one arises from Lemmas 4.1 and 9.1. Now, let's move on to approximating the second term on the right-hand side of (8.26). We note that

$$
\begin{align*}
& \left| \int_{\mathbb{G}^k} \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \prod_{j=1}^k f_{\tilde{\mathbf{x}}}(\mathbf{t}_j, \mathbf{u}_j) d\mu(\mathbf{u}_j) - \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \right| \\
&= \left| \int_{\mathbb{G}^k} \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k\sigma_j < T_n} \prod_{j=1}^k d\sigma_j \operatorname{Tr} (\sigma_j^M (\mathbf{u}_j^{-1}) \sigma_j^M (\mathbf{t}_j)) \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \right. \\
& \qquad \left. \times \prod_{j=1}^k f_{\tilde{\mathbf{x}}}(\mathbf{t}_j, \mathbf{u}_j) d\mu(\mathbf{u}_j) - \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \right| \\
&= \left| \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k\sigma_j < T_n} \int_{\mathbb{G}^k} \prod_{j=1}^k d\sigma_j \operatorname{Tr} (\sigma_j^M (\mathbf{u}_j^{-1}) \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \right. \\
& \qquad \left. \times \prod_{j=1}^k f_{\tilde{\mathbf{x}}}(\mathbf{t}_j, \mathbf{u}_j) d\mu(\mathbf{u}_j) \sigma_j^M (\mathbf{t}_j) \right) - \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \right| \\
&= \left| \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k\sigma_j < T_n} \prod_{j=1}^k \int_{\mathbb{G}^k} d\sigma_j \operatorname{Tr} \left( \int_{\mathbb{G}^k} \sigma_j^M (\mathbf{u}_j^{-1}) \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \right) \right. \\
& \qquad \left. \times \prod_{j=1}^k f_{\tilde{\mathbf{x}}}(\mathbf{t}_j, \mathbf{u}_j) d\mu(\mathbf{u}_j) \sigma_j^M (\mathbf{t}_j) \right) - \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \right| \\
&= \left| \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k\sigma_j \ge T_n} \prod_{j=1}^k d\sigma_j \operatorname{Tr} (\phi^{\tilde{r}_B^{(k)}} \sigma_j^M (\mathbf{t}_j)) - \tilde{r}_B^{(k)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) \right| \\
&= \left| \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k\sigma_j \ge T_n} \prod_{j=1}^k d\sigma_j \operatorname{Tr} (\phi^{\tilde{r}_B^{(k)}} \sigma_j^M (\mathbf{t}_j)) \right| \\
&\le T_n^{-k\hat{m}} \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k\sigma_j \ge T_n} \prod_{j=1}^k d\sigma_j (T_n/k\sigma_j)^{\hat{m}} \left| \operatorname{Tr} ((-1)^{\hat{m}} d_{\sigma_j}^{\hat{m}} \phi^{\tilde{r}_B^{(k)}} \sigma_j^M (\mathbf{t}_j)) \right| \\
&\le T_n^{-k\hat{m}} \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k\sigma_j \ge T_n} \prod_{j=1}^k d\sigma_j \left| \operatorname{Tr} (\phi^{\Delta^{\hat{m}}(\tilde{r}_B^{(k)})} \sigma_j^M (\mathbf{t}_j)) \right| = O(T_n^{-k\hat{m}}), \tag{8.30}
\end{align*}
$$

where the last inequality is a consequence of (3.1.5) in [3]. By combining (8.29) and (8.30), we readily obtain the approximation of the bias. For the variance of $\tilde{r}_n^{(k)}(\varphi, \tilde{\mathbf{t}})$, we note that the function $\zeta : \mathbb{R}^d \to \mathbb{C}$ defined by $\zeta(\mathbf{u}) = \phi^K(-\mathbf{u})/\phi^U(-\mathbf{u}/h)$ is continuous on $\mathbb{R}^d$ and it vanishes outside of $B(0_d, a)$. Hence, $\zeta$ is integrable and square integrable. We also note that $(2\pi)^d K_h^D$ is the Fourier transform of $\zeta$. This with the Plancherel theorem implies that $K_h^D$ is square-integrable and
$$ \int_{\mathbb{R}^{dk}} \left| \prod_{j=1}^{k} (2\pi)^d K_h^D(\mathbf{u}_j) \right|^2 d\mathbf{u}_j = (2\pi)^{dk} \int_{\mathbb{R}^{dk}} \left| \prod_{j=1}^{k} \zeta(\mathbf{u}_j) \right|^2 d\mathbf{u}_j. $$

We have

$$
\begin{align*}
& \mathrm{Var} \left( \bar{U}_n^{(k)} \left( \Psi, T_n, \tilde{\mathbf{t}} \right) \right) \\
& \le \frac{(n-k)!}{n!h^{2dk}} \mathbb{E} \left( \left| \Psi(Y) \tilde{K}_{T_n} (\tilde{\mathbf{t}}, Z) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - Z'}{h} \right) \right|^2 \right) \\
& = \frac{(n-k)!}{n!h^{2dk}} \int_{\mathbb{G}^k \times \mathbb{R}^{dk}} \int_{\mathbb{R}^{dk}} \left| \Psi(\mathbf{s}_1, \dots, \mathbf{s}_k) \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{v}_j}{h} \right) \right|^2 \\
& \qquad \times f_{\bar{X}Y} (\mathbf{v}_j, \mathbf{u}_j, \mathbf{s}_j) d\mathbf{v}_j d\mu(\mathbf{u}_j) d\mathbf{s}_j \\
& \le \frac{(n-k)!}{n!h^{2dk}} \mathbb{M}^2 \int_{\mathbb{G}^k \times \mathbb{R}^{dk}} \left| \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{v}_j}{h} \right) \right|^2 f_{\bar{X}} (\mathbf{v}_j, \mathbf{u}_j) d\mathbf{v}_j d\mu(\mathbf{u}_j) \\
& \le \frac{(n-k)!}{n!} \mathbb{M}^2 h^{-dk} \left\| \bar{f}_{\bar{X}} \right\|_{\infty} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k (K_h^D(\mathbf{w}_j))^2 d\mathbf{w}_j \int_{\mathbb{G}^k} \left| \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \right|^2 d\mu(\mathbf{u}_j) \\
& \le \frac{(n-k)!}{n!} \frac{\mathbb{M}^2}{(2\pi)^{dk}} h^{-dk} \left\| \bar{f}_{\bar{X}} \right\|_{\infty} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k \left| \frac{\phi^{K_h^D}(\mathbf{w}_j)}{\phi^{\mathbf{u}_j}(\mathbf{w}_j/h)} \right|^2 d\mathbf{w}_j \\
& \qquad \times \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{u}_j}(\sigma_j^M)^{-1} \right\|_{\mathrm{op}}^2 \\
& = O \left( \frac{(n-k)!}{n!} h^{-dk} \prod_{j=1}^k \left( \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi^{\mathbf{u}'_j}(\mathbf{w}_j)^{-1} \right| \right)^2 \right. \\
& \qquad \left. \times \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{u}_j}(\sigma_j^M)^{-1} \right\|_{\mathrm{op}}^2 \right). \tag{8.31}
\end{align*}
$$

Combining (8.29), (8.30) and (8.31), we obtain that

$$
\begin{align}
& \bar{U}_n^{(k)} (\Psi, T_n, \tilde{\mathbf{t}}) - \bar{r}_B^{(k)} (\varphi, \tilde{\mathbf{t}}) \nonumber \\
& = O_P \left( h^m T_n^{k \dim(\mathbb{G})/4} + T_n^{-k\hat{m}} + n^{-k^2/2} h^{-dk/2} \prod_{j=1}^k \left( \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi^{\mathbf{u}'_j}(\mathbf{w}_j)^{-1} \right| \right) \right. \nonumber \\
& \qquad \left. \times \left( \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{u}_j}(\sigma_j^M)^{-1} \right\|_{\mathrm{op}}^2 \right) \right). \tag{8.32}
\end{align}
$$

Now, we remark that
$$ \tilde{\tilde{r}}_n^{(k)}(\varphi, T_n, \tilde{\tilde{\mathbf{t}}}) - r_B^{(k)}(\varphi, \tilde{\tilde{\mathbf{t}}}) = \frac{\tilde{\tilde{U}}_n^{(k)}(\Psi, T_n, \tilde{\tilde{\mathbf{t}}}) - \tilde{\tilde{r}}_B^{(k)}(\varphi, \tilde{\tilde{\mathbf{t}}})}{\tilde{\tilde{f}}_n^{(k)}(\tilde{\tilde{\mathbf{t}}})} \quad (8.33) $$

This implies that

$$ \begin{aligned} \left| \tilde{\tilde{r}}_n^{(k)}(\varphi, T_n, \tilde{\tilde{\mathbf{t}}}) - r^{(k)}(\varphi, \tilde{\tilde{\mathbf{t}}}) \right| &\le \frac{\left| \tilde{\tilde{U}}_n^{(k)}(\Psi, T_n, \tilde{\tilde{\mathbf{t}}}) - r_B^{(k)}(\varphi, \tilde{\tilde{\mathbf{t}}}) \right|}{\left| \tilde{\tilde{f}}_n^{(k)}(\tilde{\tilde{\mathbf{t}}}) \right|} \\ &\quad + \frac{\left| r_B^{(k)}(\varphi, \tilde{\tilde{\mathbf{t}}}) \right| \left| \tilde{\tilde{f}}_n^{(k)}(\tilde{\tilde{\mathbf{t}}}) - \tilde{\tilde{f}}_{\bar{\mathbf{X}}}^{(k)}(\tilde{\tilde{\mathbf{t}}}) \right|}{\left| \tilde{\tilde{f}}_n^{(k)}(\tilde{\tilde{\mathbf{t}}}) \cdot \tilde{\tilde{f}}_{\bar{\mathbf{X}}}^{(k)}(\tilde{\tilde{\mathbf{t}}}) \right|} \end{aligned} \quad (8.34) $$

From (8.32) with $\varphi \equiv 1$, we get that

$$ \begin{aligned} \bar{f}_n^{(k)}(\tilde{\tilde{\mathbf{t}}}) - \bar{f}_{\bar{\mathbf{X}}}^{(k)}(\tilde{\tilde{\mathbf{t}}}) &= \tilde{\tilde{U}}_n^{(k)}(1, T_n, \tilde{\tilde{\mathbf{t}}}) - \bar{r}_B^{(k)}(1, \tilde{\tilde{\mathbf{t}}}) \\ &= O_P \left( h^m T_n^{k \dim(\mathbb{G})/4} + T_n^{-k \hat{m}} + n^{-k^2/2} h^{-dk/2} \prod_{j=1}^k \left( \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} |\phi^{\mathbf{U}_j}(\mathbf{w}_j)^{-1}| \right) \right. \\ &\qquad \left. \times \left( \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}} : k \sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \|\phi^{\mathbf{U}_j}(\sigma_j^M)^{-1}\|_{\text{op}}^2 \right)^2 \right). \end{aligned} $$

This suffices to complete the proof for the bounded case.

### 8.3.2. Unbounded case

According to (8.12), we can see, $\bar{U}_n^{(k)}(\bar{G}_{\varphi, T_n, \tilde{\tilde{\mathbf{t}}}})$, described in (4.7), can be decomposed as follows,

$$ \bar{U}_n^{(k)}(\varphi, T_n, \tilde{\tilde{\mathbf{t}}}) = \bar{U}_n^{(k)}(\bar{G}_{\varphi, T_n, \tilde{\tilde{\mathbf{t}}}}^{(T)}) + \bar{U}_n^{(k)}(\bar{G}_{\varphi, T_n, \tilde{\tilde{\mathbf{t}}}}^{(R)}) =: \bar{U}_n^{(T)}(\varphi, T_n, \tilde{\tilde{\mathbf{t}}}) + \bar{U}_n^{(R)}(\varphi, T_n, \tilde{\tilde{\mathbf{t}}}) \quad (8.35) $$

and we can also see

$$ \begin{aligned} \bar{r}^{(k)}(\varphi, \tilde{\tilde{\mathbf{t}}}) &= \mathbb{E} \left( \varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k) \mathbb{1}_{\{\varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k) \le \gamma_n\}} \middle| (\bar{\mathbf{Z}}_1, \dots, \bar{\mathbf{Z}}_k) = \tilde{\tilde{\mathbf{t}}} \right) \bar{f}_{\bar{\mathbf{X}}}^{(\tilde{\tilde{\mathbf{t}}})} \\ &\quad + \mathbb{E} \left( \varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k) \mathbb{1}_{\{\varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k) > \gamma_n\}} \middle| (\bar{\mathbf{Z}}_1, \dots, \bar{\mathbf{Z}}_k) = \tilde{\tilde{\mathbf{t}}} \right) \bar{f}_{\bar{\mathbf{X}}}^{(\tilde{\tilde{\mathbf{t}}})} \\ &:= \bar{r}^{(T)}(\varphi, \tilde{\tilde{\mathbf{t}}}) + \bar{r}^{(R)}(\varphi, \tilde{\tilde{\mathbf{t}}}). \end{aligned} \quad (8.36) $$

Let us begin with the first term on the right side of (8.35).

#### Truncated part

By Assumption 3.6, it holds that

$$ \bar{r}^{(T)}(\varphi, \tilde{\tilde{\mathbf{t}}}) = \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}} \prod_{j=1}^k d_{\sigma_j} \operatorname{Tr} \left( \phi^{\bar{r}^{(T)}}(\sigma_j^M) \sigma_j^M(\tilde{\mathbf{t}}_j) \right). $$

Firstly, we find the rate of the bias $\mathbb{E}(\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\tilde{\mathbf{t}}}) - \bar{r}^{(T)}(\varphi, \tilde{\tilde{\mathbf{t}}})$. We have
$$
\begin{align}
& \mathbb{E} \left( \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(T)} (\varphi, \tilde{\mathbf{t}}) \right) \nonumber \\
&= \int_{\mathbb{G}^k \times \mathbb{R}^{dk}} \int_{\mathbb{R}^{dk}} \varphi^{(T)} (\mathbf{s}_1, \dots, \mathbf{s}_k) \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \prod_{j=1}^k K_h^D \nonumber \\
&\quad \times \left( \frac{\mathbf{t}'_j - \mathbf{v}_j}{h} \right) f_{\bar{X}Y} \left( (\mathbf{v}_j, \mathbf{u}_j), \mathbf{s}_j \right) d\mathbf{v}_j d\mu(\mathbf{u}_j) d\mathbf{s}_j - \bar{r}^{(T)} (\varphi, \tilde{\mathbf{t}}) \nonumber \\
&= \int_{\mathbb{G}^k \times \mathbb{R}^{dk}} \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \prod_{j=1}^k K_h^D (\mathbf{w}_j) \bar{r}^{(T)} (\varphi, (\tilde{\mathbf{t}} - h\bar{\mathbf{w}}, \tilde{\mathbf{u}})) \nonumber \\
&\quad \times (f_{\bar{X}}(\mathbf{t}_j - h\mathbf{w}_j, \mathbf{u}_j) - f_{\bar{X}}(\mathbf{t}_j, \mathbf{u}_j)) d\mathbf{w}_j d\mu(\mathbf{u}_j) \nonumber \\
&\quad + \int_{\mathbb{G}^k} \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \bar{r}^{(T)} (\varphi, (\tilde{\mathbf{v}}, \tilde{\mathbf{u}})) f_{\bar{X}}(\mathbf{t}_j, \mathbf{u}_j) d\mu(\mathbf{u}_j) - \bar{r}^{(T)} (\varphi, \tilde{\mathbf{t}}). \tag{8.37}
\end{align}
$$

By following the same strategies as for (8.26), we obtain that

$$
\mathbb{E} \left( \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) - \bar{r}^{(T)} (\varphi, \tilde{\mathbf{t}}) = O \left( h^{mT_n^{k \dim(\mathbb{G})/4}} + T_n^{-k\hat{m}} \right). \quad (8.38)
$$

Now, let's focus on approximating the variance. Making use of the condition (F.ii), we infer that

$$
\begin{align}
& \mathrm{Var} \left( \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \nonumber \\
&\le \frac{(n-k)!}{n! h^{2dk}} \mathbb{E} \left( \left| \varphi^{(T)} (\mathbb{Y}) \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z}) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbb{Z}'}{h} \right) \right|^2 \right) \nonumber \\
&= \frac{(n-k)!}{n! h^{2dk}} \int_{\mathbb{G}^k \times \mathbb{R}^{dk}} \int_{\mathbb{R}^{dk}} \left| \varphi^{(T)} (\mathbf{s}_1, \dots, \mathbf{s}_k) \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \right. \nonumber \\
&\qquad \left. \times \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{v}_j}{h} \right) \right|^2 f_{\bar{X}Y} \left( (\mathbf{v}_j, \mathbf{u}_j), \mathbf{s}_j \right) d\mathbf{v}_j d\mu(\mathbf{u}_j) d\mathbf{s}_j \nonumber \\
&\le \frac{(n-k)!}{n!} \theta_{\delta}^{2+\delta} h^{-dk} \left\| \bar{f}_{\bar{X}} \right\|_{\infty} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k (K_h^D(\mathbf{w}_j))^2 d\mathbf{w}_j \int_{\mathbb{G}^k} \left| \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \right|^2 d\mu(\mathbf{u}_j) \nonumber \\
&\le \frac{(n-k)!}{n!} \frac{\theta_{\delta}^{2+\delta}}{(2\pi)^{dk}} h^{-dk} \left\| \bar{f}_{\bar{X}} \right\|_{\infty} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k \left| \frac{\phi_{\mathbf{U}'_j}^{K_h^D}(\mathbf{w}_j)}{\phi_{\mathbf{U}'_j}(\mathbf{w}_j/h)} \right|^2 d\mathbf{w}_j \nonumber \\
&\quad \times \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k\sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi_{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\mathrm{op}}^2 \nonumber \\
&= O \left( \frac{(n-k)!}{n!} h^{-dk} \prod_{j=1}^k \left( \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi_{\mathbf{U}'_j}(\mathbf{w}_j)^{-1} \right| \right)^2 \right. \nonumber \\
&\qquad \left. \times \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k\sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi_{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\mathrm{op}}^2 \right). \tag{8.39}
\end{align}
$$
## Remainder part

