappa) + (-1)^i \frac{1}{2}\gamma \cos\left(\frac{U(\kappa)}{\gamma}\right) \in [-\Delta, \Delta]\right) \\
&\geq \mathbb{P}\left(U(\kappa) \in [-\Delta/2, \Delta/2], (-1)^i \frac{1}{2}\gamma \cos\left(\frac{U(\kappa)}{\gamma}\right) \in [-\Delta/2, \Delta/2]\right) \\
&\geq \mathbb{P}\left(U(\kappa) \in [-\Delta/2, \Delta/2], \cos\left(\frac{U(\kappa)}{\gamma}\right) \in [-\Delta/\gamma, \Delta/\gamma]\right) \\
&\geq \mathbb{P}\left(U(\kappa) \in \left[-\frac{\Delta}{2}, \frac{\Delta}{2}\right] \cap [\arccos(\Delta), \pi - \arccos(\Delta)]\right),
\end{aligned}$$

which is positive.

For any $\Delta > 0$ define $B_{\Delta, i, \gamma}^{(1)} = m_{i, \gamma}^{-1}(A_{\Delta}^{(2)})$. Then

$$\begin{aligned}
\mathrm{Diam}(B_{\Delta, i, \gamma}^{(1)}) &= \sup_{x, y \in B_{\Delta, i, \gamma}^{(1)}} |x - y| \\
&= \sup_{x, y \in A_{\Delta}^{(2)}} |m_{i, \gamma}^{-1}(x) - m_{i, \gamma}^{-1}(y)| \\
&\leq \frac{1}{2} \sup_{x, y \in A_{\Delta}^{(2)}} \|x - y\| \leq \Delta.
\end{aligned}$$

Thus, $B_{\Delta, i, \gamma}^{(1)} \subset B_{\Delta}^{(1)}$, and

$$\mathbb{P}[(X_i(\kappa))^{(1)} \in B_{\Delta, i}^{(1)} | (X_i(\kappa))^{(2)} \in A_{\Delta}^{(2)}] \geq \mathbb{P}[(X_i(\kappa))^{(1)} \in B_{\Delta, i, \gamma}^{(1)} | (X_i(\kappa))^{(2)} \in A_{\Delta}^{(2)}] = 1.$$

Similarly, define $B_{\Delta, i, \gamma}^{(2)} = m_{i, \gamma}(A_{\Delta}^{(1)}) = \{(x + (-1)^i \gamma \frac{1}{2} \cos(\frac{x}{\gamma}), 0, \dots, 0), x \in [-\Delta, \Delta]\}$, then

$$\begin{aligned}
\mathrm{Diam}(B_{\Delta, i, \gamma}^{(2)}) &= \sup_{x, y \in A_{\Delta}^{(1)}} \left| x + (-1)^i \gamma \frac{1}{2} \cos\left(\frac{x}{\gamma}\right) - y - (-1)^i \gamma \frac{1}{2} \cos\left(\frac{y}{\gamma}\right) \right| \\
&\leq 2\Delta + \gamma \frac{1}{2} \left| \cos\left(\frac{x}{\gamma}\right) - \cos\left(\frac{y}{\gamma}\right) \right| \\
&\leq \left(\frac{1}{2} + 2\right) \Delta.
\end{aligned}$$

Thus, $B_{\Delta, i, \gamma}^{(2)} \subset B_{\Delta, i}^{(2)}$, and

$$\mathbb{P}[(X_i(\kappa))^{(2)} \in B_{\Delta, i}^{(2)} | (X_i(\kappa))^{(1)} \in A_{\Delta}^{(1)}] \geq \mathbb{P}[(X_i(\kappa))^{(2)} \in B_{\Delta, i, \gamma}^{(2)} | (X_i(\kappa))^{(1)} \in A_{\Delta}^{(1)}] = 1.$$

Thus, the distribution of $X_i(\kappa)$ satisfies (H1) and (H2). Lemma 10 follows from Lemma 8 and Theorem 2.

# E Proofs for Section 5

## E.1 Proof of Theorem 6

We shall need two technical lemmas. The following one is easily proved following the arguments at the end of the proof of Theorem 3.
Since

$$ \mathcal{F}[x \mapsto \cos(\frac{x}{\gamma})] = \frac{1}{2}\delta_{-\frac{1}{\gamma}} + \frac{1}{2}\delta_{\frac{1}{\gamma}}, $$

for all $u \in \mathbb{R}$,

$$ \underbrace{(\mathcal{F}[\tilde{g}] * \mathcal{F}[\tilde{g}] * \dots * \mathcal{F}[\tilde{g}])}_{2k+1 \text{ times}}(u) = \underbrace{\mathcal{F}[\cos(\dot{\gamma})] * \dots * \mathcal{F}[\cos(\dot{\gamma})]}_{2k+1 \text{ times}} \\ = \left(\frac{1}{2}\right)^{2k+1} \sum_{j=1}^{2k+1} \binom{2k+1}{j} \delta_{a_j}, $$

where $a_j = (2j - 2k - 1)/\gamma$. By (38),

$$ m_k(u) = \left(\frac{1}{2}\right)^{2k+1} \sum_{j=0}^{2k+1} \binom{2k+1}{j} \mathcal{F}[f_\kappa](u - a_j). $$

Therefore,

$$ \sup_{|t| \le c} |m_k(t)| \le \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_\kappa] \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right| $$

and

$$ \sup_{|t| \le c} |m'_k(t)| \le \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_\kappa]' \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right|. $$

**Assume first that** $\kappa \in (1/2, 1)$ For $\gamma$ that satisfies $\gamma \le \frac{1}{2c}$, by Lemma 4, there exist two constants $A, B$ independent of $\gamma$ and $k$ such that

$$ \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_\kappa] \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right| \le A \exp(-B\gamma^{-\frac{1}{\kappa}}) $$

and

$$ \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_\kappa]' \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right| \le A \exp(-B\gamma^{-\frac{1}{\kappa}}). $$

Thus,

$$ \sup_{|t| \le c} |m_k(t)| \le A \exp(-B\gamma^{-\frac{1}{\kappa}}), \quad (39) $$

and

$$ \sup_{|t| \le c} |m'_k(t)| \le A \exp(-B\gamma^{-\frac{1}{\kappa}}). \quad (40) $$

For all $\eta \in \{0, 1\}^2$, and $t \in [-c, c]^2$,

$$ \left(\prod_{j=1}^{2} \partial_{t_j}^{\eta_j}\right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) = \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \left[ 2i \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} m_k(t_1) \right] \\ = 2i\eta_2 \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k} \gamma^{2k+1}}{(2k)!} \partial_{t_1}^{\eta_1} m_k(t_1) + 2i(1-\eta_2) \sum_{k=0}^{\infty} \frac{(-1)^k t_2^{2k+1} \gamma^{2k+1}}{(2k+1)!} \partial_{t_1}^{\eta_1} m_k(t_1), $$

so that

$$ \left| \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1]) (t) \right| \\ \le 2 \sum_{k=0}^{\infty} \frac{|t_2|^{2k} \gamma^{2k+1}}{(2k)!} |\partial_{t_1}^{\eta_1} m_k(t_1)| + 2 \sum_{k=0}^{\infty} \frac{|t_2|^{2k+1} \gamma^{2k+1}}{(2k+1)!} |\partial_{t_1}^{\eta_1} m_k(t_1)|. $$
**Lemma 12.** Let $G$ be a probability measure with compact support $M_G$. Assume that $G \in St_{M_G}(a, d, r_0)$ for some constants $a > 0, d > 0$ and $r_0 > 0$. Recall that $\Gamma_n := \Gamma_{n,1} = \|\hat{g}_n - \bar{g}\|_{\infty}$. Then

