$$ -r^{(m)}(\varphi, \mathbf{t}, \boldsymbol{\theta}) \leq \left| \kappa_1^{-m} C_1^{-1} \frac{1}{\varphi_{\mathbf{t},\boldsymbol{\theta}}(\mathbf{h})} \mathbb{E} \left( \varphi(Y_1, \dots, Y_m) \prod_{i=1}^{m} K \left( \frac{d_{\boldsymbol{\theta}_i}(X_i, t_i)}{h_i} \right) \right) - r^{(m)}(\varphi, \mathbf{t}, \boldsymbol{\theta}) \right| \\ \leq \frac{1}{\varphi_{\mathbf{t},\boldsymbol{\theta}}(\mathbf{h}) \kappa_1^m C_1} \mathbb{E} \left( \prod_{i=1}^{m} K \left( \frac{d_{\boldsymbol{\theta}_i}(X_i, t_i)}{h_i} \right) \left| r^{(m)}(\varphi, \mathbf{X}) - r^{(m)}(\varphi, \mathbf{t}, \boldsymbol{\theta}) \right| \right). $$

Considering Hypotheses (H.1), (C.2.1), (C.3.1), and (C.6), we have $\forall \mathbf{h} = (h_1, \dots, h_m) \in \prod_{j=1}^{m} (h_{n,j}, b_j)$ and $\forall \mathbf{t} \in S_X^m$:

$$ \begin{aligned} & \left| \kappa_1^{-m} C_1^{-1} \mathbb{E}(u_n(\varphi, \mathbf{t}, \boldsymbol{\theta}, \mathbf{h})) - r^{(m)}(\varphi, \mathbf{t}, \boldsymbol{\theta}) \right| \\ & \leq \frac{C_3}{\varphi_{\mathbf{t},\boldsymbol{\theta}}(\mathbf{h}) \kappa_1^m C_1} \mathbb{E} \left( \prod_{i=1}^{m} K \left( \frac{d_{\boldsymbol{\theta}_i}(X_i, t_i)}{h_i} \right) d_{X^m}^{\gamma}(\mathbf{X}, \mathbf{t}) \right) \\ & \leq \frac{C_3}{m} (d_{\boldsymbol{\theta}_1}(X_1, t_1) + \dots + d_{\boldsymbol{\theta}_m}(X_m, t_m))^{\gamma} \\ & \leq \frac{C_3}{m} (h_1 + \dots + h_m)^{\gamma} \\ & = \frac{C_3}{m} (b_1 + \dots + b_m)^{\gamma} \leq C_3' \tilde{b}_0^{\gamma}, \end{aligned} $$

where

$$ \tilde{b}_0 := \max_{1 \leq j \leq m} b_j. $$

With this, the proof of the theorem is concluded. □

### 8.2.4. Proof of Corollary 6

In this section, we establish Corollary 6 by applying Lemma A1. Following the same reasoning as in the case of functional regression, we adopt the following notation: $S_\Omega = S_H$, $A_i = X_i$, $\varphi(B_i) = \varphi(Y_i)$,

$$ \prod_{i=1}^{m} G(H_i, (\langle t_i, \boldsymbol{\theta}_i \rangle, A_i)) = \prod_{i=1}^{m} K(H_i^{-1} d_{\boldsymbol{\theta}_i}(t_i, A_i)) $$

$$ D_{n,k}(\langle t_j, \boldsymbol{\theta}_j \rangle) = H_{n,k}(t_j, \boldsymbol{\theta}_j), \quad \forall \mathbf{t} = (t_1, \dots, t_m) \in S_H^m, \text{ and } j = 1, \dots, m, $$

$$ \mathfrak{M}_n^{(m)}(\varphi, \mathbf{t}, \boldsymbol{\theta}; \mathbf{h}_{n,k}(\mathbf{t}, \boldsymbol{\theta})) = \hat{r}_n^{* (m)}(\varphi, \mathbf{t}, \boldsymbol{\theta}; \mathbf{h}_{n,k}(\mathbf{t}, \boldsymbol{\theta})), $$

$$ \mathfrak{M}(\varphi, \mathbf{t}, \boldsymbol{\theta}) = r^{(m)}(\varphi, \mathbf{t}, \boldsymbol{\theta}). $$