Making use of Assumption 3.6, we have that

$$ \bar{r}^{(R)}(\varphi, \tilde{\mathbf{t}}) = \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}} \prod_{j=1}^{k} d_{\sigma_j} \operatorname{Tr}(\phi^{\bar{r}}(\sigma_j^M) \sigma_j^M(\tilde{\mathbf{t}}_j)). $$

For the bias $\mathbb{E}(\bar{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(R)}(\varphi, \tilde{\mathbf{t}}))$, by investigating the proof of (8.37), we may similarly show that

$$ \mathbb{E} \left( \bar{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(R)} (\tilde{\mathbf{t}}) \right) = O \left( h^m T_n^{k \dim(\mathbb{G})/4} + T_n^{-k \hat{m}} \right). \quad (8.40) $$

We turn now to approximate the variance. In the same manner, as (8.39), we can show that,

$$
\begin{align*}
& \operatorname{Var}(\bar{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}})) \\
& \le \frac{(n-k)!}{n!h^{2dk}} \mathbb{E} \left( \left| \varphi^{(R)}(\mathbb{Y}) \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbb{Z}'}{h} \right) \right|^2 \right) \\
& = \frac{(n-k)!}{n!h^{2dk}} \int_{\mathbb{G}^k \times \mathbb{R}^{dk}} \int_{\mathbb{R}^{dk}} \left| \varphi^{(R)}(\mathbf{s}_1, \dots, \mathbf{s}_k) \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbf{u}_1, \dots, \mathbf{u}_k) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{v}_j}{h} \right) \right|^2 \\
& \qquad \times f_{\tilde{\mathbf{X}}Y}(\mathbf{v}_j, \mathbf{u}_j, \mathbf{s}_j) d\mathbf{v}_j d\mu(\mathbf{u}_j) d\mathbf{s}_j \\
& \le \frac{(n-k)!}{n!} \frac{\theta_{\delta}^{2+\delta}}{(2\pi)^{dk}} h^{-dk} \left\| \tilde{f}_{\tilde{\mathbf{X}}} \right\|_{\infty} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k \left| \frac{\phi^{K_h^D}(\mathbf{w}_j)}{\phi^{\mathbf{U}'}(\mathbf{w}_j/h)} \right|^2 d\mathbf{w}_j \\
& \qquad \times \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k\sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\text{op}}^2 \\
& = O \left( \frac{(n-k)!}{n!} h^{dk} \prod_{j=1}^k \left( \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi^{\mathbf{U}'_j}(\mathbf{w}_j)^{-1} \right| \right)^2 \right. \\
& \qquad \left. \times \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k\sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\text{op}}^2 \right). \tag{8.41}
\end{align*}
$$

By combining (8.38) with (8.39), and (8.40) with (8.41), while recalling the decompositions (8.33) and (8.34), we arrive at the desired results. □

## 8.4. Proof of Theorem 4.10

We start by determining the rate of $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} |\bar{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(k)}(\varphi, \tilde{\mathbf{t}})|)$, where $\bar{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})$ is defined in (4.7). We remark that

$$
\begin{align*}
& \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \bar{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(k)} (\varphi, \tilde{\mathbf{t}}) \right| \right) \\
& \le \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \mathbb{E} \left( \bar{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(k)} (\varphi, \tilde{\mathbf{t}}) \right) \right|
\end{align*}
$$
$$+\mathbb{E}\left(\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left|\bar{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}\left(\bar{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})\right)\right|\right). \quad (8.42)$$

### 8.4.1. Bounded case

By investigating the proof of Theorem 4.9, taking into consideration the condition (F.i), we can similarly demonstrate that

$$\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \mathbb{E} \left( \bar{U}_n^{(k)} (\Psi, T_n, \tilde{\mathbf{t}}) - \bar{\tau}_B^{(k)} (\varphi, \tilde{\mathbf{t}}) \right) \right| = O(h^m T_n^{k \dim(\mathbb{G})/4} + T_n^{-k \hat{m}}). \quad (8.43)$$

Now, we find the rate of $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} |\bar{U}_n^{(k)}(\Psi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}(\bar{U}_n^{(k)}(\Psi, T_n, \tilde{\mathbf{t}}))|)$. Notice that

$$\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \bar{U}_n^{(k)}(\Psi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}\left(\bar{U}_n^{(k)}(\Psi, T_n, \tilde{\mathbf{t}})\right) \right| \\ \le \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j} \left| \mathrm{Tr} \left( \left( \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \Psi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) K_h^D \right. \right. \right. \\ \left. \left. \times \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \sigma_j^M(\mathbf{Z}_{ij}^{-1}) - \mathbb{E} \left( \Psi(\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M(\mathbb{Z}^{-1}) \right) \right) \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \sigma_j(\tilde{\mathbf{t}}) \right| \\ \le \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j} \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \sigma_j(\tilde{\mathbf{t}}) \right\|_{\mathrm{HS}} \\ \times \left\| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \Psi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \sigma_j^M(\mathbf{Z}_{ij}^{-1}) \right. \\ \left. - \mathbb{E} \left( \Psi(\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M(\mathbb{Z}^{-1}) \right) \right\|_{\mathrm{HS}} \\ \le \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^{3/2} \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\mathrm{op}} \\ \times \left\| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \Psi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \sigma_j^M(\mathbf{Z}_{ij}^{-1}) \right. \\ \left. - \mathbb{E} \left( \Psi(\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M(\mathbb{Z}^{-1}) \right) \right\|_{\mathrm{HS}},$$

where the last inequality follows from the fact that $\|AB\|_{HS} \le \|A\|_{op}\|B\|_{HS}$ for all square matrices A and B and the fact that $\sigma_j^M(\tilde{\mathbf{t}})$ is an unitary matrix. Then, we have

$$\mathbb{E}\left(\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left|\bar{U}_n^{(k)}(\Psi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}\left(\bar{U}_n^{(k)}(\Psi, T_n, \tilde{\mathbf{t}})\right)\right|\right) \\ \le \mathbb{M} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^{3/2} \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\mathrm{op}}$$
$$
\begin{align*}
& \times \left( \mathbb{E} \left( \left\| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \sigma_j^M (\mathbf{Z}_{ij}^{-1}) \right\|_{HS} \right) \right)^{1/2} \\
& - \mathbb{E} \left( K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M (\mathbf{Z}^{-1}) \right) \right)^2 \right) \right)^{1/2} \\
& \le M \frac{(n-k)!^{k/2}}{n!^{k/2}} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k\sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^{3/2} \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\text{op}} \\
& \times \left( \mathbb{E} \left( K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M (\mathbf{Z}^{-1}) \right) \right)^2 \right) \right)^{1/2} .
\end{align*}
$$

Additionally, we can see that

$$
\begin{align*}
& \prod_{j=1}^{k} \mathbb{E} \left\| \left( K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M (\mathbf{Z}^{-1}) \right) \right\|_{\mathrm{HS}}^2 \\
& \le \left\| \bar{f}_{\bar{\mathbf{X}}} \right\|_{\infty} \int_{\mathbb{R}^{dk}} \prod_{j=1}^{k} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{u}_j}{h} \right)^2 d\mathbf{u}_j \int_{\mathbb{G}^k} \prod_{j=1}^{k} \left\| \sigma_j^M (\mathbf{v}_j^{-1}) \right\|_{\mathrm{HS}}^2 d\mu(\mathbf{v}_j) \\
& = h^{dk} \left\| \bar{f}_{\bar{\mathbf{X}}} \right\|_{\infty} \int_{\mathbb{R}^{dk}} K_h^D (\mathbf{w}_j)^2 d\mathbf{w}_j \prod_{j=1}^{k} d\sigma_j \\
& = \frac{h^{dk} \left\| \bar{f}_{\bar{\mathbf{X}}} \right\|_{\infty}}{(2\pi)^{dk}} \int_{B^k(0,d,a)} \prod_{j=1}^{k} |\phi^{K_h^D}(\mathbf{w}_j)|^2 d\mathbf{w}_j \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} |\phi^{\mathbf{U}_j}(\mathbf{s}_j)^{-1}|^2 \prod_{j=1}^{k} d\sigma_j,
\end{align*}
$$

where the last equality arises from the Peter-Weyl theorem. Accordingly, we get

$$
\begin{align*}
& \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \hat{\mathbb{G}}^k} \left| \bar{U}_n^{(k)} (\Psi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} (\bar{U}_n^{(k)} (\Psi, T_n, \tilde{\mathbf{t}})) \right| \right) \\
& \le M \frac{(n-k)!^{k/2}}{n!^{k/2}} \left( \frac{h^{dk} \left\| \bar{f}_{\bar{\mathbf{X}}} \right\|_{\infty}}{(2\pi)^{dk}} \int_{B^k(0,d,a)} \prod_{j=1}^k |\phi^{K_h^D}(\mathbf{w}_j)|^2 d\mathbf{w}_j \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} |\phi^{\mathbf{U}_j}(\mathbf{w}_j)^{-1}|^2 \right)^{1/2} \\
& \times \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k\sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\text{op}} \\
& = O \left( \frac{(n-k)!^{k/2}}{n!^{k/2}} h^{dk/2} \prod_{j=1}^k \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi^{\mathbf{U}'_j}(\mathbf{w}_j)^{-1} \right| \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k\sigma_j < T_n} \right. \\
& \qquad \left. \times \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\text{op}} \right) . \tag{8.44}
\end{align*}
$$

By combining (8.43) and (8.44), we obtain
$$
\begin{align}
& \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \bar{U}_n^{(k)} (\Psi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(k)} (\Psi, \tilde{\mathbf{t}}) \right| \right) \nonumber \\
&= O \left( h^m T_n^{k \dim(\mathbb{G})/4} + T_n^{-k \hat{m}} + \frac{(n-k)!^{k/2}}{n!^{k/2}} h^{dk/2} \prod_{j=1}^k \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le a h^{-1}} \left| \phi^{\mathbf{U}_j}(\mathbf{w}_j)^{-1} \right| \right. \nonumber \\
& \qquad \left. \times \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}} : k \sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{U}_j}(\sigma_j^M)^{-1} \right\|_{\text{op}} \right). \tag{8.45}
\end{align}
$$

Likewise, for $\varphi = 1$, we can establish that the above rate is also applicable to $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} |\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \bar{f}_{\bar{\mathbf{X}}} (\tilde{\mathbf{t}})|)$. Finally, the aimed result is obtained by combining the abovementioned results while considering the decomposition (8.33) and (8.34).

### 8.4.2. Unbounded case

We will prove Theorem 4.10 under the assumption (F.ii), that is why we should truncate our U-statistics as in (8.35). Let's start by treating the truncated part $\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}})$.

#### Truncated part

We begin by finding the rate of $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} |\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(k)}(\varphi, \tilde{\mathbf{t}})|)$. Recall the decompositions (8.42) and (8.36). In a same manner as (8.43), we get that

$$ \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \mathbb{E} \left( \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(T)} (\varphi, \tilde{\mathbf{t}}) \right) \right| = O \left( h^m T_n^{k \dim(\mathbb{G})/4} + T_n^{-k \hat{m}} \right). \quad (8.46) $$

Now, we proceed to find the rate of $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} |\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}(\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}))|)$. We have

$$
\begin{align*}
& \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right| \\
&\le \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}} : k \sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j} \left| \operatorname{Tr} \left( \left( \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi^{(T)} (\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) K_h^D \right. \right. \right. \\
&\qquad \left. \left. \times \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{i_j}}{h} \right) \sigma_j^M (\mathbf{Z}_{i_j}^{-1}) - \mathbb{E} \left( \varphi^{(T)} (\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbb{Z}'}{h} \right) \sigma_j^M (\mathbb{Z}^{-1}) \right) \right) \phi^{\mathbf{U}_j} (\sigma_j^M)^{-1} \sigma_j (\tilde{\mathbf{t}}) \right| \\
&\le \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}} : k \sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j} \left\| \phi^{\mathbf{U}_j} (\sigma_j^M)^{-1} \sigma_j (\tilde{\mathbf{t}}) \right\|_{\text{HS}} \\
&\qquad \times \left\| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi^{(T)} (\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{i_j}}{h} \right) \sigma_j^M (\mathbf{Z}_{i_j}^{-1}) \right. \\
&\qquad \left. - \mathbb{E} \left( \Psi (\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbb{Z}'}{h} \right) \sigma_j^M (\mathbb{Z}^{-1}) \right) \right\|_{\text{HS}} \\
&\le \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}} : k \sigma_j < T_n} \prod_{j=1}^k d_{\sigma_j}^{3/2} \left\| \phi^{\mathbf{U}_j} (\sigma_j^M)^{-1} \right\|_{\text{op}}
\end{align*}
$$
$$ \times \left\| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi^{(T)}(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \sigma_j^M(\mathbf{Z}_{ij}^{-1}) \right. \\ \left. - \mathbb{E} \left( \Psi(\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M(\mathbf{Z}^{-1}) \right) \right\|_{\text{HS}} $$

where the last inequality stems, once more, from the property that $||AB||_{HS} \le ||A||_{op}||B||_{HS}$ for all square matrices A and B, along with the fact that $\sigma_j^M(\tilde{\mathbf{t}})$ is a unitary matrix. This yields

$$ \begin{aligned} & \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \overline{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \overline{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right| \right) \\ & \le \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^{3/2} \left\| \phi^{\mathbf{U}_j} (\sigma_j^M)^{-1} \right\|_{\text{op}} \\ & \quad \times \left( \mathbb{E} \left( \left\| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi^{(T)}(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \sigma_j^M(\mathbf{Z}_{ij}^{-1}) \right. \right. \\ & \quad \left. - \mathbb{E} \left( \varphi^{(T)}(\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M(\mathbf{Z}^{-1}) \right) \right\|_{\text{HS}} \right)^{1/2} \\ & \le \frac{(n-k)^{k/2}}{n!^{k/2}} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^{3/2} \left\| \phi^{\mathbf{U}_j} (\sigma_j^M)^{-1} \right\|_{\text{op}} \\ & \quad \times \left( \mathbb{E} \left( \left\| \varphi^{(T)}(\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M(\mathbf{Z}^{-1}) \right\|_{\text{HS}} \right)^{1/2} \right) . \end{aligned} $$

Moreover, making use of the condition (F.ii), we get

$$ \begin{aligned} & \prod_{j=1}^k \mathbb{E} \left\| \varphi^{(T)}(\mathbb{Y}) K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'}{h} \right) \sigma_j^M(\mathbf{Z}^{-1}) \right\|_{\text{HS}}^2 \\ & \le \int_{\mathbb{R}^{dk}} \int_{\mathbb{R}^{dk} \times \mathbb{G}^k} \left( \varphi^{(T)}(\mathbf{s}_1, \dots, \mathbf{s}_k) \right)^2 \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{u}_j}{h} \right)^2 \\ & \quad \times \left\| \sigma_j^M(\mathbf{v}_j^{-1}) \right\|_{\text{HS}}^2 \bar{f}_{\bar{X}Y}((\mathbf{v}_j, \mathbf{u}_j), \mathbf{s}_j) d\mathbf{u}_j d\mu(\mathbf{v}_j) d\mathbf{s}_j \\ & \le \theta_{\delta}^{\frac{2+\delta}{2}} \left\| \bar{f}_{\bar{X}} \right\|_{\infty} h^{-dk} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k K_h^D(\mathbf{w}_j)^2 d\mathbf{w}_j \prod_{j=1}^k d\sigma_j \\ & = \frac{\theta_{\delta}^{\frac{2+\delta}{2}} \left\| \bar{f}_{\bar{X}} \right\|_{\infty} h^{-dk}}{(2\pi)^{dk}} \int_{B^k(0,d,a)} \prod_{j=1}^k |\phi_h^{K_h^D}(\mathbf{w}_j)|^2 d\mathbf{w}_j \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} |\phi^{\mathbf{U}_j}(\mathbf{s}_j)^{-1}|^2 \prod_{j=1}^k d\sigma_j, \end{aligned} $$

where the last equality is derived from the Peter-Weyl theorem. Thus, we obtain
$$
\begin{align}
& \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right| \right) \nonumber \\
& \le \frac{(n-k)!^{k/2}}{n!^{k/2}} \left( \frac{\theta_{\delta}^{2+\delta} \| \bar{J}_{\bar{X}} \|_{\infty}}{(2\pi)^{dk}} \int_{B^k(0,d)} \prod_{j=1}^k |\phi_{\mathbf{w}_j}^{kD}(\mathbf{w}_j)|^2 d\mathbf{w}_j \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} |\phi_{\mathbf{U}_j}(\mathbf{w}_j)^{-1}|^2 \right)^{1/2} \nonumber \\
& \quad \times \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \| \phi_{\mathbf{U}_j}(\sigma_j^M)^{-1} \|_{\text{op}} \nonumber \\
& = O \left( \frac{(n-k)!^{k/2}}{n!^{k/2}} h^{-dk/2} \prod_{j=1}^k \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} |\phi_{\mathbf{U}_j}(\mathbf{w}_j)^{-1}| \right) \nonumber \\
& \quad \times \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \| \phi_{\mathbf{U}_j}(\sigma_j^M)^{-1} \|_{\text{op}}. \tag{8.47}
\end{align}
$$

Combining (8.46) and (8.47), we derive

$$
\begin{align}
& \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \bar{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(T)} (\varphi, \tilde{\mathbf{t}}) \right| \right) \nonumber \\
& = O \left( h^m T_n^{k \dim(\mathbb{G})/4} + T_n^{-k \hat{m}} + \frac{(n-k)!^{k/2}}{n!^{k/2}} h^{-dk/2} \prod_{j=1}^k \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} |\phi_{\mathbf{U}_j}(\mathbf{w}_j)^{-1}| \right) \nonumber \\
& \quad \times \sum_{\sigma_1, \dots, \sigma_k \in \mathbb{G}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \| \phi_{\mathbf{U}_j}(\sigma_j^M)^{-1} \|_{\text{op}}. \tag{8.48}
\end{align}
$$

## Remainder part

For the bias term, similarly to (8.46), we obtain

$$
\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \mathbb{E} \left( \bar{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(R)} (\varphi, \tilde{\mathbf{t}}) \right) \right| = O \left( h^m T_n^{k \dim(\mathbb{G})/4} + T_n^{-k \hat{m}} \right).
$$

For the term $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} |\bar{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}})| - \mathbb{E}(\bar{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}})))$, by following the same steps as in (8.48), we get the desired result. Similarly to the bounded case, we get the aimed results by combining the above results while considering the decomposition (8.33) and (8.34). Hence, the proof of the Theorem is complete. □

## 8.5. Proof of Theorem 4.11

First of all, we find the rate of $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{T}^{qk}} |\bar{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(k)}(\varphi, \tilde{\mathbf{t}})|)$. We will start with the bounded case.