(1) For any $C_1 > 0$ and $c > 0$, there exists $h_0 > 0$ such that if $h_n \le h_0$, on the event where

$$C_1 + \Gamma_n < \lambda_n < ac_A^d d_A \left(\frac{1}{h_n}\right)^{D-d} - \Gamma_n,$$

it holds

$$M_G \subset \tilde{M} \subset (M_G)_c.$$

(2) For $m_n, h_n, \lambda_n$ chosen as in Theorem 3, for all $C_1 \in (0, ac_A^d d_A)$ and $\delta' > 0$, there exist $C > 0$ and $n_0 \ge 0$ such that for all $n \ge n_0$, with probability at least $1 - 2 \exp(-n^{1/2-\delta'})$,

$$\Gamma_n^2 \le Ce^{-m_n} \quad \text{and} \quad C_1 + \Gamma_n < \lambda_n < ac_A^d d_A \left(\frac{1}{h_n}\right)^{D-d} - \Gamma_n.$$

and in particular, for all $c > 0$ and $\delta' > 0$, there exists $n'_0 \ge 0$ such that for all $n \ge n'_0$, with probability at least $1 - 2 \exp(-n^{1/2-\delta'})$,

$$M_G \subset \tilde{M} \subset (M_G)_c.$$

In particular, since $R_n \to +\infty$ and $M_G$ is compact, up to increasing $n_0$, on this event,

$$M_G \subset \tilde{M} \cap \bar{B}(0, R_n) \subset (M_G)_c.$$

In the rest of the proof of the Theorem, we lighten the notation $\tilde{M} \cap \bar{B}(0, R_n)$ into $\tilde{M}$ (equivalently, we redefine the estimator $\hat{M}$ as the intersection of the estimator of Section 4.1 with the closed euclidean ball of radius $R_n$).

**Lemma 13.** Let $G$ be a probability measure with compact support $M_G$. Assume that $G \in St_{M_G}(a, d, r_0)$ for some constants $a > 0, d > 0$ and $r_0 > 0$. Then for any $\alpha > 0, c > 0$ and $p \in [1, +\infty)$, there exists $C(\alpha, c) > 0$ such that, on the event where

$$M_G \subset \tilde{M} \subset (M_G)_c,$$

it holds

$$\|\bar{g}\|_{L_1(\mathbb{R}^D \setminus (\tilde{M})_c)} \le C(\alpha, c)h_n^\alpha \quad \text{and} \quad \int_{\mathbb{R}^D \setminus (\tilde{M})_c} \|x\|^p |\bar{g}(x)| dx \le C(\alpha, c)h_n^\alpha.$$

*Proof.* By definition,

$$\|\bar{g}\|_{L_1(\mathbb{R}^D \setminus (\tilde{M})_c)} = \frac{1}{h_n^D} \int_{x \in M_G} \int_{y \in \mathbb{R}^D \setminus (\tilde{M})_c} \psi_A \left( \frac{\|y - x\|_2}{h_n} \right) dydG(x).$$

By Point (V) of the properties of $\psi_A$, for any $A > 0$, there exists $C > 0$ such that for any $x \in M_G$ and $y \in \mathbb{R}^D \setminus (\tilde{M})_c$,

$$\psi_A \left( \frac{\|y - x\|_2}{h_n} \right) \le C \exp \left( -\beta_A \frac{\|y - x\|_2^{A/(A+1)}}{h_n^{A/(A+1)}} \right) \le C \exp \left( -\beta_A \frac{d(y, M_G)^{A/(A+1)}}{h_n^{A/(A+1)}} \right).$$

Since $M_G \subset \tilde{M}$, for all $y \in \mathbb{R}^D \setminus (\tilde{M})_c$, $d(y, M_G) \ge c$, so for any $\alpha > 0$, there exists a constant $\tilde{C} > 0$ such that

$$C \exp \left( -\beta_A \frac{d(y, M_G)^{A/(A+1)}}{h_n^{A/(A+1)}} \right) \le \tilde{C} \frac{h_n^{D+\alpha}}{d(y, M_G)^{D+\alpha}}.$$
By (39) and (40), there exists a constant $C > 0$ which depends only on $D$ and $A$ such that

$$ \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{S}_0] - \mathcal{F}[\tilde{S}_1])(t) \le C \exp(-B\gamma^{-\frac{1}{\kappa}}) \sup_{|t_2| \le c} \left( \gamma \cosh(|t_2|\gamma) + \sinh(|t_2|\gamma) \right). $$

For $\gamma$ small enough, there exists a constant $C_1 > 0$ which depends only on $D$, $A > 0$ and $C > 0$ such that

$$ \left| \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right| \le C \exp(-B\gamma^{-\frac{1}{\kappa}}). $$

Finally, using (37), there exist constants $C > 0$ and $B > 0$ which depend only on $D$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le C \exp(-B\gamma^{-\frac{1}{\kappa}}). $$

Taking $\gamma = c_\gamma(\log n)^{-\kappa}$ with $c_\gamma \le B_1^\kappa$ shows that there exists $C > 0$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le \frac{C}{n}. $$

Let us now consider the case $\kappa = 1$. For $\gamma$ that satisfies $\gamma \le \frac{1}{2c}$, by Lemma 4, for all $\delta \in (0, 1)$, there exist two constants $A > 0$, $B > 0$ independent of $\gamma$ and $k$ such that

$$ \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_1] \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right| \le A \exp(-B\gamma^{-\delta}), $$

and

$$ \sup_{|t| \le c, 0 \le j \le 2k+1} \left| \mathcal{F}[f_1]' \left( t - \frac{2j - 2k - 1}{\gamma} \right) \right| \le A \exp(-B\gamma^{-\delta}). $$

Thus, there exist constants $A > 0$ and $B > 0$ independent of $\gamma$ and $k$ such that

$$ \sup_{|t| \le c} |m_k(t)| \le A \exp(-B\gamma^{-\delta}), \quad (41) $$

and

$$ \sup_{|t| \le c} |m'_k(t)| \le A \exp(-B\gamma^{-\delta}). \quad (42) $$

Doing the same computation as in the case $\kappa \in (1/2, 1)$ shows that for all $\eta \in \{0, 1\}^2$ and $t \in [-c, c]^2$,

$$ \left| \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right| \le 2 \sum_{k=0}^{\infty} \frac{|t_2|^{2k} \gamma^{2k+1}}{(2k)!} |\partial_{t_1}^{\eta_1} m_k(t_1)| + 2 \sum_{k=0}^{\infty} \frac{|t_2|^{2k+1} \gamma^{2k+1}}{(2k+1)!} |\partial_{t_1}^{\eta_1} m_k(t_1)|. $$

By (41) and (42), there exist constants $C > 0$ and $B > 0$ which depend only on $D$ such that

$$ \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \le C \exp(-B\gamma^{-\delta}) \sup_{|t_2| \le c} \left( \gamma \cosh(|t_2|\gamma) + \sinh(|t_2|\gamma) \right). $$

For $\gamma$ small enough, there exists a constant $C > 0$ which depends only on $D$ such that

$$ \left| \left( \prod_{j=1}^{2} \partial_{t_j}^{\eta_j} \right) (\mathcal{F}[\tilde{T}_0] - \mathcal{F}[\tilde{T}_1])(t) \right| \le C \exp(-B\gamma^{-\delta}). $$
Moreover, since $\mathcal{M}_G$ is compact, $\text{Diam}(\mathcal{M}_G)$ is finite, so that on the event where $\mathcal{M}_G \subset \tilde{\mathcal{M}}$,

$$ \int_{\mathbb{R}^D \setminus (\tilde{\mathcal{M}})_c} \frac{1}{d(y, \mathcal{M}_G)^{D+\alpha}} dy \le \int_{\mathbb{R}^D \setminus (\mathcal{M}_G)_c} \frac{1}{d(y, \mathcal{M}_G)^{D+\alpha}} dy < \infty. \\ \le \int_{\mathbb{R}^D} \left( \frac{1}{c \vee (\|y\| - \text{Diam}(\mathcal{M}_G)/2)} \right)^{D+\alpha} dy < \infty. $$