We choose $D_{n,k}^-(\xi_n, \langle t, \boldsymbol{\theta} \rangle)$ and $D_{n,k}^+(\xi_n, \langle t, \boldsymbol{\theta} \rangle)$ such that for all $j = 1, \dots, m$,

$$ \phi_{t_j}(D_{n,k}^-(\xi_n, \langle t_j, \boldsymbol{\theta}_j \rangle)) = \frac{\sqrt{\xi_{n,j} k}}{n}, $$

$$ \phi_{t_j}(D_{n,k}^+(\xi_n, \langle t_j, \boldsymbol{\theta}_j \rangle)) = \frac{k}{n\sqrt{\xi_{n,j}}}, $$

where $\xi_{n,j}$ are increasing sequences that belong to $(0, 1)$ and

$$ \xi_n = \prod_{j=1}^{m} \xi_{n,j}. $$
We denote $\mathbf{h}^{-}(\mathbf{t}, \boldsymbol{\theta}) = (h_{1}^{-}(t_{1}, \boldsymbol{\theta}_{1}), \dots, h_{m}^{-}(t_{m}, \boldsymbol{\theta}_{m}))$ and $\mathbf{h}^{+}(\mathbf{t}, \boldsymbol{\theta}) = (h_{1}^{+}(t_{1}, \boldsymbol{\theta}_{1}), \dots, h_{m}^{+}(t_{m}, \boldsymbol{\theta}_{m}))$, where

$h_j^{-}(t_j, \boldsymbol{\theta}_j) = D_{n,k}^{-}(\xi_n, \langle t_j, \boldsymbol{\theta}_j \rangle)$ and $h^+(t_j, \boldsymbol{\theta}_j) = D_{n,k}^+(\xi_n, \langle t_j, \boldsymbol{\theta}_j \rangle)$, for all $j = 1, \dots, m$.

We can easily see that, for all $j = 1, \dots, m$,

$$ \phi_{t_j}^{-1} \left( \frac{\sqrt{\xi_{n,j} k_{1,n}}}{n} \right) \le h^{-}(t_j, \boldsymbol{\theta}_j) \le \phi_{t_j}^{-1} \left( \frac{\sqrt{\xi_{n,j} k_{2,n}}}{n} \right), \quad (97) $$

$$ \phi_{t_j}^{-1} \left( \frac{k_{1,n}}{n \sqrt{\xi_{n,j}}} \right) \le h^{+}(t_j, \boldsymbol{\theta}_j) \le \phi_{t_j}^{-1} \left( \frac{k_{2,n}}{n \sqrt{\xi_{n,j}}} \right). \quad (98) $$

Using Condition (8) one determines that, for all $j = 1, \dots, m$, there exist constants $0 < \mu_j \le \nu_j < \infty$ such that

$$ \mu_j \phi^{-1} \left( \frac{\rho_{n,j} k_{1,n}}{n} \right) \le \phi_{t_j, \boldsymbol{\theta}_j}^{-1} \left( \frac{\rho_{n,j} k_{1,n}}{n} \right) \quad \text{and} \quad \phi_{t_j, \boldsymbol{\theta}_j}^{-1} \left( \frac{k_{2,n}}{\rho_{n,j} n} \right) \le \nu_j \phi^{-1} \left( \frac{k_{2,n}}{\rho_{n,j} n} \right). $$