### 8.5.1. Bounded case

Makin use of the condition (F.i) and by following the steps of the proof of Theorem 4.9, we may similarly show that

$$
\sup_{\tilde{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \mathbb{E} \left( \bar{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(k)} (\varphi, \tilde{\mathbf{t}}) \right) \right| = O \left( h^m T_n^{kq/4} + T_n^{-k\hat{m}} \right).
$$
Now, we find the rate of $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{T}^{qk}} |\bar{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}(\bar{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}))|)$. Notice that

$$
\begin{align}
& \phi \bar{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) (\sigma \tilde{\mathbf{w}} \otimes \sigma \tilde{\mathbf{v}})^M \nonumber \\
&= \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi (\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \prod_{j=1}^k \int e^{\mathbf{t}'(\mathbf{w}_j, \mathbf{t}'_j)'} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{i_j}}{h} \right) d\mathbf{t}'_j \nonumber \\
&\quad \times \int_{\mathbb{T}^{qk}} \prod_{j=1}^k \sigma_{i_j}^M (\mathbf{t}_j^{-1}) \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{Z}_{i_1}, \dots, \mathbf{Z}_{i_k}) d\mu(\mathbf{t}_j) \nonumber \\
&= \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi (\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \left( \prod_{j=1}^k e^{\mathbf{t}'(\mathbf{w}_j, \mathbf{Z}'_{i_j})} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k e^{\mathbf{t}'(\mathbf{w}_j, h\mathbf{s}_j)} K_h^D(\mathbf{s}_j) d\mathbf{s}_j \right) \nonumber \\
&\quad \times \sum_{v_1', \dots, v_k' \in \mathbb{Z}^q : \|v_j'\|^2 < T_n} \prod_{j=1}^k \sigma_{v_j'}^M (\mathbf{Z}_{i_j}^{-1}) \varphi^{k_h^D}(\mathbf{w}_j h) (\varphi^{U'}(\mathbf{w}_j))^{-1} \phi^{U_j} \left( (\sigma_{v_j'}^M)^{-1} \right) \mathbf{1}\left\{ \|v_j'\|^2 < T_n \right\} \frac{(n-k)!}{n!} \nonumber \\
&= \prod_{j=1}^k \phi^{k_h^D}(\mathbf{w}_j h) (\varphi^{U'}(\mathbf{w}_j))^{-1} \phi^{U_j} \left( (\sigma_{v_j'}^M)^{-1} \right) \mathbf{1}\left\{ \|v_j'\|^2 < T_n \right\} \frac{(n-k)!}{n!} \nonumber \\
&\quad \times \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi (\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \prod_{j=1}^k e^{\mathbf{t}'(\mathbf{w}_j, \mathbf{Z}'_{i_j})} \sigma_{v_j'}^M (\mathbf{Z}_{i_j}^{-1}), \tag{8.49}
\end{align}
$$

where the final equality is derived from the Fourier inversion formula and the Peter-Weyl theorem.
Making use of the Theorem 4.33 in [54], we get that

$$
\begin{align*}
& \overline{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}(\overline{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})) \\
&= \sum_{v_1, \dots, v_k \in \mathbb{Z}^q} (2\pi)^{-dk} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} e^{\mathbf{t}'(\mathbf{w}_j, \mathbf{t}'_j)} \sigma_{v_j}^M(\mathbf{t}_j) \left( \phi \overline{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) \left( (\sigma \tilde{\mathbf{w}} \otimes \sigma \tilde{\mathbf{v}})^M \right) \right. \\
& \qquad \left. - \mathbb{E} \left[ \phi \overline{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) \left( (\sigma \tilde{\mathbf{w}} \otimes \sigma \tilde{\mathbf{v}})^M \right) \right] \right) d\mathbf{w}_j.
\end{align*}
$$

Therefore, taking into consideration the condition (F.i), we have,

$$
\begin{align*}
& \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \bar{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \bar{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right| \right) \\
&\le (2\pi)^{-dk} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} \mathbb{E} \left[ \left( \phi \bar{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) \left( (\sigma \tilde{\mathbf{w}} \otimes \sigma \tilde{\mathbf{v}})^M \right) \right. \right. \\
&\qquad \left. - \mathbb{E} \left[ \phi \bar{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) \left( (\sigma \tilde{\mathbf{w}} \otimes \sigma \tilde{\mathbf{v}})^M \right) \right] \right] d\mathbf{w}_j \\
&\le (2\pi)^{-dk} \mathbb{M} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q : \|v_j\|^2 < T_n} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} \left| \phi^{K_h^D}(\mathbf{w}_j h) (\phi^{U'}(\mathbf{w}_j))^{-1} \phi^{U_j} \left( (\sigma_{v_j}^M)^{-1} \right) \right| d\mu(\mathbf{w}_j)
\end{align*}
$$
$$
\begin{align}
& \times \mathbb{E} \left[ \left| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \prod_{j=1}^k e^{i \langle \mathbf{w}_j, \mathbf{Z}'_{ij} \rangle} \sigma_{\nu_j}^M (\mathbf{Z}_{ij}^{-1}) - \mathbb{E} \left[ \prod_{j=1}^k e^{i \langle \mathbf{w}_j, \mathbf{Z}' \rangle} \sigma_{\nu_j}^M (\mathbf{Z}^{-1}) \right] \right| \right] d\mathbf{w}_j \nonumber \\
& \le (2\pi)^{-dk} \mathbb{M} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q : \|\nu_j\|^2 < T_n} \prod_{j=1}^k \left| \phi_{\mathbf{U}_j} \left( (\sigma_{\nu_j}^M)^{-1} \right) \right| \prod_{j=1}^k \int_{\mathbb{R}^{dk}} \left| \phi_{\mathbf{U}_j}^{K_h^D} (\mathbf{w}_j h) (\phi^{\mathcal{U}'} (\mathbf{w}_j))^{-1} \right| d\mathbf{w}_j \nonumber \\
& \times \mathbb{E} \left[ \left| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \prod_{j=1}^k e^{i \langle \mathbf{w}_j, \mathbf{Z}'_{ij} \rangle} \sigma_{\nu_j}^M (\mathbf{Z}_{ij}^{-1}) - \mathbb{E} \left[ \prod_{j=1}^k e^{i \langle \mathbf{w}_j, \mathbf{Z}' \rangle} \sigma_{\nu_j}^M (\mathbf{Z}^{-1}) \right] \right|^2 \right]^{1/2} d\mathbf{w}_j \nonumber \\
& \le (2\pi)^{-dk} \mathbb{M} \frac{(n-k)!^{k/2}}{n!^{k/2}} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q : \|\nu_j\|^2 < T_n} \prod_{j=1}^k \left| \phi_{\mathbf{U}_j} \left( (\sigma_{\nu_j}^M)^{-1} \right) \right| \prod_{j=1}^k \nonumber \\
& \times \int_{\mathbb{R}^{dk}} \left| \phi_{\mathbf{U}_j}^{K_h^D} (\mathbf{w}_j h) (\phi^{\mathcal{U}'} (\mathbf{w}_j))^{-1} \right| \mathbb{E} \left[ \left| \prod_{j=1}^k e^{i \langle \mathbf{w}_j, \mathbf{Z}' \rangle} \sigma_{\nu_j}^M (\mathbf{Z}^{-1}) \right|^2 \right]^{1/2} d\mathbf{w}_j \nonumber \\
&= O \left( \frac{(n-k)!^{k/2}}{h^{dk} n!^{k/2}} \prod_{j=1}^k \left( \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi_{\mathbf{U}_j} (\mathbf{w}_j)^{-1} \right| \right) \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q : \|\nu_j\|^2 < T_n} \prod_{j=1}^k \left| \phi_{\mathbf{U}_j} \left( (\sigma_{\nu_j}^M)^{-1} \right) \right| \right). \tag{8.50}
\end{align}
$$

Hence, the proof for the bounded case is complete.

### 8.5.2. Unbounded case

Similarly to the proof of the Theorem 4.10 in the unbounded case, we must truncate our U-statistics as shown in (8.35). We will begin by training the truncated part. First, we determine the rate of $\mathbb{E}(\sup_{\tilde{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{T}^{qk}} |\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}})| - \mathbb{E}(\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}})))$. According to (8.49), we have

$$
\begin{align*}
& \phi^{\bar{U}_n^{(T)}}(\varphi, T_n, \tilde{\mathbf{t}}) (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \\
&= \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi^{(T)}(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \prod_{j=1}^k \int e^{i \langle \mathbf{w}_j, \mathbf{t}'_j \rangle} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{i_j}}{h} \right) d\mathbf{t}'_j \\
& \times \int_{\mathbb{T}^{qk}} \prod_{j=1}^k \sigma_{\mathbf{t}'_j}^M (\mathbf{t}'_j^{-1}) \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbf{Z}_{i_1}, \dots, \mathbf{Z}_{i_k}) d\mu(\mathbf{t}_j) \\
&= \prod_{j=1}^k \phi_{\mathbf{U}_j}^{K_h^D}(\mathbf{w}_j h) (\phi^{\mathcal{U}'}(\mathbf{w}_j))^{-1} \phi_{\mathbf{U}_j} \left( (\sigma_{\nu_j}^M)^{-1} \right) \mathbb{1}\left\{ \|\nu_j'\|^2 < T_n \right\} \frac{(n-k)!}{n!} \\
& \times \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi^{(T)}(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \prod_{j=1}^k e^{i \langle \mathbf{w}_j, \mathbf{Z}'_{ij} \rangle} \sigma_{\nu_j}^M (\mathbf{Z}_{ij}^{-1}).
\end{align*}
$$

By applying Theorem 4.33 from [54], we obtain that

$$
\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}(\bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}))
$$
$$ = \sum_{v_1, \dots, v_k \in \mathbb{Z}^q} (2\pi)^{-dk} \prod_{j=1}^{k} \int_{\mathbb{R}^{dk}} e^{\mathbf{t}^\top (\mathbf{w}_j, \mathbf{t}_j')} \sigma_{v_j}^{M}(\mathbf{t}_j) \left( \phi_{\bar{U}_n}^{(T)}(\varphi, T_n, \bar{\mathbf{t}}) \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) - \mathbb{E} \left[ \phi_{\bar{U}_n}^{(T)}(\varphi, T_n, \bar{\mathbf{t}}) \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right] \right) d\mathbf{w}_j. $$

Therefore, taking advantage of the condition (F.ii), we get

$$
\begin{align*}
& \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \bar{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) \right) \right| \right) \\
& \le (2\pi)^{-dk} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q} \prod_{j=1}^{k} \int_{\mathbb{R}^{dk}} \mathbb{E} \left[ \left( \phi_{\bar{U}_n}^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) - \mathbb{E} \left[ \phi_{\bar{U}_n}^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right] \right) \right] d\mathbf{w}_j \\
& \le (2\pi)^{-dk} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q; \|v_j\|^2 < T_n} \prod_{j=1}^{k} \left| \phi_{\mathbf{U}_j} \left( (\sigma_{v_j}^M)^{-1} \right) \right| \prod_{j=1}^{k} \int_{\mathbb{R}^{dk}} \left| \phi_{K_h^D}(\mathbf{w}_j h) \left( \phi_{\mathbf{U}_j}(\mathbf{w}_j) \right)^{-1} \right| \\
& \quad \times \mathbb{E} \left[ \left| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_h^k} \varphi^{(T)}(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \prod_{j=1}^{k} e^{\mathbf{t}^\top (\mathbf{w}_j, \mathbf{Z}'_j)} \sigma_{v_j}^M(\mathbf{Z}^{-1}) \right|^2 \right]^{1/2} d\mathbf{w}_j \\
& \le (2\pi)^{-dk} \frac{(n-k)^{k/2}}{n!^{k/2}} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q; \|v_j\|^2 < T_n} \prod_{j=1}^{k} \left| \phi_{\mathbf{U}_j} \left( (\sigma_{v_j}^M)^{-1} \right) \right| \\
& \quad \times \prod_{j=1}^{k} \int_{\mathbb{R}^{dk}} \left| \phi_{K_h^D}(\mathbf{w}_j h) \left( \phi_{\mathbf{U}_j}(\mathbf{w}_j) \right)^{-1} \right| \mathbb{E} \left[ \left| \varphi^{(T)}(\mathbf{Y}) \prod_{j=1}^{k} e^{\mathbf{t}^\top (\mathbf{w}_j, \mathbf{Z}'_j)} \sigma_{v_j}^M(\mathbf{Z}^{-1}) \right|^2 \right]^{1/2} d\mathbf{w}_j \\
& \le \frac{\theta_{\delta}^{\frac{1}{2+\delta}}}{(2\pi)^{dk}} \frac{(n-k)!^{k/2}}{n!^{k/2}} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q; \|v_j\|^2 < T_n} \prod_{j=1}^{k} \left| \phi_{\mathbf{U}_j} \left( (\sigma_{v_j}^M)^{-1} \right) \right| \\
& \quad \times \prod_{j=1}^{k} \int_{\mathbb{R}^{dk}} \left| \phi_{K_h^D}(\mathbf{w}_j h) \left( \phi_{\mathbf{U}_j}(\mathbf{w}_j) \right)^{-1} \right| \mathbb{E} \left[ \left| \prod_{j=1}^{k} e^{\mathbf{t}^\top (\mathbf{w}_j, \mathbf{Z}'_j)} \sigma_{v_j}^M(\mathbf{Z}^{-1}) \right|^2 \right]^{1/2} d\mathbf{w}_j \\
& = o \left( \frac{(n-k)!^{k/2}}{h^{dk} n!^{k/2}} \prod_{j=1}^{k} \left( \sup_{\mathbf{w}_j: \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi_{\mathbf{U}_j}(\mathbf{w}_j)^{-1} \right| \right) \sum_{v_1, \dots, v_k \in \mathbb{Z}^q; \|v_j\|^2 < T_n} \prod_{j=1}^{k} \left| \phi_{\mathbf{U}_j} \left( (\sigma_{v_j}^M)^{-1} \right) \right| \right).
\end{align*}
$$

For the remainder part, by following the same steps as in the truncated part, we conclude that

$$ \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \bar{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \bar{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}}) \right) \right| \right) $$
$$= O \left( \frac{(n-k)^{k/2}}{\hbar^{dk} n^{k/2}} \prod_{j=1}^{k} \left( \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi^{\mathbf{U}_j}(\mathbf{w}_j)^{-1} \right| \right) \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q : \|\nu_j\|^2 < T_n} \prod_{j=1}^{k} \left| \phi^{\mathbf{U}_j} \left( (\sigma_{\nu_j}^M)^{-1} \right) \right| \right).$$

Thus, the proof of the theorem is finished. □

## 8.6. Proof of Theorem 4.13

We start by determining the rate of $\int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \mathbb{E} (|\overline{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 d\text{Leb}_d \otimes \mu(\mathbf{t}_1) \cdots d\text{Leb}_d \otimes \mu(\mathbf{t}_k))$. It suffices to find the rate of $\mathbb{E} (\int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} |\overline{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 d\text{Leb}_d \otimes \mu(\mathbf{t}_1) \cdots d\text{Leb}_d \otimes \mu(\mathbf{t}_k))$.