Therefore, for all $c > 0$ and $\alpha > 0$, there exists $C$ depending on $A, D, c, \alpha$ and $\text{Diam}(\mathcal{M}_G)$ such that

$$ \|\tilde{g}\|_{L_1(\mathbb{R}^D \setminus (\tilde{\mathcal{M}})_c)} \le C h_n^\alpha. $$

The proof that the same holds for $\int_{\mathbb{R}^D \setminus (\tilde{\mathcal{M}})_c} \|x\|^p \tilde{g}(x) dx$ is similar. $\square$

Let $G \in \text{St}_{\mathcal{M}_G}(a, d, r_0)$ be such that if $X \sim G$, then $\Phi_X \in \mathcal{H} \cap \Upsilon_{1,S}$. Fix $p \in [1, +\infty)$. We use a bias-variance decomposition of $W_p(G, \hat{P}_{n,\eta})$ through the triangle inequality

$$ W_p(G, \hat{P}_{n,\eta}) \le W_p(G, P_{\psi_{A,h}}) + W_p(P_{\psi_{A,h}}, \hat{P}_{n,\eta}). $$

The proof is done is several steps :

(1) We first show that there exists $C > 0$ depending only on $A$ and $D$ such that the bias satisfies

$$ W_p(G, P_{\psi_{A,h_n}}) \le C h_n. $$

(2) We prove that for any $\alpha \ge 1$, on the event where

$$ \mathcal{M}_G \subset \tilde{\mathcal{M}} \subset (\mathcal{M}_G)_c, $$

there exists $C' > 0$ such that

$$ W_p(P_{\psi_{A,h_n}}, \hat{P}_{n,\eta}) \le C'(h_n^\alpha + \Gamma_n). $$

(3) We show that the choice of the parameters $m_n, h_n$ and $\lambda_n$ gives the result.

### Proof of (1)

Let $Y_\psi$ be a random variable with density $\psi_{A,h_n}$ and independent of $X$, so that the distribution of $X + Y_\psi$ is $P_{\psi_{A,h_n}}$. By definition of $W_p$,

$$ W_p^p(G, P_{\psi_{A,h_n}}) \le \mathbb{E}(\|X + Y_\psi - X\|_2^p) = \mathbb{E}(\|Y_\psi\|_2^p) = h_n^p \int_{\mathbb{R}^D} \|u\|^p \psi_{A,1}(u) du, $$

and the integral is finite by point (V) of the properties of $\psi_A$.

### Proof of (2)

If $\nu$ and $\mu$ are probability measures on $\mathbb{R}^D$ having respective densities $f$ and $g$ with respect to the Lebesgue measure, letting $\omega$ be the measure with density $\min(f, g)$ with respect to the Lebesgue measure, $a \in \mathbb{R}^D$, and $\delta_a$ the Dirac measure in $a$, by convexity of $x \mapsto x^p$,

$$ \begin{aligned} W_p^p(\mu, \nu) &\le 2^{p-1} (W_p^p(\mu, \omega + (1-\omega(\mathbb{R}^D))\delta_a) + W_p^p(\nu, \omega + (1-\omega(\mathbb{R}^D))\delta_a)) \\ &\le 2^{p-1} \int_{\mathbb{R}^D} \|x-a\|^p (f(x)-g(x)) - 2\min(f(x),g(x))) dx \\ &= 2^{p-1} \int_{\mathbb{R}^D} \|x-a\|^p |f(x)-g(x)| dx, \end{aligned} $$

so that

$$ W_p^p(\mu, \nu) \le 2^{p-1} \min_{a \in \mathbb{R}^D} \int_{\mathbb{R}^D} \|x-a\|^p |f(x)-g(x)| dx. \quad (42) $$
Finally, using (37), there exists a constant $C > 0$ which depends only on $D$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le C \exp(-B\gamma^{-\delta}). $$

Taking $\gamma = c_{\gamma}(\log n)^{-\frac{1}{\delta}}$ with $c_{\gamma} \le B^{\frac{1}{\delta+1}}$ shows that there exists $C > 0$ such that

$$ \int_{\mathbb{R}^D} |d(G_0(\kappa) * Q)(x) - d(G_1(\kappa) * Q)(x)| \le \frac{C}{n}. $$

## D.12 Proof of Theorem 5

We shall need two technical lemmas. The following one is easily proved following the arguments at the end of the proof of Theorem 2.

**Lemma 11.** Let $G$ be a probability measure with compact support $\mathcal{M}_G$. Assume $G \in \text{St}_{\mathcal{M}_G}(a, d, r_0)$ for some constants $a > 0, d > 0$ and $r_0 > 0$. Recall that $\Gamma_n := \Gamma_{n,1} = \|\hat{g}_n - \bar{g}\|_{\infty}$. Then

(1) For any $C_1 > 0$ and $c > 0$, there exists $h_0 > 0$ such that if $h_n \le h_0$, on the event where

$$ C_1 + \Gamma_n < \lambda_n < ac_A^d d_A(\frac{1}{h_n})^{D-d} - \Gamma_n, $$

it holds

$$ \mathcal{M}_G \subset \tilde{\mathcal{M}} \subset (\mathcal{M}_G)_c. $$

(2) For $m_n$, $h_n$, $\lambda_n$ chosen as in Theorem 2, for all $C_1 \in (0, ac_A^d d_A)$ and $\delta' > 0$, there exist $C > 0$ and $n_0 \ge 0$ such that for all $n \ge n_0$, with probability at least $1 - 2 \exp(-n^{1/2-\delta'})$,

$$ \Gamma_n^2 \le Ce^{-m_n} \quad \text{and} \quad C_1 + \Gamma_n < \lambda_n < ac_A^d d_A(\frac{1}{h_n})^{D-d} - \Gamma_n. $$

and in particular, for all $c > 0$ and $\delta' > 0$, there exists $n'_0 \ge 0$ such that for all $n \ge n'_0$, with probability at least $1 - 2 \exp(-n^{1/2-\delta'})$,

$$ \mathcal{M}_G \subset \tilde{\mathcal{M}} \subset (\mathcal{M}_G)_c. $$

In particular, since $R_n \to +\infty$ and $\mathcal{M}_G$ is compact, up to increasing $n_0$, on this event,

$$ \mathcal{M}_G \subset \tilde{\mathcal{M}} \cap \bar{B}(0, R_n) \subset (\mathcal{M}_G)_c. $$

In the rest of the proof of the Theorem, we lighten the notation $\tilde{\mathcal{M}} \cap \bar{B}(0, R_n)$ into $\tilde{\mathcal{M}}$ (equivalently, we redefine the estimator $\tilde{\mathcal{M}}$ as the intersection of the estimator of Section 3.2 with the closed euclidean ball of radius $R_n$).