We put $\rho_{n,j} = \sqrt{\xi_{n,j}}$, $h_{n,j} = \mu_j \phi^{-1} \left( \frac{\rho_{n,j} k_{1,n}}{n} \right)$ and $h'_{n,j} = \nu_j \phi^{-1} \left( \frac{k_{2,n}}{\rho_{n,j} n} \right)$; thus, $\mathbf{h}^{-}(\mathbf{t})$ and $\mathbf{h}^{+}(\mathbf{t})$ belong to the interval

$$ \mathcal{H}_n^{(m)} := \prod_{j=1}^{m} (h_{n,j}, h'_{n,j}). $$

We denote $\tilde{h}_n = \min_{1 \le j \le m} h_{n,j}$ and $\tilde{h}'_n = \max_{1 \le j \le m} h'_{n,j}$; therefore,

$$ h_j(t_j) \in (\tilde{h}_n, \tilde{h}'_n), \forall j = 1, \dots, m. $$

We also note, for all $\mathbf{b} = (b_1, \dots, b_m) \in (0, 1)^m$,

$$ \mathcal{H}_0^{(m)} := \prod_{j=1}^{m} (h_{n,j}, b_j), $$

and

$$ \tilde{b}_0 := \max_{1 \le j \le m} b_j. $$

Ultimately, we can choose constants $0 < \mu^* < \nu^* < \infty$ and a sequence $\{\rho_n^*\} \in (0, 1)$ while satisfying Condition (C.8) such that $\tilde{h}_n = \mu^* \phi^{-1} \left( \frac{\rho_n^* k_{1,n}}{n} \right)$, $\tilde{h}'_n = \nu^* \phi^{-1} \left( \frac{k_{2,n}}{\rho_n^* n} \right)$, and

$$ \mathcal{U}_n = \phi^{-1} \left( \frac{k_{2,n}}{\rho_n^* n} \right)^\gamma + \sqrt{\frac{\psi_{\mathcal{H}} \left( \frac{\log n}{n} \right) + \psi_\theta \left( \frac{\log n}{n} \right)}{n \phi \left( \mu^* \phi^{-1} \left( \frac{\rho_n^* k_{1,n}}{n} \right) \right)}}. $$

It is evident that $(L_0)$ is fulfilled due to Condition (C.3.1), and from (97) and (98), we can readily verify that the construction of $\mathbf{h}^{-}(\mathbf{t}, \boldsymbol{\theta})$ and $\mathbf{h}^{+}(\mathbf{t}, \boldsymbol{\theta})$ satisfies Condition $(L_1)$. $\square$
## 8.3. Proofs of Weak Convergence Results

### 8.3.1. Preliminaries of the Proofs

As previously noted, a direct approach proves ineffective when dealing with random bandwidths. Consequently, we often rely on general lemmas (see, for instance, [51]) to facilitate the application of results established for non-random bandwidths. In this section, we introduce key results from [111,112], which were obtained for a positive bandwidth $h_K$. These results are instrumental in the proofs that follow. We denote the bias term and the centered variate as the following quantities:

$$ 
\begin{aligned}
B_n(\langle t, \theta \rangle; h_K) &:= \frac{\mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle; h_K) - r^{(1)}(\varphi, t)\mathbb{E}(\hat{r}_{n,1}^{(1)}(1, \langle t, \theta \rangle; h_K))}{\mathbb{E}(\hat{r}_{n,1}^{(1)}(1, \langle t, \theta \rangle; h_K)}, \\
Q_n(\langle t, \theta \rangle; h_K) &:= (\hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle; h_K) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle; h_K))) \\
&\quad - r^{(1)}(\varphi, \langle t, \theta \rangle)(\hat{r}_{n,1}^{(1)}(1, \langle t, \theta \rangle; h_K) - \mathbb{E}(\hat{r}_{n,1}^{(1)}(1, \langle t, \theta \rangle; h_K))).
\end{aligned} 
\quad (99) $$