### 8.6.1. Bounded case

According to Theorem 4.26 in [54] and using the condition (F.i), we infer that

$$\begin{align*}
& \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \overline{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \tilde{r}^{(k)} (\varphi, \tilde{\mathbf{t}}) \right|^2 d\text{Leb}_d \otimes \mu(\mathbf{t}_1) \cdots d\text{Leb}_d \otimes \mu(\mathbf{t}_k) \right) \\
& \le M \cdot \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \hat{f}_n^{(k)} (T_n, \tilde{\mathbf{t}}) - \bar{f}_{\overline{\mathbb{X}}} (\tilde{\mathbf{t}}) \right|^2 d\text{Leb}_d \otimes \mu(\mathbf{t}_1) \cdots d\text{Leb}_d \otimes \mu(\mathbf{t}_k) \right) \\
& = \frac{M}{(2\pi)^{dk}} \cdot \mathbb{E} \left( \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} \left| \phi^{\hat{f}_n^{(k)}(T_n, \tilde{\mathbf{t}})} (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right. \right. \\
& \qquad \left. - \phi^{\bar{f}_{\overline{\mathbb{X}}} (\tilde{\mathbf{t}})} (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k \\
& = \frac{M}{(2\pi)^{dk}} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} \text{Var} \left( \phi^{\hat{f}_n^{(k)}(T_n, \tilde{\mathbf{t}})} (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) d\mathbf{w}_1 \cdots d\mathbf{w}_k \\
& \qquad + \frac{M}{(2\pi)^{dk}} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} \left| \mathbb{E} \left( \phi^{\hat{f}_n^{(k)}(T_n, \tilde{\mathbf{t}})} (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right. \\
& \qquad \left. - \phi^{\overline{\mathbb{X}}} (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k,
\end{align*}$$

where $\hat{f}_n^{(k)}(T_n, \tilde{\mathbf{t}})$ is represented in (4.5). Notice that

$$\begin{align*}
& \mathbb{E} \left( \phi^{\hat{f}_n^{(k)}(T_n, \tilde{\mathbf{t}})} (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \\
&= \prod_{j=1}^{k} \phi^{kD_h} (\mathbf{w}_j h) (\phi^{\mathbf{U}_j}(\mathbf{w}_j))^{-1} \phi^{\mathbf{U}_j} \left( (\sigma_{\nu_j}^M)^{-1} \right) \mathbb{1}\left\{ \|\nu_j\|^2 < T_n \right\} \\
& \times \mathbb{E} \left( \mathbb{E} \left( \prod_{j=1}^{k} e^{i \langle \mathbf{w}_j, \mathbf{Z}' \rangle} \sigma_{\nu_j}^M (\mathbb{Z}^{-1}) |\overline{\mathbb{X}} \right) \right)
\end{align*}$$
$$
\begin{align*}
&= \prod_{j=1}^{k} \phi_h^{K_h^D}(\mathbf{w}_j h) (\phi_{\mathbf{U}_j}^{\mathbf{U}_j}(\mathbf{w}_j))^{-1} \phi_{\mathbf{U}_j} \left( (\sigma_{\nu_j}^M)^{-1} \right) \mathbb{1}\left\{ \|\nu_j\|^2 < T_n \right\} \\
&\quad \times \mathbb{E} \left( \prod_{j=1}^{k} e^{\iota \langle \mathbf{w}_j, \mathbf{X}^{\iota} \rangle} \sigma_{\nu_j}^M (\mathbf{X}^{-1}) \right) \mathbb{E} \left( \prod_{j=1}^{k} e^{\iota \langle \mathbf{w}_j, \mathbf{U}_j^{\iota} \rangle} \sigma_{\nu_j}^M (\mathbf{U}_j^{-1}) \right) \\
&= \prod_{j=1}^{k} \phi_h^{K_h^D}(\mathbf{w}_j h) \mathbb{1}\left\{ \|\nu_j\|^2 < T_n \right\} \phi_{\bar{\mathbf{X}}} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right),
\end{align*}
$$

where the second equality follows from the fact that $\overline{\mathbb{U}} \perp \overline{\mathbb{X}}$ and the last equality follows from $\mathbb{U} \perp \mathbb{U}'$. Therefore, we get

$$
\begin{align}
& \frac{\mathbb{M}}{(2\pi)^{dk}} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q} \prod_{j=1}^{k} \int_{\mathbb{R}^{dk}} \left| \mathbb{E} \left( \phi_{\hat{J}_n^{(k)}(T_n, \bar{\mathbf{t}})} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) - \phi_{\bar{\mathbf{X}}} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right) \right|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
&= \frac{\mathbb{M}}{(2\pi)^{dk}} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q; \|\nu_j\|^2 < T_n} \prod_{j=1}^{k} \int \left| \phi_{\bar{\mathbf{X}}} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right|^2 \left| \phi_h^{K_h^D}(\mathbf{w}_j h) - 1 \right|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
&\quad + \frac{\mathbb{M}}{(2\pi)^{dk}} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q; \|\nu_j\|^2 \ge T_n} \prod_{j=1}^{k} \int \left| \phi_{\bar{\mathbf{X}}} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k. \tag{8.51}
\end{align}
$$

For the first term on the right hand of (8.51), we can see that

$$
\begin{align}
& \frac{\mathbb{M}}{(2\pi)^{dk}} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q; \|\nu_j\|^2 < T_n} \prod_{j=1}^{k} \int \left| \phi_{\bar{\mathbf{X}}} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right|^2 \left| \phi_h^{K_h^D} (\mathbf{w}_j h) - 1 \right|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
&\le \frac{\mathbb{M}}{(2\pi)^{dk}} \prod_{j=1}^{k} \sup_{\mathbf{w}_j \neq \mathbf{0}_d} \left( \|\mathbf{w}_j\|^{-2m} \left| \phi_h^{K_h^D} (\mathbf{w}_j h) - 1 \right|^2 \right) \nonumber \\
&\quad \times \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q; \|\nu_j\|^2 < T_n} \prod_{j=1}^{k} \int \left| \phi_{\bar{\mathbf{X}}} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right|^2 \|\mathbf{w}_j\|^{2m} d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
&= \frac{\mathbb{M} h^{2mk}}{(2\pi)^{dk}} \prod_{j=1}^{k} \sup_{\mathbf{w}_j \neq \mathbf{0}_d} \left( \|\mathbf{w}_j\|^{-2m} \left| \phi_h^{K_h^D} (\mathbf{w}_j) - 1 \right|^2 \right) \nonumber \\
&\quad \times \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q; \|\nu_j\|^2 < T_n} \prod_{j=1}^{k} \int \left| \phi_{\bar{\mathbf{X}}} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right|^2 \|\mathbf{w}_j\|^{2m} d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
&= O(h^{2mk}). \tag{8.52}
\end{align}
$$

For the second term on the right hand of (8.51), we have

$$
\frac{\mathbb{M}}{(2\pi)^{dk}} \sum_{\nu_1, \dots, \nu_k \in \mathbb{Z}^q; \|\nu_j\|^2 \ge T_n} \prod_{j=1}^{k} \int \left| \phi_{\bar{\mathbf{X}}} \left( (\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M \right) \right|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k
$$
$$
\begin{align}
&= \frac{\mathbb{M} T_n^{-2k\hat{m}}}{(2\pi)^{dk}} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q : \|v_j\|^2 \ge T_n} \prod_{j=1}^k \frac{T_n^{2\hat{m}}}{\|v_j\|^{2\hat{m}}} \int_{\mathbb{R}^{dk}} |(-1)^{\hat{m}} \|v_j\|^{4\hat{m}} \phi_{\bar{X}} ((\sigma_{\tilde{w}} \otimes \sigma_{\tilde{v}})^M)|^2 \nonumber \\
&\quad \times d\mathbf{w}_1 \cdots d\mathbf{w}_k \\
&= \frac{\mathbb{M} T_n^{-2k\hat{m}}}{(2\pi)^{dk}} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q : \|v_j\|^2 \ge T_n} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} |\phi^{\Delta\hat{m}}(\bar{f}_{\bar{X}}) ((\sigma_{\tilde{w}} \otimes \sigma_{\tilde{v}})^M)|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
&\le \mathbb{M} T_n^{-2k\hat{m}} \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \prod_{j=1}^k |\Delta^{\hat{m}}(\bar{f}_{\bar{X}})(\tilde{\mathbf{t}}_j)|^2 d\mathrm{Leb}_d \otimes \mu(\tilde{\mathbf{t}}_1) \cdots d\mathrm{Leb}_d \otimes \mu(\tilde{\mathbf{t}}_k) \nonumber \\
&= o\left(T_n^{-2k\hat{m}}\right). \tag{8.53}
\end{align}
$$

On the other side, we infer that

$$
\begin{align}
& \frac{\mathbb{M}}{(2\pi)^{dk}} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} \mathrm{Var} \left( \phi^{\hat{f}_n^{(k)}(T_n, \bar{\mathbf{t}})} \left( (\sigma_{\tilde{w}} \otimes \sigma_{\tilde{v}})^M \right) \right) d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
& \le \frac{\mathbb{M}(n-k)!}{(2\pi)^{dk}n!} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q} \prod_{j=1}^k \int \nonumber \\
& \qquad \times \mathrm{E} \left( \left| \phi^{K_h^D}(\mathbf{w}_j) e^{\mathrm{i}(\mathbf{w}_j, \mathbb{Z})} (\phi^{\mathbf{U}_j}(\mathbf{w}_j))^{-1} \sigma_{v_j}^M (\mathbb{Z}^{-1}) \phi^{\mathbf{U}_j} \left( \left( \sigma_{v_j}^M \right)^{-1} \right) \right|^2 \right) d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
& \le \frac{\mathbb{M}(n-k)!}{(2\pi)^{dk}n!} \sum_{v_1, \dots, v_k \in \mathbb{Z}^q} \prod_{j=1}^k \left| \phi^{\mathbf{U}_j} \left( \left( \sigma_{v_j}^M \right)^{-1} \right) \right|^2 \int_{\mathbb{R}^{dk}} \left| \phi^{K_h^D}(\mathbf{w}_j) (\phi^{\mathbf{U}_j}(\mathbf{w}_j))^{-1} \right|^2 d\mathbf{w}_1 \cdots d\mathbf{w}_k \nonumber \\
& = o \left( \frac{(n-k)!}{n!} h^{-dk} \prod_{j=1}^k \left( \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi^{\mathbf{U}_j}(\mathbf{w}_j)^{-1} \right|^2 \right) \right. \nonumber \\
& \qquad \left. \times \sum_{v_1, \dots, v_k \in \mathbb{Z}^q : \|v_j\|^2 < T_n} \prod_{j=1}^k \left| \phi^{\mathbf{U}_j} \left( \left( \sigma_{v_j}^M \right)^{-1} \right) \right|^2 \right). \tag{8.54}
\end{align}
$$

8.6.2. *Unbounded case*

Notice that

$$
\begin{align*}
& \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \bar{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \bar{r}^{(k)} (\varphi, \tilde{\mathbf{t}}) \bar{f}_{\bar{X}} (\tilde{\mathbf{t}}) \right|^2 d\mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d\mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right) \\
&= \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in I_n^k} \varphi(Y_{\mathbf{i}}) \tilde{K}_{T_n}(\tilde{\mathbf{t}}, Z_{\mathbf{i}}) \prod_{j=1}^k K_h^D \left( \frac{\tilde{\mathbf{t}}_j - Z_{\mathbf{i}_j}'}{h} \right) - \bar{r}^{(k)} (\varphi, \tilde{\mathbf{t}}) \bar{f}_{\bar{X}} (\tilde{\mathbf{t}}) \right|^2 \right. \\
& \qquad \left. \times d\mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d\mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right)
\end{align*}
$$
$$
\begin{align}
&\le 2\mathbb{E}\left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in \mathcal{I}_h^k} r^{(k)}(\varphi, \overline{\mathbb{Z}}_i) \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_i) \prod_{j=1}^k K_h^D\left(\frac{\mathbf{t}'_j - \mathbf{Z}'_{i_j}}{h}\right) - \bar{r}^{(T)}(\varphi, \tilde{\mathbf{t}}) \bar{f}_{\overline{\mathbb{X}}}(\tilde{\mathbf{t}}) \right|^2 \right. \nonumber \\
&\qquad \left. \times d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right) \nonumber \\
&\quad + 2\mathbb{E}\left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in \mathcal{I}_n^k} (\varphi(\mathbb{Y}_i) - \bar{r}^{(k)}(\varphi, \overline{\mathbb{Z}}_i)) \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_i) \prod_{j=1}^k K_h^D\left(\frac{\mathbf{t}'_j - \mathbf{Z}'_{i_j}}{h}\right) \right|^2 \right. \nonumber \\
&\qquad \left. \times d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right) \nonumber \\
&=: 2(\mathfrak{B}_{1,n} + \mathfrak{B}_{2,n}). \tag{8.55}
\end{align}
$$

Let us begin with the second term on the right-hand side of (8.55). Making use of the condition (F.ii),
we infer that

$$
\begin{align*}
\mathfrak{B}_{2,n} &= \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in \mathcal{I}_n^k} (\varphi(\mathbb{Y}_i) - \bar{r}^{(k)}(\varphi, \overline{\mathbb{Z}_i})) \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_i) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{i_j}}{h} \right) \right|^2 \right. \\
&\qquad \left. \times d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right) \\
&\le \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \frac{(n-k)!}{n!} \mathbb{E} \left| \varphi(\mathbb{Y}) \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_j}{h} \right) \right|^2 \\
&\qquad \times d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \\
&\le \frac{(n-k)!}{n!} \theta_{\delta}^{\frac{2}{2+\delta}} \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left( \int_{\mathbb{T}^{qk} \times \mathbb{R}^{dk}} \left| \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \tilde{\mathbf{u}}) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{v}_j}{h} \right) \right|^2 f_{\bar{\mathbf{X}}}(\mathbf{v}_j, \mathbf{u}_j) d\mathbf{v}_j d\mu(\mathbf{u}_j) \right) \\
&\qquad \times d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \\
&\le \frac{(n-k)!}{n!} \theta_{\delta}^{\frac{2}{2+\delta}} h^{-dk} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k \left| \frac{\phi_{\mathbf{u}_j}^{K_h^D}(\mathbf{w}_j)}{\phi^{\mathbf{u}_j}(\mathbf{w}_j/h)} \right|^2 d\mathbf{w}_j \sum_{\sigma_1, \dots, \sigma_k \in \mathcal{G}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{u}_j}(\sigma_j^M) - 1 \right\|_{\mathrm{op}}^2 \\
&\qquad \times \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \prod_{j=1}^k f_{\bar{\mathbf{X}}}(\mathbf{t}'_j, \mathbf{u}_j) d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \\
&= \frac{(n-k)!}{n!} \theta_{\delta}^{\frac{2}{2+\delta}} h^{-dk} \int_{\mathbb{R}^{dk}} \prod_{j=1}^k \left| \frac{\phi_{\mathbf{u}_j}^{K_h^D}(\mathbf{w}_j)}{\phi^{\mathbf{u}_j}(\mathbf{w}_j/h)} \right|^2 d\mathbf{w}_j \sum_{\sigma_1, \dots, \sigma_k \in \mathcal{G}: k_{\sigma_j} < T_n} \prod_{j=1}^k d_{\sigma_j}^2 \left\| \phi^{\mathbf{u}_j}(\sigma_j^M) - 1 \right\|_{\mathrm{op}}^2 \\
&\qquad \times \sum_{\mathbf{v}_1, \dots, \mathbf{v}_k \in \mathbb{Z}: \|\mathbf{v}_j\|^2 < T_n} \prod_{j=1}^k \int_{\mathbb{R}^{dk}} \phi^{\overline{\mathbb{X}}}((\sigma_{\tilde{\mathbf{w}}} \otimes \sigma_{\tilde{\mathbf{v}}})^M) d\mathbf{w}_1 \cdots d\mathbf{w}_k
\end{align*}
$$
$$ = O \left( \frac{(n-k)!}{n!} h^{-dk} \prod_{j=1}^{k} \left( \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi^{\mathbf{U}_j}(\mathbf{w}_j)^{-1} \right| \right) \right. \\ \left. \times \sum_{v_1, \dots, v_k \in \mathbb{Z}^q : \|v_j\|^2 < T_n} \prod_{j=1}^{k} \left| \phi^{\mathbf{U}_j} \left( \left( \sigma_{v_j}^{M} \right)^{-1} \right) \right| \right). \quad (8.56) $$

Now, let's consider the first term on the right-hand side of (8.55). In a manner similar to the second term, and following Assumption 4.8, we obtain

$$
\begin{align*}
\mathfrak{B}_{1,n} &\le 2\mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in I_h^k} \{ \tilde{r}^{(k)}(\varphi, \bar{Z}_\mathbf{i}) - \tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}}) \} \Re(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_\mathbf{i})) \right. \right. \\
&\qquad \left. \times \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \right|^2 d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right. \\
&\qquad \left. + 2\mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left\{ \tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}}) \right\}^2 \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in I_h^k} \Re(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_\mathbf{i})) \right. \right. \\
&\qquad \left. \left. \times \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) - \tilde{f}_{\bar{X}}(\tilde{\mathbf{t}}) \right|^2 d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right) \right. \\
&\le 2 \sup_{\tilde{\mathbf{t}} \in \mathbb{G}^k} \left| \tilde{r}^{(k)}(\varphi, \bar{Z}_\mathbf{i}) - \tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}}) \right|^2 \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in I_h^k} \Re(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_\mathbf{i})) \right. \right. \\
&\qquad \left. \left. \times \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \right|^2 d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right) \right. \\
&\qquad \left. + 2 \sup_{\tilde{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{G}^k} \left| \tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}}) \right|^2 \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in I_h^k} \Re(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_\mathbf{i})) \right. \right. \\
&\qquad \left. \left. \times \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) - \tilde{f}_{\bar{X}}(\tilde{\mathbf{t}}) \right|^2 d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right) \right. \\
&\le 2h^{2\mathfrak{R}} \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in I_h^k} \Re(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_\mathbf{i})) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) \right|^2 d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right. \\
&\qquad \left. + 2C \mathbb{E} \left( \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left| \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in I_h^k} \Re(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}_\mathbf{i})) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{Z}'_{ij}}{h} \right) - \tilde{f}_{\bar{X}}(\tilde{\mathbf{t}}) \right|^2 d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_1) \cdots d \mathrm{Leb}_d \otimes \mu(\bar{\mathbf{t}}_k) \right) \right)
\end{align*}
$$
$$
\begin{align}
&\le 2 \frac{(n-k)!}{n!} h^{2\Re} \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left( \int_{\mathbb{T}^{qk} \times \mathbb{R}^{dk}} \left| \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \tilde{\mathbf{u}}) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{v}_j}{h} \right) \right|^2 \right. \nonumber \\
&\qquad \left. \times f_{\bar{\mathbf{x}}}(\mathbf{v}_j, \mathbf{u}_j) d\mathbf{v}_j d\mu(\mathbf{u}_j) \right) d\text{Leb}_d \otimes \mu(\tilde{\mathbf{t}}_1) \cdots d\text{Leb}_d \otimes \mu(\tilde{\mathbf{t}}_k) \nonumber \\
&\quad + 2C \frac{(n-k)!}{n!} \int_{\mathbb{R}^{dk} \times \mathbb{T}^{qk}} \left( \int_{\mathbb{T}^{qk} \times \mathbb{R}^{dk}} \left| \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \tilde{\mathbf{u}}) \prod_{j=1}^k K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{v}_j}{h} \right) \right|^2 f_{\bar{\mathbf{x}}}(\mathbf{v}_j, \mathbf{u}_j) \right. \nonumber \\
&\qquad \left. \times d\mathbf{v}_j d\mu(\mathbf{u}_j) \right) d\text{Leb}_d \otimes \mu(\tilde{\mathbf{t}}_1) \cdots d\text{Leb}_d \otimes \mu(\tilde{\mathbf{t}}_k) \nonumber \\
&= O \left( \frac{(n-k)!}{n!} h^{-dk+2\Re} \prod_{j=1}^k \left( \sup_{\mathbf{w}_j : \|\mathbf{w}_j\| \le ah^{-1}} \left| \phi^{\mathbf{U}_j}(\mathbf{w}_j)^{-1} \right| \right) \right. \nonumber \\
&\qquad \left. \times \sum_{v_1, \dots, v_k \in \mathbb{Z}^q : \|v_j\|^2 < T_n} \prod_{j=1}^k \left| \phi^{\mathbf{U}_j} \left( (\sigma_{v_j}^M)^{-1} \right) \right| \right). \tag{8.57}
\end{align}
$$

Hence, the proof is complete. □

8.7. *Proof of Theorem 5.3*

We will begin by treating the first term in (5.2).