**Lemma 12.** Let $G$ be a probability measure with compact support $\mathcal{M}_G$. Assume $G \in \text{St}_{\mathcal{M}_G}(a, d, r_0)$ for some constants $a > 0, d > 0$ and $r_0 > 0$. Then for any $\alpha > 0, c > 0$ and $p \in [1, +\infty)$, there exists $C(\alpha, c) > 0$ such that, on the event where

$$ \mathcal{M}_G \subset \tilde{\mathcal{M}} \subset (\mathcal{M}_G)_c, $$

it holds

$$ \|\bar{g}\|_{L_1(\mathbb{R}^D \setminus (\tilde{\mathcal{M}})_c)} \le C(\alpha, c)h_n^\alpha \quad \text{and} \quad \int_{\mathbb{R}^D \setminus (\tilde{\mathcal{M}})_c} \|x\|^p |\bar{g}(x)| dx \le C(\alpha, c)h_n^\alpha. $$

*Proof.* By definition,

$$ \|\bar{g}\|_{L_1(\mathbb{R}^D \setminus (\tilde{\mathcal{M}})_c)} = \frac{1}{h_n^D} \int_{x \in \mathcal{M}_G} \int_{y \in \mathbb{R}^D \setminus (\tilde{\mathcal{M}})_c} \psi_A \left( \frac{\|y-x\|_2}{h_n} \right) dydG(x). $$
This entails

$$
\begin{align}
W_p^p(P_{\psi_{A,h_n}}, \hat{P}_{n,\eta}) &\le 2^{p-1} \min_{a \in \mathbb{R}^D} \int_{\mathbb{R}^D} \|x-a\|^p |\bar{g}(x) - c_n \hat{g}_n^+(x) 1_{(\widehat{\mathcal{M}})_n})(x)|dx \nonumber \\
&\le 2^{p-1} \int_{(\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x) - c_n \hat{g}_n^+(x)|dx + 2^{p-1} \int_{\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_n} \|x\|^p \bar{g}(x)dx. \tag{43}
\end{align}
$$

For $S$ compact subset of $\mathbb{R}^D$, write $M_S = \sup_{x \in S} \|x\|^p$ and $\mathrm{Vol}(S)$ for the Lebesgue measure of $S$, then

$$
\begin{align*}
\int_{(\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x) - c_n \hat{g}_n^+(x)| dx &\le M_{(\widehat{\mathcal{M}})_n} \int_{(\widehat{\mathcal{M}})_n} |\hat{g}_n^+(x) - \bar{g}(x)| dx + M_{(\widehat{\mathcal{M}})_n} \frac{|c_n - 1|}{c_n} \\
&\le M_{(\widehat{\mathcal{M}})_n} \mathrm{Vol}((\widehat{\mathcal{M}})_n) \Gamma_n + M_{(\widehat{\mathcal{M}})_n} \frac{|c_n - 1|}{c_n}.
\end{align*}
$$

We also have

$$
\frac{|c_n - 1|}{c_n} = \left| \frac{1}{c_n} - 1 \right| = \left| \int_{(\widehat{\mathcal{M}})_n} (\hat{g}_n^+(y) - \bar{g}(y)) dy - \int_{\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_n} \bar{g}(y) dy \right| \\
\le \|\hat{g}_n - \bar{g}\|_{L_1((\widehat{\mathcal{M}})_n)} + \|\bar{g}\|_{L_1(\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_n)}.
$$

Using Hölder's inequality,

$$
\| \hat{g}_n - \bar{g} \|_{L_1((\widehat{\mathcal{M}})_n)} \le \mathrm{Vol}((\widehat{\mathcal{M}})_n) \Gamma_n.
$$

By Lemma 13, for any $\alpha > 0$, there exists $C$ such that

$$
\int_{(\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x) - c_n \hat{g}_n^+(x)| dx \le 2M_{(\widehat{\mathcal{M}})_n} \mathrm{Vol}((\widehat{\mathcal{M}})_n) \Gamma_n + M_{(\widehat{\mathcal{M}})_n} Ch_n^\alpha.
$$

For any $c > 0$, when $\widehat{\mathcal{M}} \subset (\mathcal{M}_G)_c$, one has $(\widehat{\mathcal{M}})_\eta \subset (\mathcal{M}_G)_{\eta+c}$. This inclusion entails $M_{(\widehat{\mathcal{M}})_\eta} \le$
$M_{(\mathcal{M}_G)_{\eta+c}}$ and $\mathrm{Vol}((\widehat{\mathcal{M}})_\eta) \le \mathrm{Vol}((\mathcal{M}_G)_{\eta+c})$. Therefore, for any $c > 0$,

$$
\int_{(\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x) - c_n \hat{g}_n^+(x)| dx \le 2M_{(\mathcal{M}_G)_{\eta+c}} \mathrm{Vol}((\mathcal{M}_G)_{\eta+c}) \Gamma_n + M_{(\mathcal{M}_G)_{\eta+c}} Ch_n^\alpha. \quad (44)
$$

Again by Lemma 13, on the event where $\mathcal{M}_G \subset \widehat{\mathcal{M}} \subset (\mathcal{M}_G)_\eta$,

$$
\int_{\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x)| dx \le C' h_n^\alpha. \quad (45)
$$

Finally, using (43), (44) and (45), for any $\alpha \ge 1$, there exists $C > 0$ such that

$$
W_p^p(P_{\psi_{A,h_n}}, \hat{P}_{n,\eta}) \le C(h_n^\alpha + \Gamma_n).
$$

Proof of (3)

Using (1) and (2), for sequences $h_n$, $m_n$ and $\lambda_n$ satisfying the assumptions of Theorem 6,
on the event where $M_G \subset \widehat{M} \subset (M_G)_\eta$, for any $\alpha \ge p$, there exists $C > 0$ such that

$$
W_p(G, \hat{P}_{n,\eta}) \le C(h_n + (h_n^\alpha + \Gamma_n)^{1/p}) \le 2C(h_n + \Gamma_n^{1/p}).
$$

We may assume $h_n \le 1$ for all $n$ without loss of generality. As stated in Lemma 12, for any $\delta' > 0$, there exist $C'$ and $n_0$ such that for all $n \ge n_0$, with probability at least $1 - 2 \exp(-n^{1/2-\delta'})$, $\Gamma_n^{1/p} \le C' e^{-m_n/(2p)}$ and $M_G \subset \widehat{M} \subset (M_G)_\eta$, and therefore

$$
W_p(G, \hat{P}_{n,\eta}) \le C m_n^{-1}
$$
By (6), for any $A > 0$, there exists $C > 0$ such that for any $x \in \mathcal{M}_G$ and $y \in \mathbb{R}^D \setminus (\widehat{\mathcal{M}})_c$,

$$ \psi_A \left( \frac{\|y - x\|_2}{h_n} \right) \le C \exp \left( -\beta_A \frac{\|y - x\|_2^{A/(A+1)}}{h_n^{A/(A+1)}} \right) \le C \exp \left( -\beta_A \frac{d(y, \mathcal{M}_G)^{A/(A+1)}}{h_n^{A/(A+1)}} \right). $$

Since $\mathcal{M}_G \subset \widehat{\mathcal{M}}$, for all $y \in \mathbb{R}^D \setminus (\widehat{\mathcal{M}})_c$, $d(y, \mathcal{M}_G) \ge c$, so for any $\alpha > 0$, there exists a constant $\tilde{C} > 0$ such that

$$ C \exp \left( -\beta_A \frac{d(y, \mathcal{M}_G)^{A/(A+1)}}{h_n^{A/(A+1)}} \right) \le \tilde{C} \frac{h_n^{D+\alpha}}{d(y, \mathcal{M}_G)^{D+\alpha}}. $$

Moreover, since $\mathcal{M}_G$ is compact, $\mathrm{Diam}(\mathcal{M}_G)$ is finite, so that on the event where $\mathcal{M}_G \subset \widehat{\mathcal{M}}$,

$$ \int_{\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_c} \frac{1}{d(y, \mathcal{M}_G)^{D+\alpha}} dy \le \int_{\mathbb{R}^D \setminus (\mathcal{M}_G)_c} \frac{1}{d(y, \mathcal{M}_G)^{D+\alpha}} dy < \infty. $$ $$ \le \int_{\mathbb{R}^D} \left( \frac{1}{c \vee (\|y\| - \mathrm{Diam}(\mathcal{M}_G)/2)} \right)^{D+\alpha} dy < \infty. $$