Decomposition (99) plays a pivotal role in our proof. Following the methodology employed in [111], we show that convergence in the quadratic mean to one is achieved and that the bias satisfies

$$ B_n(\langle t, \theta \rangle; h_K) = o(1) \text{ as } n \to \infty. $$

### Proof of Theorem 11

By applying the Cramér–Wold device, it is sufficient to demonstrate the convergence of the one-dimensional distribution to establish Theorem 11. In fact, leveraging the linearity of $v_n(\psi; k_K | t)$ is enough to show that

$$ v_n(\Phi; H_{n,k}(t, \theta) | \langle t, \theta \rangle) \to N(0, \sigma^2(\Phi, t, \theta)) $$

for all $\Phi$ of the form

$$ \Phi = \sum_{p=1}^{L} c_p \psi_p, \quad c_1, \dots, c_L \in \mathbb{R}, \quad \psi_1, \dots, \psi_L \in \mathcal{F}\mathfrak{R}_{\theta}. $$

Hence, we solely illustrate the convergence in a single dimension. Recall that we are dealing with

$$ 
\begin{aligned}
v_n(\psi; H_{n,k}(t, \theta) | \langle t, \theta \rangle) &= \sqrt{k}(\hat{r}_n^{(1)}(\psi, \langle t, \theta \rangle; H_{n,k}(t, \theta)) - r^{(1)}(\psi, \langle t, \theta \rangle)) \\
&= \sqrt{k}\left(\frac{\sum_{i=1}^n \varphi(Y_i)K(d_{\theta}(X_i, t)/H_{n,k}(t, \theta))}{\sum_{i=1}^n K(d_{\theta}(X_i, t)/H_{n,k}(t, \theta))} - r^{(1)}(\psi, \langle t, \theta \rangle)\right).
\end{aligned} 
\quad (100) $$

We set

$$ \varphi_M(Y) := \varphi(Y)1_{\{F(Y)<M\}}. $$

To obtain the desired result, we write

$$ \hat{r}_n^{(1)}(\varphi, \langle t, \theta \rangle; H_{n,k}(t, \theta)) - r^{(1)}(\varphi, \langle t, \theta \rangle) = \frac{1}{\hat{r}_{n,1}^{(1)}(1, t; H_{n,k}(t, \theta))} [I_1(\langle t, \theta \rangle; H_{n,k}) + I_2(t; H_{n,k})], $$

where

$$ I_1(\langle t, \theta \rangle; H_{n,k}) = \underbrace{(\hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle; H_{n,k}(t, \theta)) - \hat{r}_{n,2}^{(1)}(\varphi_M, \langle t, \theta \rangle; H_{n,k}(t, \theta)))}_{S_1(\langle t, \theta \rangle; H_{n,k})} - \mathbb{E}(S_1(\langle t, \theta \rangle; H_{n,k})) $$
$$ + \underbrace{\mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t; H_{n,k}(\langle t, \theta \rangle, \theta)) - r^{(1)}(\varphi, \langle t, \theta \rangle))}_{S_2(t; H_{n,k})}, \quad (101) $$

and

$$ I_2(\langle t, \theta \rangle; H_{n,k}) = \left( \hat{r}_{n,2}^{(1)}(\varphi_M, \langle t, \theta \rangle; H_{n,k}(t, \theta)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, \langle t, \theta \rangle; H_{n,k}(t, \theta)) \right. \\ \left. - r^{(1)}(\varphi, t) (\hat{r}_{n,1}^{(1)}(1, \langle t, \theta \rangle; H_{n,k}(t, \theta)) - 1) \right). \quad (102) $$

To derive the desired results, we adopt the strategy of [213].

### Proof of Theorem 12

In this section, we utilize the same methodology as in [112] and previously in [109]. This approach involves applying the blocking method, which partitions a strictly stationary sequence $(X_1, \dots, X_n)$ into $2v_n$ equal-sized blocks, each of length $n - 2v_n a_n$. It is important to refer to the notation outlined in the proof of Lemma A2. The primary objective is to establish the asymptotic equicontinuity of the conditional empirical process

$$ \{ v_n(\psi | \langle t, \theta \rangle) = \sqrt{k}(\hat{r}_n^{(1)}(\psi, \langle t, \theta \rangle; H_{n,k}(t, \theta)) - r^{(1)}(\varphi, \langle t, \theta \rangle)), \psi \in \mathcal{F} \}_{\mathcal{R}_{\theta}, \varphi \in \mathcal{F}}. $$