Linear term of (5.2)

We remark that

$$
\frac{k U_n^{(1)} (\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}})}{\hat{f}_n^{(k)} (\tilde{\mathbf{t}})} - r^{(k)} (\varphi, \tilde{\mathbf{t}}) = \frac{kn^{-1}}{\hat{f}_n^{(k)} (\tilde{\mathbf{t}})} \sum_{i=1}^{n} \left( \begin{aligned}
& \Re (K_{T_n} (\mathbf{t}_1, \mathbf{Z}_i) \tilde{\varphi}_n (\mathbf{Y}_i) \\
& - \mathbb{E} (\Re (K_{T_n} (\mathbf{t}_1, \mathbf{Z}_i) \tilde{\varphi}_n (\mathbf{Y}_i)))
\end{aligned} \right) - r^{(k)} (\varphi, \tilde{\mathbf{t}}). \quad (8.58)
$$

According to (8.58), we can see that

$$
\begin{align}
& \frac{k U_n^{(1)} (\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}})}{\hat{f}_n^{(k)} (\tilde{\mathbf{t}})} - r^{(k)} (\varphi, \tilde{\mathbf{t}}) \nonumber \\
&= \frac{1}{\prod_{j=1}^k f_{\mathbf{x}}(\mathbf{t}_j)} \cdot kn^{-1} \sum_{i=1}^n w_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i) \left( 1 + \frac{\prod_{j=1}^k f_{\mathbf{x}}(\mathbf{t}_j)}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} - 1 \right) \nonumber \\
&= \frac{1}{\prod_{j=1}^k f_{\mathbf{x}}(\mathbf{t}_j)} \cdot kn^{-1} \sum_{i=1}^n w_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i) \nonumber \\
&\quad + \frac{1}{\prod_{j=1}^k f_{\mathbf{x}}(\mathbf{t}_j)} \cdot kn^{-1} \sum_{i=1}^n w_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i) \cdot \frac{\prod_{j=1}^k f_{\mathbf{x}}(\mathbf{t}_j) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})}. \tag{8.59}
\end{align}
$$

Thus, we obtain that

$$
\begin{align*}
& \frac{k \sqrt{n} U_n^{(1)} (\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}}) / \hat{f}_n^{(k)} (\tilde{\mathbf{t}}) - r^{(k)} (\varphi, \tilde{\mathbf{t}}) - \mathbb{E} (w_n (\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y})) / \prod_{j=1}^k f_{\mathbf{x}} (\mathbf{t}_j)}{\sqrt{\mathrm{Var} (w_n (\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))} / \prod_{j=1}^k f_{\mathbf{x}} (\mathbf{t}_j)} \\
&= k \sqrt{n} \cdot \frac{n^{-1} \sum_{i=1}^n w_n (\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i) - \mathbb{E} (w_n (\mathbf{t}_1, \mathbf{Z}, \mathbf{Y}))}{\sqrt{\mathrm{Var} (w_n (\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))}}
\end{align*}
$$
$$ + k\sqrt{n} \cdot \frac{n^{-1} \sum_{i=1}^{n} w_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i)}{\sqrt{\mathrm{Var}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))}} \cdot \frac{\prod_{j=1}^{k} f_{\mathbf{X}}(\mathbf{t}_j) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} \quad (8.60) $$

So, it suffices to prove that

$$ k\sqrt{n} \cdot \frac{n^{-1} \sum_{i=1}^{n} w_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i) - \mathbb{E}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))}{\sqrt{\mathrm{Var}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))}} \xrightarrow{d} N(0, 1), \quad (8.61) $$

and

$$ k\sqrt{n} \cdot \frac{\mathbb{E}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))}{\sqrt{\mathrm{Var}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))}} \cdot \left( \prod_{j=1}^{k} f_{\mathbf{X}}(\mathbf{t}_j) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) \right) = o_{\mathbb{P}}(1). \quad (8.62) $$

Given that $\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \prod_{j=1}^k f_{\mathbf{X}}(\mathbf{t}_j) = o_{\mathbb{P}}(1)$ according to Proposition 4.2 and $\prod_{j=1}^k f_{\mathbf{X}}(\mathbf{t}_j) > 0$, it follows that (8.62) holds. Now, to prove (8.61), we only need to verify the Lyapunov condition

$$ \frac{\mathbb{E}[|w_n(\mathbf{t}_1, \mathbf{Z}, \mathbf{Y}) - \mathbb{E}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))|^{2+\delta}]}{n^{\delta/2} (\mathrm{Var}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y})))^{1+\delta/2}} \to 0, \quad (8.63) $$

for $\delta > 0$ in (F.ii). We can see that, the numerator in (8.63) is bounded by $C \cdot \mathbb{E}(|w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y})|^{2+\delta})$, and it holds that

$$ \begin{aligned} \mathbb{E}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y})) &= \mathbb{E} \left[ \mathrm{Re} (K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1)) \left( \prod_{j=1}^{k-1} f_{\mathbf{X}}(\mathbf{Z}_j) r^{(k)}(\varphi, \tilde{\mathbf{Z}}) - \prod_{j=1}^{k-1} f_{\mathbf{X}}(\mathbf{t}_j) r^{(k)}(\varphi, \tilde{\mathbf{t}}) \right) \right] \\ &= o(1). \end{aligned} $$

We also not

$$ \begin{aligned} \mathbb{E}(|w_n(\mathbf{t}_1, \mathbf{Z}, \mathbf{Y})|^{2+\delta}) &\le C \cdot \mathbb{E}(|\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))|^{2+\delta}), \\ \mathbb{E}((w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))^2) &\ge C \cdot \mathbb{E}((\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1)))^2) \to \infty. \end{aligned} $$

Therefore, (8.63) follows if we establish

$$ \frac{\mathbb{E}[|\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))|^{2+\delta}]}{n^{\delta/2} (\mathbb{E}((\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1)))^2))^{1+\delta/2}} \to 0. \quad (8.64) $$

We first remark that

$$ \mathbb{E}(|K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1)|^{2+\delta}) \le \sup_{\mathbf{t}_1 \in \mathbb{G}} f_{\mathbf{X}}(\mathbf{t}_1) \int_{\mathbb{G}} |K_{T_n}(\mathbf{t}_1, \mathbf{u})|^{2+\delta} d\mu(\mathbf{u}), \quad (8.65) $$

and

$$ \begin{aligned} |K_{T_n}(\mathbf{t}_1, \mathbf{u})|^\delta &\le \left( \sum_{\sigma_1 \in \hat{\mathbb{G}}: k\sigma_1 < T_n} d_{\sigma_1} \left| \mathrm{Tr} \left( \sigma_1^M (\mathbf{u}^{-1}) \phi \mathbf{u}_1 (\sigma_1^M)^{-1} \sigma_1^M (\mathbf{t}_1) \right) \right| \right)^\delta \\ &\le \left( \sum_{\sigma_1 \in \hat{\mathbb{G}}: k\sigma_1 < T_n} d_{\sigma_1}^{3/2} \left\| \sigma_1^M (\mathbf{u}^{-1}) \phi \mathbf{u}_1 (\sigma_1^M)^{-1} \right\|_{\mathrm{HS}} \right)^\delta \\ &\le \left( \sum_{\sigma_1 \in \hat{\mathbb{G}}: k\sigma_1 < T_n} d_{\sigma_1}^2 \left\| \phi \mathbf{u}_1 (\sigma_1^M)^{-1} \right\|_{\mathrm{op}} \right)^\delta, \end{aligned} \quad (8.66) $$
where the third inequality arises from Lemma 9.2-(ii) using the matrix Hölder's inequality, while the last inequality is a consequence of Lemma 9.2-(ii) with the fact that $\|AB\|_{HS} \le \|A\|_{op}\|B\|_{HS}$ for all $d_{\sigma_1} \times d_{\sigma_1}$ complex matrices A and B. Additionally, we note that for all $\mathbf{t}_1 \in \mathbb{G}$,

$$ 
\begin{aligned}
\int_{\mathbb{G}} |K_{T_n}(\mathbf{t}_1, \mathbf{u})|^2 d\mu(\mathbf{u}) &= \int_{\mathbb{G}} |K_{T_n}(\mathbf{t}_1, \mathbf{u}^{-1})|^2 d\mu(\mathbf{u}) \\
&= \sum_{\sigma_1 \in \hat{\mathbb{G}}: k_{\sigma_1} < T_n} d_{\sigma_1} \left\| \phi_{\mathbf{U}_1}(\sigma_1^M)^{-1} \sigma_1^M(\mathbf{t}_1) \right\|_{\text{HS}}^2 \\
&\le \sum_{\sigma_1 \in \hat{\mathbb{G}}: k_{\sigma_1} < T_n} d_{\sigma_1} \left\| \phi_{\mathbf{U}_1}(\sigma_1^M)^{-1} \right\|_{\text{op}}^2 \left\| \sigma_1^M(\mathbf{t}_1) \right\|_{\text{HS}}^2 \\
&= \sum_{\sigma_1 \in \hat{\mathbb{G}}: k_{\sigma_1} < T_n} d_{\sigma_1}^2 \left\| \phi_{\mathbf{U}_1}(\sigma_1^M)^{-1} \right\|_{\text{op}}^2.
\end{aligned} 
\quad (8.67) $$

Here, the first equality stems from the property that $\int_{\mathbb{G}} f(g)d\mu(g) = \int_{\mathbb{G}} f(g^{-1})d\mu(g)$ for all integrable functions $f: \mathbb{G} \to \mathbb{C}$, and the second one is a consequence of Lemma 9.1. By combining (8.65), (8.66), and (8.67) and utilizing Lemma 4.1, we arrive at:

$$ 
\begin{aligned}
& \mathbb{E}\left[\left|(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))\right|^{2+\delta}\right] \\
& \le \begin{cases}
C \cdot T_n^{(2+\delta)\beta+(1+\delta)\dim(\mathbb{G})/2 & \text{for (S1)-(i),} \\
C \cdot T_n^{(2+\delta)\alpha+(1+\delta)\dim(\mathbb{G})/2} \exp\left((2+\delta)\gamma \cdot T_n^{\beta}\right) & \text{for (S2)-(i),} \\
C \cdot T_n^{(2+\delta)\alpha+(1+\delta)\dim(\mathbb{G})/2} \exp\left((2+\delta)\gamma \cdot T_n^{\beta} (\log T_n - \xi_1)\right) & \text{for (S3)-(i).}
\end{cases} 
\end{aligned} 
\quad (8.68) $$

This yields the same upper bound for $\mathbb{E}[|\text{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))|^{2+\delta}] \le \mathbb{E}[|(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))|^{2+\delta}]$. For the verification of (8.62), we denote that

$$ 
\begin{aligned}
& \left| \mathbb{E} \left( \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \prod_{j=1}^{k} f_{\mathbf{X}}(\mathbf{t}_j) \right) \right| \\
& \le \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} \ge T_n} \prod_{j=1}^{k} d_{\sigma_j} \left| \text{Tr} \left( \phi^{\mathbf{X}}(\sigma_j^M) \sigma_j^M(\mathbf{t}_j) \right) \right| \\
& = T_n^{-k\hat{m}} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} \ge T_n} \prod_{j=1}^{k} d_{\sigma_j} \left| \text{Tr} \left( (-1)^{\hat{m}} k_{\sigma_j}^{\hat{m}} \phi^{\mathbf{X}}(\sigma_j^M) \sigma_j^M(\mathbf{t}_j) \right) \right| \\
& \le T_n^{-k\hat{m}} \sum_{\sigma_1, \dots, \sigma_k \in \hat{\mathbb{G}}: k_{\sigma_j} \ge T_n} \prod_{j=1}^{k} d_{\sigma_j} \left| \text{Tr} \left( \phi^{\Delta\hat{m}}(\tilde{f}_{\mathbf{X}})(\sigma_j^M) \sigma_j^M(\mathbf{t}_j) \right) \right| = o(T_n^{-k\hat{m}}),
\end{aligned} 
\quad (8.69) $$

where the second inequality is derived from (3.1.5) in [3] and the last equality follows from the absolute convergence of the Fourier series of $\Delta^{\hat{m}}(\tilde{f}(\cdot))$. Lemma 4.1 and Proposition 4.2 also demonstrate that

$$ \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \mathbb{E}(\hat{f}_n^{(k)}(\tilde{\mathbf{t}})) = O_P \left( n^{-k^2} T_n^{k\beta + \frac{k \dim(\mathbb{G})}{2}} \right). $$

As a result, we obtain that,
$$ \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_{\mathbb{X}}(\tilde{\mathbf{t}}) = o(T_n^{-k\hat{m}}) + O_P\left(n^{-k^2} T_n^{k\beta + \frac{k\dim(\mathbb{G})}{2}}\right). $$

Now, let us recall $\tilde{r}^{(k)}(\tilde{\mathbf{t}}) := r^{(k)}(\varphi, \tilde{\mathbf{t}}) \tilde{f}_{\mathbb{X}}(\tilde{\mathbf{t}})$. Consequently, we have

$$
\begin{aligned}
& \left|\mathbb{E}(w_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))\right| \\
&= \left|\mathbb{E}\left[\text{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{z}_1)) \left(\tilde{\varphi}_n(\mathbf{y}) - \prod_{j=1}^{k-1} f_{\mathbf{x}}(\mathbf{t}_j) r^{(k)}(\varphi, \tilde{\mathbf{t}})\right)\right]\right| \\
&= \left|\mathbb{E}\left[\text{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{z}_1)) \tilde{\varphi}_n(\mathbf{y})\right] - \mathbb{E}\left[\text{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{z}_1)) \prod_{j=1}^{k-1} f_{\mathbf{x}}(\mathbf{t}_j) r^{(k)}(\varphi, \tilde{\mathbf{t}})\right]\right| \\
&\le \sum_{\sigma_1 \in \hat{\mathbb{G}}: k_{\sigma_1} \ge T_n} \left( \left| r^{(k)}(\varphi, \mathbf{t}_1) \right| \cdot d_{\delta_1} \left| \text{Tr}(\phi^{\mathbf{x}}(\sigma_1^M) \sigma_1^M(\mathbf{t}_1)) \right| + d_{\delta_1} \left| \text{Tr}(\phi^{\tilde{r}^{(k)}}(\sigma_1^M) \sigma_1^M(\mathbf{t}_1)) \right| \right) \\
&\le T_n^{-\hat{m}} \sum_{\sigma_1 \in \hat{\mathbb{G}}: k_{\sigma_1} \ge T_n} \left( \left| r^{(k)}(\varphi, \mathbf{t}_1) \right| \cdot d_{\delta_1} \left| \text{Tr}(\phi^{\Delta\hat{m}}(f_{\mathbf{x}})(\sigma_1^M) \sigma_1^M(\mathbf{t}_1)) \right| \right. \\
&\qquad \left. + d_{\delta_1} \left| \text{Tr}(\phi^{\Delta\hat{m}(\tilde{r}^{(k)})}(\sigma_1^M) \sigma_1^M(\mathbf{t}_1)) \right| \right) \\
&= o(T_n^{-\hat{m}}),
\end{aligned}
$$

where the second inequality stems from (3.1.5) in [3] and the final equality is a consequence of the absolute convergence of the Fourier series of $\Delta^{\hat{m}}(f_{\mathbf{x}}(\cdot))$ and $\Delta^{\hat{m}}(\tilde{r}^{(k)}(\varphi, \cdot))$. Therefore, (8.62) follows if

$$ \sqrt{n} T_n^{-\hat{m}} \left( T_n^{-k\hat{m}} + n^{-k^2/2} T_n^{k\beta + \frac{k\dim(\mathbb{G})}{2}} \right) \cdot T_n^{-k\beta + \frac{-k\dim(\mathbb{G})}{4}} = o(1), \quad (8.70) $$

which is satisfied by (T1''). This completes the proof. □

**Remark 8.1.** We provide a detailed explanation for why we do not cover the supersmooth and log-supersmooth scenarios. For (S2)-(i), we need

$$ \sqrt{n} \cdot T_n^{k\alpha + \frac{k\dim(\mathbb{G})}{2}} \exp(k \cdot \gamma \cdot T_n^\beta) = o(1), \quad (8.71) $$

for the consistency of $\hat{f}_n^{(k)}(\tilde{\mathbf{t}})$. Under (S2)-(i)+(B2), we also need

$$
\begin{aligned}
& \sqrt{n} \cdot T_n^{-\hat{m}} \left( T_n^{-k\hat{m}} + n^{-k^2/2} T_n^{k\beta + \frac{k\dim(\mathbb{G})}{2}} \exp(k \cdot \gamma \cdot T_n^\beta) \right) \cdot T_n^{-k\alpha - q/2} \exp(-k \cdot \gamma (\eta \cdot T_n)^\beta) \\
&= O(1), \qquad (8.72)
\end{aligned}
$$

for (8.70). We require a log-type rate for $T_n$ to fulfill (8.71), whereas (8.72) fails to hold with any log-type rate for $T_n$. The scenario of log-supersmoothness encounters a similar issue.