Therefore, for all $c > 0$ and $\alpha > 0$, there exists $C$ depending on $A$, $D$, $c$, $\alpha$ and $\mathrm{Diam}(\mathcal{M}_G)$ such that

$$ \|\tilde{g}\|_{L_1(\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_c)} \le C h_n^\alpha. $$

The proof that the same holds for $\int_{\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_c} \|x\|^p \tilde{g}(x) dx$ is similar. $\square$

Let $G \in \mathrm{St}_{\mathcal{M}_G}(a, d, r_0)$ be such that if $X \sim G$, then $\Phi_X \in \mathcal{H} \cap \Upsilon_{1,S}$. Fix $p \in [1, +\infty)$. We use a bias-variance decomposition of $W_p(G, \hat{P}_{n,\eta})$ through the triangle inequality

$$ W_p(G, \hat{P}_{n,\eta}) \le W_p(G, P_{\psi_{A,h_n}}) + W_p(P_{\psi_{A,h_n}}, \hat{P}_{n,\eta}). $$

The proof is done is several steps :

(1) We first show that there exists $C > 0$ depending only on $A$ and $D$ such that the bias satisfies

$$ W_p(G, P_{\psi_{A,h_n}}) \le C h_n. $$

(2) We prove that for any $\alpha \ge 1$, on the event where

$$ \mathcal{M}_G \subset \widehat{\mathcal{M}} \subset (\mathcal{M}_G)_c, $$

there exists $C' > 0$ such that

$$ W_p(P_{\psi_{A,h_n}}, \hat{P}_{n,\eta}) \le C'(h_n^\alpha + \Gamma_n). $$

(3) We show that the choice of the parameters $m_n$, $h_n$ and $\lambda_n$ gives the result.

**Proof of (1)** Let $Y_\psi$ be a random variable with density $\psi_{A,h_n}$ and independent of $X$, so that the distribution of $X + Y_\psi$ is $P_{\psi_{A,h_n}}$. By definition of $W_p$,

$$ W_p^p(G, P_{\psi_{A,h_n}}) \le \mathbb{E}(\|X + Y_\psi - X\|_2^p) = \mathbb{E}(\|Y_\psi\|_2^p) = h_n^p \int_{\mathbb{R}^D} \|u\|^p \psi_{A,1}(u) du, $$

and the integral is finite by point (V) of the properties of $\psi_A$.
on this event, up to changing the constant C.

On the event of probability at most $2 \exp(-n^{1/2-\delta'})$ where this does not hold, since the support of $\hat{P}_{n,\eta^*}$ is a subset of $\bar{B}(0, R_n)$, $W_p(G, \hat{P}_{n,\eta^*}) \le 2R_n$.

Therefore, taking $\delta' < \delta$ where $\delta$ is as defined in the statement of the Theorem, there exists $C > 0$ such that for $n \ge n_0$,

$$ \mathbb{E}_{(G*\mathbb{Q})^{\otimes n}}[W_p(G, \hat{P}_{n,\eta^*})] \le Cm_n^{-1}, $$

which concludes the proof.

## E.2 Proof of Theorem 7

Let $\hat{P}_n$ be an estimator of $G$. According to [27],

$$ \sup_{\substack{G \in \mathrm{St}_K(a,d,r_0) \cap \mathcal{L}(1,S,\mathcal{H}_1^*) \\ \mathbb{Q} \in \mathcal{Q}^{(D)}(\nu,c(\nu),E)}} \mathbb{E}_{(G*\mathbb{Q})^{\otimes n}}[W_p(G, \hat{P}_n)] \\ \ge \frac{1}{2} W_p(G_0(1), G_1(1))(1 - \|G_0(1) * Q) - G_1(1) * Q\|_1)^n. $$

Using the same two distributions $G_0(1)$, $G_1(1)$ and the same set $\mathcal{H}_1^*$ as in Theorem 5. We have shown in Theorem 5 that there exists a constant $C > 0$ such that

$$ \|G_0(1) * Q - G_1(1) * Q\|_{TV} \le \frac{C}{n}, $$

taking $\gamma$ of the form $c \log(n)^{-1-\delta}$ for any $\delta > 0$ and $c$ small enough, which implies that the minimax risk is lower bounded by $W_p(G_0(1), G_1(1))$. We show that there exist constants $c > 0$ and $n_0 > 0$ such that for $n \ge n_0$

$$ W_p(G_0(1), G_1(1)) \ge c\gamma. $$

Let $\mathcal{U}_\gamma$ be the set of $u \in \mathbb{R}$ such that $|\cos(\frac{u}{\gamma})| \ge 1/2$, that is $\mathcal{U}_\gamma = \bigcup_{k \in \mathbb{Z}} [k\pi\gamma - \frac{\pi\gamma}{3}, k\pi\gamma + \frac{\pi\gamma}{3}]$. For each $k \in \mathbb{Z}$, let $I_{k,\gamma} := [k\pi\gamma - \frac{\pi\gamma}{2}, k\pi\gamma + \frac{\pi\gamma}{2}]$. Let us also define, for any two sets $A$ and $B$ of $\mathbb{R}^d$, $d(A, B) = \inf_{x \in A, y \in B} \|x - y\|_2$. We first show that

$$ d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma)) \ge \gamma(\frac{1}{4\sqrt{2}} \wedge \frac{\pi}{6}). $$

Let $x \in M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1})$ and $y \in M_1(\gamma)$. There exists $k \in \mathbb{Z}$ such that $x \in M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1})$. If $y \in I_{k,\gamma} \times \mathbb{R}^{D-1}$ (that is, if the first coordinate of $x$ and $y$ are in the same interval $I_{k,\gamma}$), then

$$ \|x - y\|_2 \ge d(M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1}), M_1(\gamma) \cap (I_{k,\gamma} \times \mathbb{R}^{D-1})). $$

All points of $M_0(\gamma)$ are of the form $(u, u + \frac{1}{2}\gamma \cos(\frac{u}{\gamma}), 0, \dots, 0)^\top$ and the distance between $(u, u + \frac{1}{2}\gamma \cos(\frac{u}{\gamma}), 0, \dots, 0)^\top$ and the diagonal defined by $\mathcal{D} := \{(u, u, 0, \dots, 0)^\top : u \in \mathbb{R}\}$ is $\frac{1}{4\sqrt{2}}\gamma |\cos(\frac{u}{\gamma})|$. Since the sets $M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1})$ and $M_1(\gamma) \cap (I_{k,\gamma} \times \mathbb{R}^{D-1})$ are on opposite sides of the diagonal $\mathcal{D}$,

$$ d(M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1}), M_1(\gamma) \cap (I_{k,\gamma} \times \mathbb{R}^{D-1})) \\ \ge d(M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1}), \mathcal{D}) \\ = \frac{1}{4\sqrt{2}}\gamma, $$

so that $\|x - y\|_2 \ge \frac{1}{4\sqrt{2}}\gamma$. If now $y \notin I_{k,\gamma} \times \mathbb{R}^{D-1}$,

$$ d(M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1}), M_1(\gamma) \cap ((\mathbb{R} \setminus I_{k,\gamma}) \times \mathbb{R}^{D-1})) \ge d(\mathcal{U}_\gamma \cap I_{k,\gamma}, \mathbb{R} \setminus I_{k,\gamma}) \\ = \frac{\pi\gamma}{6}, $$
**Proof of (2)** If $\nu$ and $\mu$ are probability measures on $\mathbb{R}^D$ having respective densities $f$ and $g$ with respect to the Lebesgue measure, letting $\omega$ be the measure with density $\min(f, g)$ with respect to the Lebesgue measure, $a \in \mathbb{R}^D$, and $\delta_a$ the Dirac measure in $a$, by convexity of $x \mapsto x^p$,

$$
\begin{aligned}
W_p^p(\mu, \nu) &\le 2^{p-1} (W_p^p(\mu, \omega + (1-\omega(\mathbb{R}^D))\delta_a) + W_p^p(\nu, \omega + (1-\omega(\mathbb{R}^D))\delta_a)) \\
&\le 2^{p-1} \int_{\mathbb{R}^D} \|x-a\|^p (f(x)-g(x)-2\min(f(x),g(x)))dx \\
&= 2^{p-1} \int_{\mathbb{R}^D} \|x-a\|^p |f(x)-g(x)|dx,
\end{aligned}
$$

so that

$$ W_p^p(\mu, \nu) \le 2^{p-1} \min_{a \in \mathbb{R}^D} \int_{\mathbb{R}^D} \|x-a\|^p |f(x)-g(x)|dx. \quad (43) $$