Let us introduce, for any $\varphi K \in \mathcal{F} \mathcal{R}_{\theta}$ and $t \in \mathcal{X}$,

$$ W_n(\langle t, \theta \rangle, \varphi; H_{n,k}(t, \theta)) := \sum_{i=1}^{n} \varphi(Y_i) K \left( \frac{d_{\theta}(X_i, t)}{H_{n,k}(t, \theta)} \right) - n \mathbb{E} \left\{ \varphi(Y_1) K \left( \frac{d_{\theta}(X_1, t)}{H_{n,k}(t, \theta)} \right) \right\}, \quad (103) $$

$$ \begin{aligned} v_n(\varphi | \langle t, \theta \rangle) &= \sqrt{k}(\hat{r}_n^{(1)}(\varphi, \langle t, \theta \rangle; H_{n,k}(t, \theta)) - r^{(1)}(\varphi, \langle t, \theta \rangle)) \\ &:= \sqrt{k}(\hat{r}_n^{(1)}(\psi, \langle t, \theta \rangle; H_{n,k}(t, \theta)) - r^{(1)}(\varphi, \langle t, \theta \rangle)). \end{aligned} \quad (104) $$

Then, we have

$$ \begin{aligned} v_n(\psi | \langle t, \theta \rangle) &= \sqrt{k}(\hat{r}_n^{(1)}(\psi, t; H_{n,k}(t, \theta)) - r^{(1)}(\varphi, t)) \\ &= \sqrt{k} \left( \frac{\sum_{i=1}^{n} \varphi(Y_i) K(d_{\theta}(X_i, t) / H_{n,k}(t, \theta))}{\sum_{i=1}^{n} K(d_{\theta}(X_i, t) / H_{n,k}(t, \theta))} - r^{(1)}(\varphi, \langle t, \theta \rangle) \right) \\ &= \frac{1}{\sqrt{k}} \frac{1}{\mathcal{R}_{n,1}^{(1)}(\varphi, \langle t, \theta \rangle; H_{n,k}(t, \theta))} \mathcal{W}_n(\langle t, \theta \rangle, \varphi; H_{n,k}(t, \theta)) \\ &\quad - \frac{\mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle; H_{n,k}(t, \theta)))}{\mathcal{R}_{n,1}^{(1)}(\varphi, \langle t, \theta \rangle) \mathbb{E}(\hat{r}_{n,1}^{(1)}(\varphi, \langle t, \theta \rangle; H_{n,k}(t, \theta)))} \frac{1}{\sqrt{k}} \mathcal{W}_n(\langle t, \theta \rangle, 1; H_{n,k}(t, \theta)) \\ &\quad - \sqrt{k} B_n(\langle t, \theta \rangle; H_{n,k}(t, \theta)), \end{aligned} \quad (105) $$

where for $h_K > 0$, we have

$$ \check{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle, h_K) := \frac{1}{n\varphi(h_K(t))} \sum_{i=1}^{n} \varphi(Y_i) \Delta_i(t, h_K), $$

$$ \check{r}_{n,1}^{(1)}(1, \langle t, \theta \rangle, h_K) := \frac{1}{n\varphi(h_K(t))} \sum_{i=1}^{n} \Delta_i(t, \theta, h_K). $$
We investigate the asymptotic equicontinuity of each of the preceding terms. For a given class of functions $\mathcal{G}$, we let $\alpha_n(\cdot)$ denote an empirical process based on $(X_1, Y_1), \dots, (X_n, Y_n)$, indexed by $\mathcal{G}$:

$$ \alpha_n(g) = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \{g(X_i, Y_i) - \mathbb{E}(g(X_i, Y_i))\}, \quad \text{with} \quad \|\alpha_n(g)\|_{\mathcal{G}} = \sup_{g \in \mathcal{G}} |\alpha_n(g)|, $$

and for a measurable function $\varphi(\cdot)$ and $t \in \mathcal{H}, \theta \in \theta$, we set

$$ \eta_{n,t,\theta,\varphi,K}(u,v,h_K) = \varphi(v)K\left(\frac{d_{\theta}(u,t)}{h_K}\right), \quad \text{for } u,v \in \mathcal{X}. $$