### The other terms of (5.2)

Now, we must move on to the nonlinear part of the Hoeffding decomposition. Accordingly, the goal, as in [132], is to prove that:

$$ (nT_n)^{1/2} \left[ \tilde{\mu}_n(\varphi, T_n, \tilde{\mathbf{t}}) - k\tilde{U}_n^{(1)} \left( \pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}} \right) \right] \to 0 \quad \text{in } L^2. \quad (8.73) $$

To achieve this, we require the variance formula for a (centered) U-statistic
$$V_n(\varphi, T_n, \tilde{\mathbf{t}}) = \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in I_n^k} \frac{\tilde{G}_{\varphi, T_n, \mathbf{t}}(\mathbf{Z}_i, \mathbf{Y}_i)}{N}, \qquad (8.74)$$

where $\tilde{G}_{\varphi, T_n, \tilde{\mathbf{t}}}(\mathbf{z}, \mathbf{y}) = G_{\varphi, T_n, \tilde{\mathbf{t}}}(\mathbf{z}, \mathbf{y}) - \mathbb{E}(G_{\varphi, T_n, \tilde{\mathbf{t}}}(\mathbf{z}, \mathbf{y}))$ and $N = \prod_{j=1}^k \mathbb{E}K_{T_n}(\mathbf{t}_j, \mathbf{Z}_j)$, which is given by

$$\mathrm{Var}(V_n) = \left( \frac{(n-k)!}{n!} \right)^2 \sum_{s=1}^{k} \frac{(n-s)!}{(n-2k+s)!} \sum_{|\Lambda_1|=s=|\Lambda_2|}^{(s)} \mathbf{I}(\Lambda_1, \Lambda_2), \qquad (8.75)$$

where $\sum^{(s)}$ denotes summation over all positions $\Lambda_1$ and $\Lambda_2$ of length $l$,

$$\mathbf{I}(\Lambda_1, \Lambda_2) = \mathbb{E} \left[ \tilde{G}_{\varphi, T_n, \tilde{\mathbf{t}}} \left( (\mathbf{Z}_{i_1}, \mathbf{Y}_{i_1}), \dots, (\mathbf{Z}_{i_k}, \mathbf{Y}_{i_k}) \right) \tilde{G}_{\varphi, T_n, \tilde{\mathbf{t}}} \left( (\mathbf{Z}_{j_1}, \mathbf{Y}_{j_1}), \dots, (\mathbf{Z}_{j_k}, \mathbf{Y}_{j_k}) \right) \right], \quad (8.76)$$

and the $i$'s in position $\Lambda_1$ correspond to the $j$'s in position $\Lambda_2$. All $i$'s and $j$'s are otherwise pairwise distinct. When applied to $V_n = \tilde{\mu}_n - k\tilde{U}_n^{(1)}(\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}})$, we get

$$\sum_{i=1}^{n} \mathbf{I}(\Lambda_1, \Lambda_2) = 0.$$

Moreover, we can also assert that

$$N^{-2}\mathbf{I}(\Lambda_1, \Lambda_2) = O(T_n^{-s}) \quad \text{for each} \quad 2 \le s \le k.$$

This implies that

$$\begin{aligned} nT_n \mathrm{Var}(\tilde{\mu}_n - k\tilde{U}_n^{(1)}(\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}})) &= O\left[ nT_n \sum_{l=2}^{k} \binom{n}{s}^{-1} \binom{k}{s} \binom{n-k}{k-s} T_n^{-s} \right] \\ &= O\left[ \sum_{s=2}^{k} (nT_n)^{1-s} \right] \\ &= O\left[ (nT_n)^{-1} \right] = o(1). \end{aligned} \qquad (8.77)$$

Hence, the proof is complete. □

## 8.8. Proof of Theorem 5.1

The theorem follows from Theorem 5.3 and Lemma 9.3.

## 8.9. Proof of Theorem 5.6

Similarly to the proof of Theorem 5.1, it is sufficient to demonstrate that:

$$k\sqrt{n} \cdot \frac{n^{-1} \sum_{i=1}^{n} \tilde{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_i, \mathbf{Y}_i) - \mathbb{E}(\tilde{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}, \mathbf{Y}))}{\sqrt{\mathrm{Var}(\tilde{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}, \mathbf{Y}))}} \xrightarrow{d} N(0, 1), \qquad (8.78)$$

and

$$k\sqrt{n} \cdot \frac{\mathbb{E}(\tilde{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}))}{\sqrt{\mathrm{Var}(\tilde{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}))}} \cdot \left( \prod_{j=1}^{k} f_{\bar{\mathbf{x}}}(\tilde{\mathbf{t}}_j) - \bar{f}_n^{(k)}(\tilde{\mathbf{t}}) \right) = o_P(1). \qquad (8.79)$$
In order to prove (8.78), since $\mathbb{E}[|\check{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}) - \mathbb{E}(\check{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}))|^{2+\delta}] \le \mathbb{E}[|\check{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y})|^{2+\delta}]$, it is enough to show that the following Lyapunov condition holds:

$$ \frac{\mathbb{E}\left[\left|\check{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y})\right|^{2+\delta}\right]}{n^{\delta/2} (\mathrm{Var}(\check{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y})))^{1+\delta/2}} \to 0. \quad (8.80) $$

Let

$$ A_n(\beta_0) = h^{-d(2\beta_0+1)} \quad \text{and} \quad B_n(\beta^*) = T_n^{2\beta^*+\dim(G)/2}. \quad (8.81) $$

Define

$$ \check{K}(\tilde{\mathbf{u}}) = \lim_{n} h^{2d\beta_0} (K_h^D(\tilde{\mathbf{u}}))^2 = \prod_{j=1}^{d} \frac{1}{(2\pi |a_j|)^2} \left| \int_{\mathbb{R}} e^{-i\mathbf{u}\cdot\mathbf{s}_j} \phi^{K_j}(\mathbf{s}_j) \mathbf{s}_j^{\beta_0} d\mathbf{s}_j \right|^2. $$

Notice that

$$
\begin{aligned}
& \mathbb{E} \left( \left( \check{\tilde{w}}_n(\tilde{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}) \right)^2 \right) \\
&= \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 \left( \bar{\varphi}_n(\mathbf{s}) - \prod_{j=1}^{k-1} f_{\bar{\mathbf{Z}}}(\tilde{\mathbf{t}}_j) \bar{\tau}^{(k)} \left( \varphi, \tilde{\bar{\mathbf{t}}} \right) \right)^2 \\
&\quad \times f_{\bar{\mathbf{Z}}_Y}((\mathbf{u}, \mathbf{w}), \mathbf{s}) du d\mu(\mathbf{w}) ds \\
&= \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 \\
&\quad \times \left( \int_{\mathbb{R}^{dk}} \int_{\mathbb{R}^{d(k-1)} \times \mathbb{G}^{k-1}} \varphi(\mathbf{s}, \mathbf{s}_2, \dots, \mathbf{s}_k) \prod_{j=2}^k \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{z}'_j}{h} \right) \mathrm{Re} \left( K_{T_n}(\mathbf{t}_j, \mathbf{z}_j) \right) \right. \\
&\quad \left. \times \frac{f_{\bar{\mathbf{Z}}_Y}(\bar{\mathbf{z}}_j, \mathbf{y}_j) f_{\bar{\mathbf{Z}}}(\bar{\mathbf{z}}_j)}{f_{\bar{\mathbf{Z}}}(\bar{\mathbf{z}}_j)} d\mathbf{z}'_j d\mu(\mathbf{z}_j) d\mathbf{s}_j - \prod_{j=1}^{k-1} f_{\bar{\mathbf{Z}}}(\tilde{\mathbf{t}}_j) \bar{\tau}^{(k)} \left( \varphi, \tilde{\bar{\mathbf{t}}} \right) \right)^2 f_{\bar{\mathbf{Z}}_Y}((\mathbf{u}, \mathbf{w}), \mathbf{s}) du d\mu(\mathbf{w}) ds \\
&= \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 \\
&\quad \times \left( \int_{\mathbb{R}^{d(k-1)} \times \mathbb{G}^{k-1}} \mathbb{E} \left[ \varphi(\mathbf{s}, \mathbf{Y}_2, \dots, \mathbf{Y}_k) \mid \bar{\mathbf{Z}}_2 = \tilde{\mathbf{t}}_2, \dots, \bar{\mathbf{Z}}_k = \tilde{\mathbf{t}}_k \right] \right. \\
&\quad \left. \times \prod_{j=2}^k \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{z}'_j}{h} \right) \mathrm{Re} \left( K_{T_n}(\mathbf{t}_j, \mathbf{z}_j) \right) f_{\bar{\mathbf{X}}}(\bar{\mathbf{z}}_j) d\mathbf{z}'_j d\mu(\mathbf{z}_j) - \prod_{j=1}^{k-1} f_{\bar{\mathbf{Z}}}(\tilde{\mathbf{t}}_j) \bar{\tau}^{(k)} \left( \varphi, \tilde{\bar{\mathbf{t}}} \right) \right)^2 \\
&\quad \times f_{\bar{\mathbf{X}}_Y}((\mathbf{u}, \mathbf{w}), \mathbf{s}) du d\mu(\mathbf{w}) ds \\
&= \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 \left( \bar{\tau}^{(k)} \left( \varphi, (\mathbf{s}, \tilde{\mathbf{t}}_2, \dots, \tilde{\mathbf{t}}_k) \right) \prod_{j=2}^k f_{\bar{\mathbf{Z}}}(\tilde{\mathbf{t}}_j) \right)^2.
\end{aligned}
$$
$$
\begin{align*}
& - \left( \prod_{j=1}^{k-1} f_{\bar{Z}(\bar{\mathbf{t}}_j)} \bar{r}^{(k)} (\varphi, \bar{\tilde{\mathbf{t}}}) + o(1) \right)^2 f_{\bar{X}Y((\mathbf{u}, \mathbf{w}), \mathbf{s})} du d\mu(\mathbf{w}) ds \\
& \geq \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 (\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})))^2 \left( \bar{r}^{(k)} (\varphi, (\mathbf{s}, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{Z}(\bar{\mathbf{t}}_j)} \right. \\
& \qquad \left. - \prod_{j=1}^{k-1} f_{\bar{Z}(\bar{\mathbf{t}}_j)} \bar{r}^{(k)} (\varphi, \bar{\tilde{\mathbf{t}}}) + o(1) \right)^2 (f_{\bar{X}Y((\mathbf{u}, \mathbf{w}), \mathbf{s})} - f_{\bar{X}Y((\mathbf{t}'_1, \mathbf{w}), \mathbf{s})}) du d\mu(\mathbf{w}) ds \\
& \qquad + \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 (\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})))^2 \left( \bar{r}^{(k)} (\varphi, (\mathbf{s}, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{Z}(\bar{\mathbf{t}}_j)} \right. \\
& \qquad \left. - \prod_{j=1}^{k-1} f_{\bar{Z}(\bar{\mathbf{t}}_j)} \bar{r}^{(k)} (\varphi, \bar{\tilde{\mathbf{t}}}) + o(1) \right)^2 \\
& \geq \inf_{\bar{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \left( \bar{r}^{(k)} (\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{Z}(\bar{\mathbf{t}}_j)} \right)^2 \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \\
& \qquad \times (\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})))^2 (f_{\bar{Z}(\mathbf{u}, \mathbf{w})} - f_{\bar{Z}(\mathbf{t}'_1, \mathbf{w})}) du d\mu(\mathbf{w}) \\
& \qquad + \inf_{\bar{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \inf_{\mathbf{w} \in \mathbb{G}} \left[ \left( \bar{r}^{(k)} (\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{Z}(\bar{\mathbf{t}}_j)} \right)^2 f_{\bar{X}(\mathbf{t}'_1, \mathbf{w})} \right] \\
& \qquad \times \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 (\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})))^2 du d\mu(\mathbf{w}) \\
& \geq \inf_{\bar{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \left( \bar{r}^{(k)} (\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{Z}(\bar{\mathbf{t}}_j)} \right)^2 \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \\
& \qquad \times (\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})))^2 (f_{\bar{Z}(\mathbf{u}, \mathbf{w})} - f_{\bar{Z}(\mathbf{t}'_1, \mathbf{w})}) du d\mu(\mathbf{w}) \\
& \qquad + \inf_{\bar{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \inf_{\mathbf{w} \in \mathbb{G}} \left[ \left( \bar{r}^{(k)} (\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{Z}(\bar{\mathbf{t}}_j)} \right)^2 f_{\bar{X}(\mathbf{t}'_1, \mathbf{w})} \right] \\
& \qquad \times \left( A_n(\beta_0) \left( \int_{\mathbb{R}^d \times \mathbb{G}} h^{2d\beta_0} (K_h^D(\mathbf{v}))^2 d\mathbf{v} - \int_{\mathbb{R}^d} \check{K}(\mathbf{v}) d\mathbf{v} \right) + A_n(\beta_0) \int_{\mathbb{R}^d} \check{K}(\mathbf{v}) d\mathbf{v} \right) B_n(\beta^*) \\
& = \inf_{\bar{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \left( \bar{r}^{(k)} (\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{Z}(\bar{\mathbf{t}}_j)} \right)^2 \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \\
& \qquad \times (\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})))^2 (f_{\bar{Z}(\mathbf{u}, \mathbf{w})} - f_{\bar{Z}(\mathbf{t}'_1, \mathbf{w})}) du d\mu(\mathbf{w})
\end{align*}
$$
$$ + \inf_{\tilde{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \inf_{\mathbf{w} \in \mathbb{G}} \left[ \left( \tilde{\mathbf{r}}^{(k)} (\varphi, (\tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2, \dots, \tilde{\mathbf{t}}_k)) \prod_{j=2}^k f_{\tilde{\mathbf{Z}}}(\tilde{\mathbf{t}}_j) \right)^2 f_{\tilde{\mathbf{X}}}(\mathbf{t}'_1, \mathbf{w}) \right] A_n(\beta_0) \\ \times \left( \int_{\mathbb{R}^d} \check{K}(\mathbf{v}) \, d\mathbf{v} + o(1) \right) B_n(\beta^*), $$

where the last inequality follows from the fact that $\inf_{\mathbf{w} \in \mathbb{G}} f_{\tilde{\mathbf{Z}}}(\mathbf{t}'_1, \mathbf{w}) \ge \inf_{\mathbf{w} \in \mathbb{G}} f_{\tilde{\mathbf{X}}}(\mathbf{t}'_1, \mathbf{w})$, and the last equality results from Lemma 9.5 in Appendix with $k=2$, along with the Lebesgue-dominated convergence theorem. Notice that, we can find some constant $C > 0$, in such a way that

$$ \inf_{\tilde{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \left| \tilde{\mathbf{r}}^{(k)} (\varphi, (\tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2, \dots, \tilde{\mathbf{t}}_k)) \right| = C. \quad (8.82) $$

In addition, we note that

$$ \begin{aligned} & \sup_{\mathbf{w} \in \mathbb{G}} \left| \int_{\mathbb{R}^d} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \left( f_{\tilde{\mathbf{X}}}(\mathbf{u}, \mathbf{w}) - f_{\tilde{\mathbf{X}}}(\mathbf{t}'_1, \mathbf{w}) \right) d\mathbf{u} \right| \\ & \le \sup_{\mathbf{w} \in \mathbb{G}} \left| \int_{\mathbb{R}^d} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 f_{\tilde{\mathbf{X}}}(\mathbf{u}, \mathbf{w}) d\mathbf{u} - f_{\tilde{\mathbf{X}}}(\mathbf{t}'_1, \mathbf{w}) A_n(\beta_0) \int_{\mathbb{R}^d} \check{K}(\mathbf{v}) d\mathbf{v} \right| \\ & \quad + \sup_{\mathbf{w} \in \mathbb{G}} \left| f_{\tilde{\mathbf{X}}}(\mathbf{t}'_1, \mathbf{w}) \int_{\mathbb{R}^d} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 d\mathbf{u} - A_n(\beta_0) \int_{\mathbb{R}^d} \check{K}(\mathbf{v}) d\mathbf{v} \right|. \end{aligned} \quad (8.83) $$