This entails

$$
\begin{aligned}
W_p^p(P_{\psi_{A,h_n}}, \hat{P}_{n,\eta}) &\le 2^{p-1} \min_{a \in \mathbb{R}^D} \int_{\mathbb{R}^D} \|x-a\|^p |\bar{g}(x) - c_n \hat{g}_n^+(x) 1_{(\hat{\mathcal{M}})_n})(x)|dx \\
&\le 2^{p-1} \int_{(\hat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x) - c_n \hat{g}_n^+(x)|dx + 2^{p-1} \int_{\mathbb{R}^D \setminus (\hat{\mathcal{M}})_n} \|x\|^p \bar{g}(x)dx. \quad (44)
\end{aligned}
$$

For $S$ compact subset of $\mathbb{R}^D$, write $M_S = \sup_{x \in S} \|x\|^p$ and $\mathrm{Vol}(S)$ for the Lebesgue measure of $S$, then

$$
\begin{aligned}
\int_{(\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x) - c_n \widehat{g}_n^+(x)| dx &\le M_{(\widehat{\mathcal{M}})_n} \int_{(\widehat{\mathcal{M}})_n} |\widehat{g}_n^+(x) - \bar{g}(x)| dx + M_{(\widehat{\mathcal{M}})_n} \frac{|c_n - 1|}{c_n} \\
&\le M_{(\widehat{\mathcal{M}})_n} \mathrm{Vol}((\widehat{\mathcal{M}})_n) \Gamma_n + M_{(\widehat{\mathcal{M}})_n} \frac{|c_n - 1|}{c_n}.
\end{aligned}
$$

We also have

$$
\begin{aligned}
\frac{|c_n - 1|}{c_n} &= \left| \frac{1}{c_n} - 1 \right| = \left| \int_{(\widehat{\mathcal{M}})_n} (\widehat{g}_n^+(y) - \bar{g}(y)) dy - \int_{\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_n} \bar{g}(y) dy \right| \\
&\le \|\widehat{g}_n - \bar{g}\|_{L_1((\widehat{\mathcal{M}})_n)} + \|\bar{g}\|_{L_1(\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_n)}.
\end{aligned}
$$

Using Hölder's inequality,

$$ \|\widehat{g}_n - \bar{g}\|_{L_1((\widehat{\mathcal{M}})_n)} \le \mathrm{Vol}((\widehat{\mathcal{M}})_n) \Gamma_n. $$

By Lemma 12, for any $\alpha > 0$, there exists $C$ such that

$$ \int_{(\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x) - c_n \widehat{g}_n^+(x)| dx \le 2M_{(\widehat{\mathcal{M}})_n} \mathrm{Vol}((\widehat{\mathcal{M}})_n) \Gamma_n + M_{(\widehat{\mathcal{M}})_n} Ch_n^\alpha. $$

For any $c > 0$, when $\widehat{\mathcal{M}} \subset (\mathcal{M}_G)_c$, one has $(\widehat{\mathcal{M}})_\eta \subset (\mathcal{M}_G)_{\eta+c}$. This inclusion entails $M_{(\widehat{\mathcal{M}})_\eta} \le M_{(\mathcal{M}_G)_{\eta+c}}$ and $\mathrm{Vol}((\widehat{\mathcal{M}})_\eta) \le \mathrm{Vol}((\mathcal{M}_G)_{\eta+c})$. Therefore, for any $c > 0$,

$$ \int_{(\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x) - c_n \widehat{g}_n^+(x)| dx \le 2M_{(\mathcal{M}_G)_{\eta+c}} \mathrm{Vol}((\mathcal{M}_G)_{\eta+c}) \Gamma_n + M_{(\mathcal{M}_G)_{\eta+c}} Ch_n^\alpha. \quad (45) $$

Again by Lemma 12, on the event where $\mathcal{M}_G \subset \widehat{\mathcal{M}} \subset (\mathcal{M}_G)_\eta$,

$$ \int_{\mathbb{R}^D \setminus (\widehat{\mathcal{M}})_n} \|x\|^p |\bar{g}(x)| dx \le C' h_n^\alpha. \quad (46) $$

Finally, using (44), (45) and (46), for any $\alpha \ge 1$, there exists $C > 0$ such that

$$ W_p^p(P_{\psi_{A,h_n}}, \hat{P}_{n,\eta}) \le C(h_n^\alpha + \Gamma_n). $$
so that $\|x - y\|_2 \ge \frac{\pi\gamma}{6}$, and thus $d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma)) \ge \gamma(\frac{1}{4\sqrt{2}} \wedge \frac{\pi}{6})$.

Now, let us show that $W_p(G_0(1), G_1(1)) \ge \gamma(\frac{1}{8\sqrt{2}} \wedge \frac{\pi}{12})$. Let $\pi$ be a transport plan between $G_0(1)$ and $G_1(1)$, then

$$ 
\begin{aligned}
& \int_{M_0(\gamma) \times M_1(\gamma)} \|x - y\|_2^p d\pi(x, y) \\
\ge & \int_{M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}) \times M_1(\gamma)} \|x - y\|_2^p d\pi(x, y) \\
\ge & d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma))^p \pi(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}) \times M_1(\gamma)) \\
= & d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma))^p G_0(1)(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1})) \\
= & d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma))^p \mathbb{P}[U(1) \in \mathcal{U}_\gamma]
\end{aligned}
$$

since $G_1(1)$ has support $M_1(\gamma)$ and by definition of $G_0(1)$. Therefore, by taking the infimum on all transport plans between $G_0(1)$ and $G_1(1)$,

$$ W_p(G_0(1), G_1(1)) \ge \gamma(\frac{1}{4\sqrt{2}} \wedge \frac{\pi}{6})\mathbb{P}[U(1) \in \mathcal{U}_\gamma]^{1/p} $$