That implies

$$ \frac{1}{\sqrt{n\varphi(h_K)}} W_n(\langle t, \theta \rangle, \varphi, h_K) = \frac{1}{\sqrt{\varphi(h_K)}} \alpha_n(\eta_{n,t,\theta,K}). $$

Again, keeping in mind that $1_{\{D_n^- \le H_{n,k}(t,\theta) \le D_n^+\}} \xrightarrow{a.co} 1$ when $\frac{k}{n} \to 0$, we establish the asymptotic equi-continuity of

$$ \left\{ \sqrt{\frac{n}{k}} \alpha_n(\eta_{n,t,\theta,\varphi,K}) : \eta_{n,t,\theta,\varphi,K} \in \mathcal{F}\mathcal{R}_{\theta} \right\}, $$

which means, for every $\varepsilon > 0$, that

$$ \lim_{b \to 0} \limsup_{n \to \infty} \mathbb{P} \left\{ \sqrt{\frac{n}{k}} \left\| \alpha_n \left( \eta_{n,t,\theta,\varphi,K} \right) \right\|_{\mathcal{F}\mathcal{R}_{\theta}(b, \|\cdot\|_p)} > \varepsilon \right\} = 0, $$

where

$$ \mathcal{F}\mathcal{R}_{\theta}(b, \|\cdot\|_p) = \left\{ \eta_{n,t,\theta,\varphi_1,K_1} - \eta_{n,t,\theta,\varphi_2,K_2} : \left\| \eta_{n,t,\theta,\varphi_1,K_1} - \eta_{n,t,\theta,\varphi_2,K_2} \right\|_p < b, \eta_{n,t,\theta,\varphi_1,K_1}, \eta_{n,t,\theta,\varphi_2,K_2} \in \mathcal{F}\mathcal{R}_{\theta} \right\}. $$

This approach entails working with the independent block sequence $\{\zeta_j = (\zeta_{j,1}, \zeta_{j,2})\}_{j=1}^{\infty}$ in place of the dependent sequence, facilitated by the results in [119]. Accordingly, we obtain

$$ \mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{n} \left( \varphi(Y_j) K \left( \frac{d_{\theta}(X_j, t)}{H_{n,k}(t, \theta)} \right) - \mathbb{P}(\eta_{n,t,\theta,\varphi,K}(H_{n,k}(t, \theta))) \right) \right\|_{\mathcal{F}\mathcal{R}_{\theta}(b, \|\cdot\|_p)} > \delta \right\} \\ \le 2\mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \sum_{i \in H_j} \left( \varphi(\zeta_i) K \left( \frac{d_{\theta}(\zeta_i, t)}{H_{n,k}(t, \zeta)} \right) - \mathbb{P}(\eta_{n,t,\theta,\varphi,K}(H_{n,k}(t, \zeta))) \right) \right\|_{\mathcal{F}\mathcal{R}_{\theta}(b, \|\cdot\|_p)} > \delta' \right\} \\ + 2(v_n - 1)\beta_{a_n}, \qquad (106) $$

where $H_{n,k}(t, \zeta)$ is defined by

$$ H_{n,k}(t, \zeta) = \min \left\{ h \in \mathbb{R}^{+} : \sum_{i=1}^{n} 1_{B_{\theta}(t,h)}(\zeta_i) = k \right\}, \qquad (107) $$

We choose

$$ a_n = [(\log n)^{-1} (n^{p-2} \phi^p(h_K))^{1/2(p-1)}] \text{ and } v_n = \left[ \frac{n}{2a_n} \right] - 1. $$