We begin with the first term on the right-hand side of (8.83). We have

$$ \begin{aligned} & \sup_{\mathbf{w} \in \mathbb{G}} \left| \int_{\mathbb{R}^d} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 f_{\tilde{\mathbf{Z}}}(\mathbf{u}, \mathbf{w}) d\mathbf{u} - f_{\tilde{\mathbf{Z}}}(\mathbf{t}'_1, \mathbf{w}) A_n(\beta_0) \int_{\mathbb{R}^d} \check{K}(\mathbf{v}) d\mathbf{v} \right| \\ & = A_n(\beta_0) \sup_{\mathbf{w} \in \mathbb{G}} \left| h^{-d} \int_{\mathbb{R}^d} \left( h^{2d\beta_0} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 f_{\tilde{\mathbf{Z}}}(\mathbf{u}, \mathbf{w}) d\mathbf{u} - f_{\tilde{\mathbf{Z}}}(\mathbf{t}'_1, \mathbf{w}) \int_{\mathbb{R}^d} \check{K}(\mathbf{v}) d\mathbf{v} \right| \\ & = o(A_n(\beta_0)), \end{aligned} \quad (8.84) $$

where the equality follows by applying Lemma 9.5 with $k=2$, and Lemma 9.4 with $W_n = h^{2d\beta_0}(K_h^D)^2$, $S = \mathbb{G}$, $f = f_{\tilde{\mathbf{Z}}}$ and $a_n = h^d$. In fact, $f_{\tilde{\mathbf{Z}}}(\cdot)$ satisfies the conditions on $f(\cdot)$ in Lemma 9.4. This can be observed by noting that $\|f_{\tilde{\mathbf{Z}}}\|_{\infty} \le \|f_{\tilde{\mathbf{X}}}\|_{\infty} < \infty$, and $\|f_{\mathbf{w}}\|_{\infty} \le \|f_{\mathbf{x}}\|_{\infty} < \infty$. Hence, $\sup_{z \in \mathbb{G}} f_{\tilde{\mathbf{Z}}}(\mathbf{t}, \mathbf{z}) \le \|f_{\tilde{\mathbf{Z}}}\|_{\infty} < \infty$ and $\sup_{z \in \mathbb{G}} \int_{\mathbb{R}^d} f_{\tilde{\mathbf{Z}}}(\mathbf{u}, \mathbf{z}) d\mathbf{u} = \|f_{\tilde{\mathbf{Z}}}\|_{\infty} < \infty$. Now, for the second term on the right-hand side of (8.83), we also get that

$$ \sup_{\mathbf{w} \in \mathbb{G}} f_{\tilde{\mathbf{Z}}}(\mathbf{t}'_1, \mathbf{w}) \left| \int_{\mathbb{R}^d} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 d\mathbf{u} - A_n(\beta_0) \int_{\mathbb{R}^d} \check{K}(\mathbf{v}) d\mathbf{v} \right| = o(A_n(\beta_0)). \quad (8.85) $$

By combining (8.84) and (8.85), we arrive at
$$ 
\begin{aligned}
& \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 (\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})))^2 (f_{\bar{\mathbf{x}}}(\mathbf{u}, \mathbf{w}) - f_{\bar{\mathbf{x}}}(\mathbf{t}'_1, \mathbf{w})) du d\mu(\mathbf{w}) \\
&= o(A_n(\beta_0) B_n(\beta^*)).
\end{aligned}
\quad (8.86)
$$

As a result, we obtain that

$$ \mathbb{E} \left( \left( \tilde{\tilde{w}}_n(\bar{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}) \right)^2 \right) \geq C A_n(\beta_0) B_n(\beta^*). \quad (8.87) $$

On the other side, we can see that

$$ 
\begin{aligned}
& \left( \mathbb{E} \left[ \tilde{\tilde{w}}_n(\bar{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}) \right] \right)^2 \\
&= \left( \mathbb{E} \left[ K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{z}'_1}{h} \right) \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{z}_1)) \left( \bar{\varphi}_n(\mathbf{y}) - \prod_{j=1}^{k-1} f_{\bar{\mathbf{x}}}(\bar{\mathbf{t}}_j) \bar{\tau}^{(k)}(\varphi, \tilde{\tilde{\mathbf{t}}}) \right) \right] \right)^2 \\
&\le \left( \mathbb{E} \left[ K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{z}'_1}{h} \right) \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{z}_1)) (\bar{\varphi}_n(\mathbf{y})) \right] \right)^2 \\
&= \left( \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})) \int_{\mathbb{R}^{dk}} \int_{\mathbb{R}^{d(k-1)} \times \mathbb{G}^{k-1}} \varphi(\mathbf{s}, \mathbf{s}_2, \dots, \mathbf{s}_k) \right. \\
&\qquad \left. \times \prod_{j=2}^k \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{z}'_j}{h} \right) \mathrm{Re}(K_{T_n}(\mathbf{t}_j, \mathbf{z}_j)) \frac{f_{\bar{\mathbf{X}}}(\bar{\mathbf{z}}_j, \mathbf{y}_j) f_{\bar{\mathbf{X}}}(\bar{\mathbf{z}}_j)}{f_{\bar{\mathbf{X}}}(\bar{\mathbf{z}}_j)} d\mathbf{z}'_j d\mu(\mathbf{z}_j) d\mathbf{s}_j \right. \\
&\qquad \left. \times f_{\bar{\mathbf{X}}}(\mathbf{u}, \mathbf{w}), \mathbf{s} \right) du d\mu(\mathbf{w}) ds \right)^2 \\
&= \left( \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})) \int_{\mathbb{R}^{d(k-1)} \times \mathbb{G}^{(k-1)}} \bar{\tau}^{(k)}(\varphi, (\mathbf{s}, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \right. \\
&\qquad \left. \times \prod_{j=2}^k \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{z}'_j}{h} \right) \mathrm{Re}(K_{T_n}(\mathbf{t}_j, \mathbf{z}_j)) f_{\bar{\mathbf{X}}}(\bar{\mathbf{z}}_j) d\mathbf{z}'_j d\mu(\mathbf{z}_j) \times f_{\bar{\mathbf{X}}}(\mathbf{u}, \mathbf{w}) du d\mu(\mathbf{w}) ds \right)^2 \\
&\le \sup_{\bar{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{G}^k} \bar{\tau}^{(k)}(\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \left( \int_{\mathbb{R}^d \times \mathbb{G}} \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})) \int_{\mathbb{R}^{d(k-1)} \times \mathbb{G}^{(k-1)}} \right. \\
&\qquad \left. \times \prod_{j=2}^k \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{z}'_j}{h} \right) \mathrm{Re}(K_{T_n}(\mathbf{t}_j, \mathbf{z}_j)) f_{\bar{\mathbf{X}}}(\bar{\mathbf{z}}_j) d\mathbf{z}'_j d\mu(\mathbf{z}_j) \times f_{\bar{\mathbf{X}}}(\mathbf{u}, \mathbf{w}) du d\mu(\mathbf{w}) \right)^2 \\
&\le \sup_{\bar{\mathbf{t}} \in \mathbb{R}^{dk} \times \mathbb{G}^k} \bar{\tau}^{(k)}(\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \left\| f_{\bar{\mathbf{x}}} \right\|_{\infty}^k \left( \int_{\mathbb{R}^{dk}} \prod_{j=i}^k \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_j - \mathbf{z}'_j}{h} \right) d\mathbf{z}'_j \right)^2 \\
&\qquad \times \left( \int_{\mathbb{G}^k} \mathrm{Re}(K_{T_n}(\mathbf{t}_j, \mathbf{z}_j)) d\mu(\mathbf{z}_j) \right)^2
\end{aligned}
$$
$$= O \left( T_n^{k \dim(\mathbb{G})/2} \right) = o \left( A_n(\beta_0) B_n(\beta^*) \right). \qquad (8.88)$$

Combining this with (8.87) gives that

$$\mathrm{Var}(\tilde{\mathring{w}}_n(\bar{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y})) \geq C \cdot A_n(\beta_0) B_n(\beta^*).$$

Making use of the condition (F.ii), we infer that

$$
\begin{aligned}
& \mathbb{E} \left( \left| \tilde{\mathring{w}}_n(\bar{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}) \right|^{2+\delta} \right) \\
& \leq \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left| K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{v}) \right) \bar{\varphi}_n(\mathbf{s}) \right|^{2+\delta} f_{\bar{\mathbf{X}}\mathbf{Y}}((\mathbf{u}, \mathbf{v}), \mathbf{s}) \, du \, d\mu(\mathbf{v}) \, ds \\
& = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left| \bar{\varphi}_n(\mathbf{s}) \right|^{2+\delta} \left| K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{v}) \right) \right|^{2+\delta} \frac{f_{\bar{\mathbf{X}}\mathbf{Y}}((\mathbf{u}, \mathbf{v}), \mathbf{s})}{f_{\bar{\mathbf{X}}}(\mathbf{u}, \mathbf{v})} f_{\bar{\mathbf{X}}}(\mathbf{u}, \mathbf{v}) \, du \, d\mu(\mathbf{v}) \, ds \\
& \leq \frac{\theta_\delta \| f_{\bar{\mathbf{X}}} \|_\infty}{h^{d(1+\delta)+d\beta_0(2+\delta)}} \int_{\mathbb{R}^d} h^{d\beta_0(2+\delta)} \left| K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right|^{2+\delta} \, du \cdot \sup_{\mathbf{v} \in \mathbb{G}} \left| \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{v}) \right) \right|^\delta \\
& \qquad \times \int_{\mathbb{G}} \left| \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{v}) \right) \right|^2 d\mu(\mathbf{v}) \\
& \leq \frac{\theta_\delta \| f_{\bar{\mathbf{X}}} \|_\infty}{h^{d(1+\delta)+d\beta_0(2+\delta)}} \int_{\mathbb{R}^d} \left( \check{K}(\mathbf{w}) \right)^{1+\delta/2} d\mathbf{w} \cdot \sup_{\mathbf{v} \in \mathbb{G}} \left| \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{v}) \right) \right|^\delta \int_{\mathbb{G}} \left| \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{v}) \right) \right|^2 d\mu(\mathbf{v}) \\
& \leq \frac{C}{h^{d(1+\delta)+d\beta_0(2+\delta)}} \prod_{\ell=1}^d \left( \sup_{\mathbf{w}_\ell \in \mathbb{R}} \left| \int_{\mathbb{R}} e^{-i \mathbf{w}_\ell \cdot \mathbf{s}_\ell} \phi^K(\mathbf{s}_\ell) \, d\mathbf{s}_\ell \right|^{\delta} \int_{\mathbb{R}} \left| \int_{\mathbb{R}} e^{-i \mathbf{w}_\ell \cdot \mathbf{s}_\ell} \phi^K(\mathbf{s}_\ell) \, d\mathbf{s}_\ell \right|^2 \, d\mathbf{w}_\ell \right) \\
& \qquad \cdot T_n^{\beta^*(2+\delta)+(1+\delta) \dim(\mathbb{G})/2} \\
& \leq \frac{C}{h^{d(1+\delta)+d\beta_0(2+\delta)}} \prod_{\ell=1}^d \int_{\mathbb{R}} \left| \phi^K(\mathbf{w}_\ell) \mathbf{w}_\ell^{\beta_0} \right|^2 \, d\mathbf{w}_\ell \cdot T_n^{\beta^*(2+\delta)+(1+\delta) \dim(\mathbb{G})/2} \\
& = O \left( h^{-d(1+\delta)-d\beta_0(2+\delta)} T_n^{\beta^*(2+\delta)+(1+\delta) \dim(\mathbb{G})/2} \right),
\end{aligned}
$$

where the third inequality can be derived from Lemma 9.5 using $k = 2 + \delta$ combined with the Lebesgue-dominated convergence theorem. The fourth inequality can be inferred from the demonstration in Theorem 5.3, while the last inequality stems from applying the Plancherel theorem. Therefore, (8.80) follows if

$$h^{-d(1+\delta)-d\beta_0(2+\delta)} T_n^{\beta^*(2+\delta)+(1+\delta) \dim(\mathbb{G})/2} / n^{\delta/2} A_n(\beta_0) B_n(\beta^*) \to 0, \qquad (8.89)$$

which is equivalent to $n^{(-1/2+dp/2+\dim(\mathbb{G})p^*/4)\delta} \to 0$. Since $2dp + \dim(\mathbb{G})p^* < 2$. This completes the proof of the linear part.

### The other terms of (5.10)

We now turn to the nonlinear part of the Hoeffding decomposition described in (5.10). In light of this, the objective is to demonstrate that:

$$(nT_n)^{1/2} \left[ \bar{\mu}_n(\varphi, T_n, \tilde{\mathbf{t}}) - k\bar{U}_n^{(1)}(\pi_1 \bar{G}_{\varphi, T_n, \tilde{\mathbf{t}}}) \right] \to 0 \quad \text{in } L^2.$$
We obtain the aimed result by following the same procedures of treating the nonlinear part of (5.2). For (8.79), similar to Proposition 4.2, we can establish that $\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \prod_{j=1}^k f_{\bar{\mathbf{x}}}(\tilde{\mathbf{t}}_j) = o_{\mathbb{P}}(1)$. Based on this result and $\prod_{j=1}^k f_{\bar{\mathbf{x}}}(\tilde{\mathbf{t}}_j) > 0$, it follows that (8.79) holds. Hence, the proof of the Theorem is complete. $\square$

**Remark 8.2.** In the proof of Theorem 5.6, establishing (8.87) is crucial. To achieve this, we used the fact that the upper bound and lower bound of $\int_{\mathbb{G}} (\text{Re}(K_{T_n}(\mathbf{t}, \mathbf{u})))^2 d\mu(\mathbf{u})$ are of the same order. Yet, this does not hold for (S2*). Indeed, we could only demonstrate that the upper bound has an order $T_n^{2v+\dim(\mathbb{G})/2} \exp(2\gamma^* \cdot T_n^{\beta^*})$, while the lower bound is of order $T_n^{2v+\dim(\mathbb{G})/2} \exp(2\gamma^* (\eta \cdot T_n)^{\beta^*})$ for arbitrary $0 < \eta < 1$. Thus, we cannot ensure that (8.86) has a negligible order. A similar challenge arises in the case of (S3*).

## 8.10. Proof of Theorem 5.7

First, let us recall the definition

$$ \check{w}_n(\bar{\mathbf{t}}_1, \bar{\mathbf{z}}_1, \mathbf{y}) = K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{z}'_1}{h} \right) \text{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{z}_1) \right) \left( \bar{\varphi}_n(\mathbf{y}) - \prod_{j=1}^{k-1} f_{\bar{\mathbf{x}}}(\bar{\mathbf{t}}_j) \bar{\tau}^{(k)}(\varphi, \tilde{\mathbf{t}}) \right). $$

As in the proof of Theorem 5.6, it suffices to prove that (8.80) holds. We first find a lower bound for $K_{\ell,h}^D(\cdot)$ over any strictly compact subset A of $[-\pi/2, \pi/2]$. We consider sequences $b_n = h^{\beta_0/(2(k+2))}$ and $a_n = b_n + b_n^{k+2}$. Using the fact that $\phi_{\ell,h}^{K_{\ell,h}^D}(\cdot)$ vanishes outside of $[-1, 1]$, $K_{\ell,h}^D(\mathbf{u}_\ell)$ can be expressed as

$$ \frac{1}{2\pi} \int_{|\mathbf{s}_\ell| \le 1-a_n} e^{-i\mathbf{s}_\ell \cdot \mathbf{u}_\ell} \frac{\phi_{\ell,h}^{K_{\ell,h}^D}(\mathbf{s}_\ell)}{\phi_{\ell,h}^{U_\ell}(\mathbf{s}_\ell/h)} d\mathbf{s}_\ell + \frac{1}{2\pi} \int_{1-a_n < |\mathbf{s}_\ell| \le 1-b_n} e^{-i\mathbf{s}_\ell \cdot \mathbf{u}_\ell} \frac{\phi_{\ell,h}^{K_{\ell,h}^D}(\mathbf{s}_\ell)}{\phi_{\ell,h}^{U_\ell}(\mathbf{s}_\ell/h)} d\mathbf{s}_\ell \\ + \frac{1}{2\pi} \int_{1-b_n < |\mathbf{s}_\ell| \le 1} e^{-i\mathbf{s}_\ell \cdot \mathbf{u}_\ell} \frac{\phi_{\ell,h}^{K_{\ell,h}^D}(\mathbf{s}_\ell)}{\phi_{\ell,h}^{U_\ell}(\mathbf{s}_\ell/h)} d\mathbf{s}_\ell. \quad (8.90) $$

We show that the third term of (8.90) dominates the first and second terms. We note $\exp(\lambda_0((1-b_n)/h)^{\beta_0})b_n^{k+1} = c_n$. Since the third term on the right-hand side of (8.90), $\phi_{\ell,h}^{K_{\ell,h}^D}(\cdot)$ and $\phi_{\ell,h}^{U_\ell}(\cdot)$ are real-valued, with $\phi_{\ell,h}^{K_{\ell,h}^D}(\cdot)$ being nonnegative and $\phi_{\ell,h}^{U_\ell}(\cdot)$ being positive, we have, for $\mathbf{u}_\ell \in A$,

$$ \begin{aligned} & \frac{1}{2\pi} \int_{1-b_n < |\mathbf{s}_\ell| \le 1} e^{-i\mathbf{s}_\ell \cdot \mathbf{u}_\ell} \frac{\phi_{\ell,h}^{K_{\ell,h}^D}(\mathbf{s}_\ell)}{\phi_{\ell,h}^{U_\ell}(\mathbf{s}_\ell/h)} d\mathbf{s}_\ell \\ = & \frac{1}{2\pi} \int_{1-b_n < |\mathbf{s}_\ell| \le 1} \cos(\mathbf{s}_\ell \cdot \mathbf{u}_\ell) \frac{\phi_{\ell,h}^{K_{\ell,h}^D}(\mathbf{s}_\ell)}{\phi_{\ell,h}^{U_\ell}(\mathbf{s}_\ell/h)} d\mathbf{s}_\ell \\ \ge & \frac{1}{2\pi} \inf_{a \in A} \cos(a) \cdot \inf_{1-b_n < |\mathbf{s}_\ell| \le 1} \left(\phi_{\ell,h}^{U_\ell}(\mathbf{s}_\ell/h)\right)^{-1} \int_{1-b_n < |\mathbf{s}_\ell| \le 1} \phi_{\ell,h}^{K_{\ell,h}^D}(\mathbf{s}_\ell) d\mathbf{s}_\ell \\ \ge & C \cdot \exp\left(\lambda_0 \left(\frac{1-b_n}{h}\right)^{\beta_0}\right) \int_{1-b_n < |\mathbf{s}_\ell| \le 1} (1-\mathbf{s}_\ell)^k d\mathbf{s}_\ell = C \cdot c_n, \end{aligned} $$

for sufficiently large $n$. Therefore, the absolute value of the first term on the right-hand side of (8.90) is bounded by
$$ \sup_{|\mathbf{s}_\ell| \le 1 - a_n} \left| \left( \phi_{\mathbf{U}_\ell} (\mathbf{s}_\ell / h) \right)^{-1} \right| \cdot O(1) = O \left( \exp \left( \lambda_0 \left( \frac{1 - a_n}{h} \right)^{\beta_0} \right) \right). $$