$U(1)$ admits a density $f_1$ with respect to Lebesgue measure that is supported on $[-1, 1]$ and continuous. Let us write $\omega$ one of its modulus of continuity. We have

$$ 
\begin{aligned}
\mathbb{P}[U(1) \in \mathcal{U}_\gamma] &= \int_{[-1,1]} f_1(x)1_{\mathcal{U}_\gamma}(x)dx \\
&= \sum_{k \in [-1/(\pi\gamma), 1/(\pi\gamma)]} \int_{[k\pi\gamma - \frac{\pi\gamma}{3}, k\pi\gamma + \frac{\pi\gamma}{3}]} f_1(x)dx \\
&\le \sum_{k \in [-1/(\pi\gamma), 1/(\pi\gamma)]} \left( \int_{[k\pi\gamma - \frac{\pi\gamma}{3}, k\pi\gamma + \frac{\pi\gamma}{3}]} f_1(k\pi\gamma)dx + \frac{2\pi\gamma}{3}\omega(\pi\gamma/3) \right) \\
&\le \sum_{k \in [-1/(\pi\gamma), 1/(\pi\gamma)]} \frac{2}{3} \int_{[k\pi\gamma - \frac{\pi\gamma}{2}, k\pi\gamma + \frac{\pi\gamma}{2}]} f_1(k\pi\gamma)dx + \frac{3}{\pi\gamma} \frac{2\pi\gamma}{3}\omega(\pi\gamma/3) \\
&\le \sum_{k \in [-1/(\pi\gamma), 1/(\pi\gamma)]} \frac{2}{3} \int_{[k\pi\gamma - \frac{\pi\gamma}{2}, k\pi\gamma + \frac{\pi\gamma}{2}]} f_1(x)dx \\
&\quad + \frac{3}{\pi\gamma} \left( \frac{2\pi\gamma}{3}\omega(\pi\gamma/3) + \pi\gamma\omega(\pi\gamma/2) \right) \\
&\le \frac{2}{3} \int_{[-1,1]} f_1(x)dx + 3 \left( \frac{2}{3}\omega(\pi\gamma/3) + \omega(\pi\gamma/2) \right) \\
&\xrightarrow{\gamma \to 0} \frac{2}{3} \int_{[-1,1]} f_1(x)dx = \frac{2}{3}.
\end{aligned}
$$

Therefore, there exists $n_0$ such that for all $n \ge n_0$, $W_p(G_0(1), G_1(1)) \ge \gamma(\frac{1}{8\sqrt{2}} \wedge \frac{\pi}{12})$.

## References

[1] Eddie Aamari, Catherine Aaron, and Clément Levrard. Minimax boundary estimation and estimation with boundary. *Bernoulli*, 29(4):3334–3368, 2023.

[2] Eddie Aamari and Clément Levrard. Nonasymptotic rates for manifold, tangent space and curvature estimation. *Ann. Statist.*, 47(1):177–204, 2019.

[3] Yariv Aizenbud and Barak Sober. Non-parametric estimation of manifolds from noisy data, 2021.
**Proof of (3)** Using (1) and (2), for sequences $h_n, m_n$ and $\lambda_n$ satisfying the assumptions of Theorem 5, on the event where $\mathcal{M}_G \subset \tilde{\mathcal{M}} \subset (\mathcal{M}_G)_\eta$, for any $\alpha \ge p$, there exists $C > 0$ such that

$$W_p(G, \hat{P}_{n,\eta}) \le C(h_n + (h_n^\alpha + \Gamma_n)^{1/p}) \le 2C(h_n + \Gamma_n^{1/p}).$$

We may assume $h_n \le 1$ for all $n$ without loss of generality. As stated in Lemma 11, for any $\delta' > 0$, there exist $C'$ and $n_0$ such that for all $n \ge n_0$, with probability at least $1 - 2 \exp(-n^{1/2-\delta'})$, $\Gamma_n^{1/p} \le C'e^{-m_n/(2p)}$ and $\mathcal{M}_G \subset \tilde{\mathcal{M}} \subset (\mathcal{M}_G)_\eta$, and therefore

$$W_p(G, \hat{P}_{n,\eta^*}) \le Cm_n^{-1}$$

on this event, up to changing the constant $C$.

On the event of probability at most $2 \exp(-n^{1/2-\delta'})$ where this does not hold, since the support of $\hat{P}_{n,\eta^*}$ is a subset of $\tilde{B}(0, R_n)$, $W_p(G, \hat{P}_{n,\eta^*}) \le 2R_n$.

Therefore, taking $\delta' < \delta$ where $\delta$ is as defined in the statement of the Theorem, there exists $C > 0$ such that for $n \ge n_0$,

$$\mathbb{E}_{(G*\mathbb{Q})^{\otimes n}}[W_p(G, \hat{P}_{n,\eta^*})] \le Cm_n^{-1},$$

which concludes the proof.

## D.13 Proof of Theorem 6

Let $\hat{P}_n$ be an estimator of $G$. According to [Yu97],

$$\sup_{\substack{G \in \text{St}_K(a,d,r_0) \cap \mathcal{L}(1,S, \mathcal{H}_1^*) \\ \mathbb{Q} \in \mathcal{Q}^{(D)}(\nu, c(\nu), E)}} \mathbb{E}_{(G*\mathbb{Q})^{\otimes n}}[W_p(G, \hat{P}_n)] \ge \frac{1}{2} W_p(G_0(1), G_1(1))(1 - \|G_0(1)*Q\| - G_1(1)*Q\|_1)^n.$$

Using the same two distributions $G_0(1), G_1(1)$ and the same set $\mathcal{H}_1^*$ as in Theorem 4. We have shown in Theorem 4 that there exists a constant $C > 0$ such that

$$\|G_0(1) * Q - G_1(1) * Q\|_{TV} \le \frac{C}{n},$$

taking $\gamma$ of the form $c \log(n)^{-1-\delta}$ for any $\delta > 0$ and $c$ small enough, which implies that the minimax risk is lower bounded by $W_p(G_0(1), G_1(1))$. We show that there exist constants $c > 0$ and $n_0 > 0$ such that for $n \ge n_0$

$$W_p(G_0(1), G_1(1)) \ge c\gamma.$$

Let $\mathcal{U}_\gamma$ be the set of $u \in \mathbb{R}$ such that $|\cos(\frac{u}{\gamma})| \ge 1/2$, that is $\mathcal{U}_\gamma = \bigcup_{k \in \mathbb{Z}} [k\pi\gamma - \frac{\pi\gamma}{3}, k\pi\gamma + \frac{\pi\gamma}{3}]$. For each $k \in \mathbb{Z}$, let $I_{k,\gamma} := [k\pi\gamma - \frac{\pi\gamma}{2}, k\pi\gamma + \frac{\pi\gamma}{2}]$. Let us also define, for any two sets $A$ and $B$ of $\mathbb{R}^d$, $d(A, B) = \inf_{x \in A, y \in B} \|x - y\|_2$. We first show that

$$d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma)) \ge \gamma(\frac{1}{4\sqrt{2}} \wedge \frac{\pi}{6}).$$

Let $x \in M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1})$ and $y \in M_1(\gamma)$. There exists $k \in \mathbb{Z}$ such that $x \in M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1})$. If $y \in I_{k,\gamma} \times \mathbb{R}^{D-1}$ (that is, if the first coordinate of $x$ and $y$ are in the same interval $I_{k,\gamma}$), then

$$\|x - y\|_2 \ge d(M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1}), M_1(\gamma) \cap (I_{k,\gamma} \times \mathbb{R}^{D-1})).$$

All points of $M_0(\gamma)$ are of the form $(u, u + \frac{1}{2}\gamma \cos(\frac{u}{\gamma}), 0, \dots, 0)^\top$ and the distance between $(u, u + \frac{1}{2}\gamma \cos(\frac{u}{\gamma}), 0, \dots, 0)^\top$ and the diagonal defined by $\mathcal{D} := \{(u, u, 0, \dots, 0)^\top : u \in \mathbb{R}\}$ is $\frac{1}{4\sqrt{2}}\gamma |\cos(\frac{u}{\gamma})|$. Since the sets $M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1})$ and $M_1(\gamma) \cap (I_{k,\gamma} \times \mathbb{R}^{D-1})$ are on opposite sides of the diagonal $\mathcal{D}$,

$$d(M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1}), M_1(\gamma) \cap (I_{k,\gamma} \times \mathbb{R}^{D-1})) \ge d(M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1}), \mathcal{D}) \\ = \frac{1}{4\sqrt{2}}\gamma,$$
[4] Luigi Ambrosio, Nicola Fusco, and Diego Pallara. *Functions of bounded variation and free discontinuity problems*. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.

[5] Clément Berenfeld, Paul Rosa, and Judith Rousseau. Estimating a density near an unknown manifold: a bayesian nonparametric approach. *The Annals of Statistics*, 52(5):2081–2111, 2024.

[6] Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. *Geometric and topological inference*. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2018.

[7] Victor-Emmanuel Brunel, Jason M. Klusowski, and Dana Yang. Estimation of convex supports from noisy measurements. *Bernoulli*, 27(2):772–793, 2021.

[8] Jérémie Capitao-Miniconi and Élisabeth Gassiat. Deconvolution of spherical data corrupted with unknown noise. *Electron. J. Stat.*, 17(1):607–649, 2023.

[9] Antonio Cuevas and Ricardo Fraiman. A plug-in approach to support estimation. *Ann. Statist.*, 25(6):2300–2312, 1997.

[10] Jérôme Dedecker, Aurélie Fischer, and Bertrand Michel. Improved rates for Wasserstein deconvolution with ordinary smooth error in dimension one. *Electron. J. Stat.*, 9(1):234–265, 2015.

[11] Jérôme Dedecker and Bertrand Michel. Minimax rates of convergence for Wasserstein deconvolution with supersmooth errors in any dimension. *J. Multivariate Anal.*, 122:278–291, 2013.

[12] Sylvain Delattre and Aurélie Fischer. Estimation via length-constrained generalized empirical principal curves under small noise, 2019.

[13] Luc Devroye and Gary L. Wise. Detection of abnormal behavior via nonparametric estimation of the support. *SIAM J. Appl. Math.*, 38(3):480–488, 1980.

[14] Vincent Divol. Measure estimation on manifolds: an optimal transport approach. *Probab. Theory Related Fields*, 183(1-2):581–647, 2022.

[15] Herbert Federer. Curvature measures. *Trans. Amer. Math. Soc.*, 93:418–491, 1959.

[16] Nicolás García Trillos, Daniel Sanz-Alonso, and Ruiyi Yang. Local regularization of noisy point clouds: improved global geometric estimates and data analysis. *J. Mach. Learn. Res.*, 20:Paper No. 136, 37, 2019.

[17] Élisabeth Gassiat, Sylvain Le Corff, and Luc Lehéricy. Deconvolution with unknown noise distribution is possible for multivariate signals. *Ann. Statist.*, 50(1):303–323, 2022.

[18] Élisabeth Gassiat, Sylvain Le Corff, and Luc Lehéricy. Supplementary to “deconvolution with unknown noise distribution is possible for multivariate signals”. *Ann. Statist.*, 2022.

[19] Élisabeth Gassiat, Luc Lehéricy, and Sylvain Le Corff. Erratum: Deconvolution with unknown noise distribution is possible for multivariate signals. preprint, 2025.

[20] Christopher R. Genovese, Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman. Manifold estimation and singular deconvolution under Hausdorff loss. *Ann. Statist.*, 40(2):941–963, 2012.

[21] Alexander Goldenshluger and Oleg Lepski. Universal pointwise selection rule in multivariate function estimation. *Bernoulli*, 14(4):1150–1190, 2008.

[22] Alexander Meister. Estimating the support of multivariate densities under measurement error. *J. Multivariate Anal.*, 97(8):1702–1717, 2006.
so that $\|x - y\|_2 \ge \frac{1}{4\sqrt{2}}\gamma$. If now $y \notin I_{k,\gamma} \times \mathbb{R}^{D-1}$,

$$d(M_0(\gamma) \cap ((\mathcal{U}_\gamma \cap I_{k,\gamma}) \times \mathbb{R}^{D-1}), M_1(\gamma) \cap ((\mathbb{R} \setminus I_{k,\gamma}) \times \mathbb{R}^{D-1})) \ge d(\mathcal{U}_\gamma \cap I_{k,\gamma}, \mathbb{R} \setminus I_{k,\gamma}) \\ = \frac{\pi\gamma}{6},$$

so that $\|x - y\|_2 \ge \frac{\pi\gamma}{6}$, and thus $d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma)) \ge \gamma(\frac{1}{4\sqrt{2}} \wedge \frac{\pi}{6})$.

Now, let us show that $W_p(G_0(1), G_1(1)) \ge \gamma(\frac{1}{8\sqrt{2}} \wedge \frac{\pi}{12})$. Let $\pi$ be a transport plan between $G_0(1)$ and $G_1(1)$, then

$$\int_{M_0(\gamma) \times M_1(\gamma)} \|x - y\|_2^p d\pi(x, y) \ge \int_{M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}) \times M_1(\gamma)} \|x - y\|_2^p d\pi(x, y) \\ \ge d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma))^p \pi(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}) \times M_1(\gamma)) \\ = d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma))^p G_0(1)(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1})) \\ = d(M_0(\gamma) \cap (\mathcal{U}_\gamma \times \mathbb{R}^{D-1}), M_1(\gamma))^p \mathbb{P}[U(1) \in \mathcal{U}_\gamma]$$

since $G_1(1)$ has support $M_1(\gamma)$ and by definition of $G_0(1)$. Therefore, by taking the infimum on all transport plans between $G_0(1)$ and $G_1(1)$,

$$W_p(G_0(1), G_1(1)) \ge \gamma(\frac{1}{4\sqrt{2}} \wedge \frac{\pi}{6})\mathbb{P}[U(1) \in \mathcal{U}_\gamma]^{1/p}.$$

$U(1)$ admits a density $f_1$ with respect to Lebesgue measure that is supported on $[-1, 1]$ and continuous. Let us write $\omega$ one of its modulus of continuity. We have

$$\begin{aligned} \mathbb{P}[U(1) \in \mathcal{U}_\gamma] &= \int_{[-1,1]} f_1(x)1_{\mathcal{U}_\gamma}(x)dx \\ &= \sum_{k \in [-1/(\pi\gamma), 1/(\pi\gamma)]} \int_{[k\pi\gamma - \frac{\pi\gamma}{3}, k\pi\gamma + \frac{\pi\gamma}{3}]} f_1(x)dx \\ &\le \sum_{k \in [-1/(\pi\gamma), 1/(\pi\gamma)]} \left( \int_{[k\pi\gamma - \frac{\pi\gamma}{3}, k\pi\gamma + \frac{\pi\gamma}{3}]} f_1(k\pi\gamma)dx + \frac{2\pi\gamma}{3}\omega(\pi\gamma/3) \right) \\ &\le \sum_{k \in [-1/(\pi\gamma), 1/(\pi\gamma)]} \frac{2}{3} \int_{[k\pi\gamma - \frac{\pi\gamma}{2}, k\pi\gamma + \frac{\pi\gamma}{2}]} f_1(k\pi\gamma)dx + \frac{3}{\pi\gamma} \frac{2\pi\gamma}{3}\omega(\pi\gamma/3) \\ &\le \sum_{k \in [-1/(\pi\gamma), 1/(\pi\gamma)]} \frac{2}{3} \int_{[k\pi\gamma - \frac{\pi\gamma}{2}, k\pi\gamma + \frac{\pi\gamma}{2}]} f_1(x)dx + \frac{3}{\pi\gamma} \left( \frac{2\pi\gamma}{3}\omega(\pi\gamma/3) + \pi\gamma\omega(\pi\gamma/2) \right) \\ &\le \frac{2}{3} \int_{[-1,1]} f_1(x)dx + 3 \left( \frac{2}{3}\omega(\pi\gamma/3) + \omega(\pi\gamma/2) \right) \\ &\xrightarrow{\gamma \to 0} \frac{2}{3} \int_{[-1,1]} f_1(x)dx = \frac{2}{3}. \end{aligned}$$

Therefore, there exists $n_0$ such that for all $n \ge n_0$, $W_p(G_0(1), G_1(1)) \ge \gamma(\frac{1}{8\sqrt{2}} \wedge \frac{\pi}{12})$.

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