In addition, since $(1 - a_n)^{\beta_0} = (1 - b_n - b_n^{k+2})^{\beta_0} \le (1 - b_n)^{\beta_0} - \beta_0 b_n^{k+2/2}$, for sufficiently large $n$, we obtain

$$ O \left( \exp \left( \lambda_0 \left( \frac{1 - a_n}{h} \right)^{\beta_0} \right) \right) = O \left( \exp \left( \lambda_0 \left( \frac{1 - b_n}{h} \right)^{\beta_0} \right) \cdot \exp \left( -\frac{\lambda_0 \beta_0}{2} \cdot h^{-\beta_0/2} \right) \right) = o(c_n). $$

Furthermore, we can also see that the absolute value of the second term on the right-hand side of (8.90) is bounded by

$$ \sup_{1-a_n < |\mathbf{s}_\ell| \le 1-b_n} \left| \left( \phi_{\mathbf{U}_\ell} (\mathbf{s}_\ell / h) \right)^{-1} \right| \cdot O(a_n - b_n) = O \left( \exp \left( \lambda_0 \left( \frac{1-b_n}{h} \right)^{\beta_0} \right) \cdot b_n^{k+2} \right) = o(c_n). $$

Hence, $\inf_{\mathbf{u}_\ell} |K_{\ell,h}^D(\mathbf{u}_\ell)| \ge C \cdot c_n$, for sufficiently large $n$. Notice that

$$
\begin{align*}
& \mathbb{E} \left( \left( \check{\mathring{w}}_n (\bar{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}) \right)^2 \right) \\
& \le \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 \bar{\phi}_n^2(\mathbf{s}) f_{\bar{\mathbf{X}}Y}((\mathbf{u}, \mathbf{w}), \mathbf{s}) du d\mu(\mathbf{w}) ds \\
& = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 \\
& \qquad \times f_{\bar{\mathbf{X}}Y}((\mathbf{u}, \mathbf{w}), \mathbf{s}) du d\mu(\mathbf{w}) ds \\
& = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left( \frac{1}{h^d} K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \right)^2 \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 \left( \bar{\tau}^{(k)} (\varphi, (\mathbf{s}, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{\mathbf{X}}}(\bar{\mathbf{t}}_j) + o(1) \right)^2 \\
& \qquad \times f_{\bar{\mathbf{X}}Y}((\mathbf{u}, \mathbf{w}), \mathbf{s}) du d\mu(\mathbf{w}) ds \\
& \ge \inf_{\bar{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \left( \bar{\tau}^{(k)} (\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{\mathbf{X}}}(\bar{\mathbf{t}}_j) \right)^2 \frac{1}{h^d} \int_{\mathbb{G}} \int_{A^d} \left( K_h^D(\mathbf{v}) \right)^2 \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 \\
& \qquad \times (f_{\bar{\mathbf{X}}}(\mathbf{t}'_1 - h\mathbf{v}, \mathbf{w}) - f_{\bar{\mathbf{X}}}(\mathbf{t}'_1, \mathbf{w})) dv d\mu(\mathbf{w}) \\
& \qquad + \inf_{\bar{\mathbf{t}}_1 \in \mathbb{R}^d \times \mathbb{G}} \inf_{\mathbf{w} \in \mathbb{G}} \left[ \left( \bar{\tau}^{(k)} (\varphi, (\bar{\mathbf{t}}_1, \bar{\mathbf{t}}_2, \dots, \bar{\mathbf{t}}_k)) \prod_{j=2}^k f_{\bar{\mathbf{X}}}(\bar{\mathbf{t}}_j) \right)^2 f_{\bar{\mathbf{X}}}(\mathbf{t}'_1, \mathbf{w}) \right] \frac{1}{h^d} \int_A \left( K_h^D(\mathbf{v}) \right)^2 dv \\
& \qquad \times \int_{\mathbb{G}} \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{w}) \right) \right)^2 d\mu(\mathbf{w})
\end{align*}
$$
$$
\begin{align*}
&= C \cdot \frac{1}{h^d} \int_{A^d} (K_h^D(\mathbf{v}))^2 d\mathbf{v} \int_{\mathbb{G}} (\mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{w})))^2 d\mu(\mathbf{w}) (1+o(1)) \\
&\geq C \cdot \frac{c_n^{2d}}{h^d} B_n (\beta^*) (1+o(1)),
\end{align*}
$$

where the last equality follows from the fact that $f_{\bar{\mathbf{x}}}(\cdot)$ is uniformly continuous and $B_n$ is defined in (8.81). We have also

$$
(\mathbb{E}[\check{w}_n(\bar{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y})])^2 = O(T_n^{k \dim(\mathbb{G})/2}) = o(c_n^{2d} h^{-d} B_n(\beta^*))
$$

and

$$
(\mathrm{Var}(\check{w}_n(\bar{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y})))^{1+\delta/2} \geq C \cdot c_n^{d(2+\delta)} h^{-d(1+\delta/2)} (B_n(\beta^*))^{1+\delta/2},
$$

for sufficiently large $n$. Now, let us consider the numerator in (8.80). Taking Advantage of the condition (F.ii), we observe that

$$
\begin{align*}
& \mathbb{E} \left( \left| \check{w}_n(\bar{\mathbf{t}}_1, \bar{\mathbf{Z}}_1, \mathbf{Y}) \right|^{2+\delta} \right) \\
&\le \int_{\mathbb{R}^d} \int_{\mathbb{R}^d \times \mathbb{G}} \left| K_h^D \left( \frac{\mathbf{t}'_1 - \mathbf{u}}{h} \right) \Re (K_{T_n}(\mathbf{t}_1, \mathbf{v})) \bar{\varphi}(\mathbf{s}) \right|^{2+\delta} f_{\bar{\mathbf{X}}\mathbf{Y}} ((\mathbf{u}, \mathbf{v}), \mathbf{s}) \, d\mathbf{u} d\mathbf{v} \, d\mathbf{s} \\
&\le \frac{\theta_\delta \| f_{\bar{\mathbf{X}}} \|_\infty}{h^{d(1+\delta)}} \int_{\mathbb{R}^d} \left| K_h^D(\mathbf{w}) \right|^{2+\delta} d\mathbf{w} \int_{\mathbb{G}} \left| \Re (K_{T_n}(\mathbf{t}_1, \mathbf{v})) \right|^{2+\delta} d\mu(\mathbf{v}) \\
&\le \frac{\theta_\delta \| f_{\bar{\mathbf{X}}} \|_\infty}{h^{d(1+\delta)}} \exp \left( d\lambda_0 (2+\delta)/h^{\beta_0} \right) \int_{\mathbb{R}^d} \left| K_h^D(\mathbf{w}) \right|^{2+\delta} d\mathbf{w} \\
&\qquad \times \sup_{\mathbf{v} \in \mathbb{G}} \left| \Re (K_{T_n}(\mathbf{t}_1, \mathbf{v})) \right|^{\delta} \int_{\mathbb{G}} \left( \Re (K_{T_n}(\mathbf{t}_1, \mathbf{v})) \right)^2 d\mu(\mathbf{v}) \\
&\le \frac{\theta_\delta \| f_{\bar{\mathbf{X}}} \|_\infty}{h^{d(1+\delta)}} \exp \left( d\lambda_0 (2+\delta)/h^{\beta_0} \right) \int_{\mathbb{R}^d} \exp \left( -2d\lambda_0/h^{\beta_0} \right) \left( K_h^D(\mathbf{w}) \right)^2 d\mathbf{w} \exp \left( -d\delta\lambda_0/h^{\beta_0} \right) \\
&\qquad \times \sup_{\mathbf{w} \in \mathbb{R}^d} \left| K_h^D(\mathbf{w}) \right|^{\delta} \\
&\qquad \times \sup_{\mathbf{v} \in \mathbb{G}} \left| K_{T_n}(\mathbf{t}_1, \mathbf{v}) \right|^{\delta} \int_{\mathbb{G}} \left| K_{T_n}(\mathbf{t}_1, \mathbf{v}) \right|^2 d\mu(\mathbf{v}) \\
&\le \frac{C}{h^{d(1+\delta)}} \exp \left( d\lambda_0 (2+\delta)/h^{\beta_0} \right) \cdot T_n^{\beta^*(2+\delta)+(1+\delta) \dim(\mathbb{G})/2}.
\end{align*}
$$

Therefore, (8.80) is derived if

$$
n^{-\delta/2} h^{-d\delta/2} \exp \left( d\lambda_0(2+\delta) \left( \frac{1}{h^{\beta_0}} - \left( \frac{1-b_n}{h} \right)^{\beta_0} \right) \right) b_n^{-d(2+\delta)(k+1)} T_n^{\dim(\mathbb{G})\delta/4} \to 0. \quad (8.91)
$$

This holds since, for large enough values of $n$ and any $\kappa > 0$,

$$
\exp \left( d\lambda_0(2+\delta) \left( \frac{1}{h^{\beta_0}} - \left( \frac{1-b_n}{h} \right)^{\beta_0} \right) \right)
$$
$$
\begin{align*}
&\le \exp \left( d\lambda_0(2 + \delta) \left( \frac{1}{h^{\beta_0}} - \frac{1}{h^{\beta_0}} + \frac{2\beta_0 b_n}{h^{\beta_0}} \right) \right) \\
&\le \exp \left( d\lambda_0(4 + 2\delta) h^{-\beta_0(1 - 1/(2(k+2)))} \right) \\
&\le \exp \left( d\lambda_0(4 + 2\delta) C^{-\beta_0(1 - 1/(2(k+2)))} (\log n)^{1 - 1/(2(k+2))} \right) = O(n^\kappa).
\end{align*}
$$

With an appropriate choice of $p^*$, the result above leads to (8.91), thus concluding the proof of the linear term. For the nonlinear terms in (5.10), we reach the desired results by using analogous reasoning as for (5.2). Consequently, the proof of the theorem is achieved. □

9. Appendix

This appendix contains supplementary information that is an essential part of providing a more
comprehensive understanding of the paper.

**Lemma 9.1 ([76]).** For any subset $S$ of $\hat{\mathbb{G}}$ with finite cardinality and $d_{\sigma} \times d_{\sigma}$ complex matrices $A^{\sigma}$, it holds that

$$
\int_{\mathbb{G}} \left| \sum_{\sigma \in S} d_{\sigma} \operatorname{Tr}(A^{\sigma} \sigma^{M}(\mathbf{u})) \right|^2 d\mu(\mathbf{u}) = \sum_{\sigma \in S} d_{\sigma} \|A^{\sigma}\|_{\mathrm{HS}}^2.
$$

**Lemma 9.2 ([76]).** For all $\sigma \in \hat{\mathbb{G}}$ and $g \in \mathbb{G}$, $\sigma^M(g)$ are unitary matrices, i.e., $(\sigma^M(g)^{-1})$ equals the conjugate transpose of $\sigma^M(g)$, so that

$$
(i) \quad \|A\sigma^M(g)\|_{HS} = \|\sigma^M(g)A\|_{HS} = \|A\|_{HS} \text{ for any } d_\sigma \times d_\sigma \text{ complex matrices } A;
$$

$$
(ii) \quad \| \sigma^M(g) \|_{HS} = d_{\sigma}^{1/2}.
$$

**Lemma 9.3 ([76]).** Assume that either (G1) or (G2) and the condition (A2) hold. Then, for each $j \in \{1, 2, 3\}$, under (Sj) – (ii), (Bj) holds with $q = \dim(\mathbb{G})/2$.

**Lemma 9.4 ([75]).** Let $d \in \mathbb{N}$ and $W_n : \mathbb{R}^d \to \mathbb{C}$ be a sequence of Borel functions such that $W_n(\mathbf{w}) \to W(\mathbf{w})$ for all $t \in \mathbb{R}^d$ and $\sup_n |W_n(\mathbf{w})| \le W^*(\mathbf{w})$, where $W : \mathbb{R}^d \to \mathbb{C}$ and $W^* : \mathbb{R}^d \to [0, \infty)$ are functions satisfying $\int_{\mathbb{R}^d} W^*(\mathbf{w})d\mathbf{w} < \infty$ and $\lim_{\|\mathbf{w}\| \to \infty} \|\mathbf{w}\|^d W^*(\mathbf{w}) = 0$. For given point $\mathbf{t}_1 \in \mathbb{R}^d$ and set $S$, let $f : \mathbb{R}^d \times S \to \mathbb{C}$ be a function such that $\sup_{\mathbf{s} \in S} |f(\mathbf{t}_1, \mathbf{s})| < \infty$, $\sup_{\mathbf{s} \in S} \int_{\mathbb{R}^d} |f(\mathbf{w}, \mathbf{s})|d\mathbf{w} < \infty$ and $\sup_{\mathbf{s} \in S} \sup_{\mathbf{w} \in \overline{B(0, \delta)}} |f(\mathbf{t}_1 + \mathbf{w}, \mathbf{s}) - f(\mathbf{t}_1, \mathbf{s})| \to 0$ as $\delta \to 0$. Then, for any sequence $a_n \to 0$, it holds that

$$
\sup_{\mathbf{s} \in S} \left| \frac{1}{a_n^d} \int_{\mathbb{R}^d} f(\mathbf{w}, \mathbf{s}) W_n \left( \frac{\mathbf{t}_1 - \mathbf{w}}{a_n} \right) d\mathbf{w} - f(\mathbf{t}_1, \mathbf{s}) \int_{\mathbb{R}^d} W(\mathbf{w}) d\mathbf{w} \right| \to 0.
$$

**Lemma 9.5 ([75]).** Assume that $\phi^{\mathbf{U}_j}(\cdot)$ satisfies (S1) with $\beta = \beta_0$ and that $\phi^{K_j} = 0$ on $\mathbb{R}\setminus[-1, 1]$. Then, there exists a constant $C_1 > 0$ depending only on $k$, $\phi^{\mathbf{U}_j}(\cdot)$ and $\phi^{K_j}(\cdot)$ such that

$$
\sup_{\mathbf{w}_j \in \mathbb{R}^n} \sup h^{k\beta_0} |K_{j,h}^D(\mathbf{w}_j)|^k \le C_1
$$

If we further assume that $\phi^{\mathbf{U}_j}(\cdot)$ and $\phi^{K_j}(\cdot)$ are continuously differentiable and that

$$
h^{k(\beta_0 - 1)} \sup_{\mathbf{w}_j \in \mathbb{R}: |\mathbf{w}_j| \le h^{-1}} \left| \left( (\phi^{\mathbf{U}_j})^{-1} \right)' (\mathbf{w}_j) \right|^k = O(1),
$$
then there exists a constant $C_2 > 0$ depending only on $k$, $\phi^{U_j}(\cdot)$ and $\phi^{K_j}(\cdot)$ such that

$$ \sup_n h^{k\beta_0} |K_{j,h}^D(\mathbf{w}_j)|^k \le C_2 \cdot |\mathbf{w}_j|^{-k}, $$

for all $\mathbf{w}_j \in \mathbb{R}$.

**Proposition 9.6.** [[76]] Assume that the condition (A2) holds. Then, for each $j \in \{1, 2, 3\}$, under (Sj)-(ii), $E(|K_{T_n}(\mathbf{t}_1, \mathbf{z})|^2)$ attains the lower bound given in (Bj) with $q = 0$.

**Proposition 9.7.** Let $L: \mathbb{R} \to \mathbb{R}$ be an $(m+1)$-times continuously differentiable symmetric function such that $L = 0$ on $\mathbb{R}\setminus[-1, 1]$, $L(0) = 1$, $L^{(k)}(0) = 0$ for all $1 \le k \le m-1$ and $L^{(m+1)}$ is absolutely continuous on $[-1, 1]$. Then, the function $K_j$ defined by $K_j(\mathbf{w}_j) = (2\pi)^{-1}\phi_L(\mathbf{w}_j)$ is an $m$th-order kernel and $\phi^{K_j}$ equals $L$.

## CRediT authorship contribution statement

**Nourelhouda Taachouche:** Conceptualization, Methodology, Investigation, Writing – Original Draft, Writing – Review & Editing.

**Salim Bouzebda:** Conceptualization, Methodology, Investigation, Writing – Original Draft, Writing – Review & Editing.

Both authors contributed equally to this work.

## Funding

This research received no external funding.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The authors extend their sincere gratitude to the Editor-in-Chief, the Associate Editor, and the referee for their invaluable feedback.

## Data availability

Not applicable.

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