Note that in our context, $a_n$ is given by the following expression:
$$a_n = (\log n)^{-1} (n^{-2}k^p)^{1/2(p-1)}.$$

Utilizing Condition (C.5.1), we have $(v_n - 1)\beta_{a_n} \to 0$ as $n \to \infty$. Consequently, we focus on the right-hand side term of (106). We begin by assuming that the blocks are independent. Symmetrization is performed using a sequence $\{\epsilon_j\}_{j \in \mathbb{N}^*}$ of i.i.d. Rademacher variables, i.e., random variables satisfying $\mathbb{P}(\epsilon_j = 1) = \mathbb{P}(\epsilon_j = -1) = 1/2$. It is important to note that the sequence $\{\epsilon_j\}_{j \in \mathbb{N}^*}$ is independent of the sequence $\{\zeta_i = (\zeta_i, \zeta_i)\}_{i \in \mathbb{N}^*}$. Thus, we need to establish the following for all $\delta > 0$:

$$\lim_{b \to 0} \limsup_{n \to \infty} \mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \epsilon_j \sum_{i \in H_j} \left( \varphi(\zeta_i) K \left( \frac{d(\zeta_i, t)}{H_{n,k}(t, \zeta)} \right) \right) \right\|_{\mathcal{F}\mathcal{R}_{\theta(b, \|\cdot\|_p)}} > \delta \right\} = 0.$$

Again, using the fact that $\mathbb{1}_{\{D_n^- \le H_{n,k}(t,\zeta) \le D_n^+\}} \xrightarrow{a.co} 1$ when $\frac{k}{n} \to 0$ and [155], it suffices to show that

$$\lim_{b \to 0} \limsup_{n \to \infty} \mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \epsilon_j \sum_{i \in H_j} \left( \varphi(\zeta_i) K \left( \frac{d_{\theta}(\zeta_i, t)}{D_n^-} \right) \right) \right\|_{\mathcal{F}\mathcal{R}_{\theta(b, \|\cdot\|_p)}} > \delta \right\} = 0.$$

Since the $p^{th}$-conditional moment satisfies (C.4.2), we can truncate and obtain, for each $\lambda > 0$, as $n \to \infty$,

$$k^{-1/2} \sum_{j=1}^{n} \mathbb{E} \left( \kappa_2 F(\zeta_i) \mathbb{1}_{\{F(\zeta_i) \ge \lambda(M_n)^{1/2(p-1)}\}} \right) \quad (108)$$

$$\begin{aligned}
&= k^{-1/2} \int_{0}^{\infty} \mathbb{P} \left( \kappa_2 F \mathbb{1}_{\{F \ge \lambda(M_n)^{1/2(p-1)}\}} \ge t \right) dt \\
&= k^{-1/2} \int_{0}^{\lambda(M_n)^{1/2(p-1)}} \mathbb{P} \left( F \ge \lambda(M_n)^{1/2(p-1)} \right) dt \\
&\quad + k^{-1/2} \int_{\lambda(M_n)^{1/2(p-1)}}^{\infty} \mathbb{P} \left( F \ge t \right) dt \\
&\xrightarrow{n \to \infty} 0.
\end{aligned} \quad (109)$$

Hence,

$$\exists \lambda_n \xrightarrow{n \to \infty} 0: \quad k^{-1/2} \mathbb{E} \left( \kappa_2 F \mathbb{1}_{\{F \ge \lambda_n(M_n)^{1/2(p-1)}\}} \right) \xrightarrow{n \to \infty} 0.$$

Then, it suffices to show

$$\lim_{b \to 0} \limsup_{n \to \infty} \mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \epsilon_j \sum_{i \in H_j} \left( \varphi(\zeta_i) K \left( \frac{d(\zeta_i, t)}{D_n^-} \right) \right) \mathbb{1}_{\{\kappa_2 F(\zeta_i) \le \lambda_n M_n^{1/2(p-1)}\}} \right\|_{\mathcal{F}\mathcal{R}_{\theta(b, \|\cdot\|_p)}} > \delta \right\} = 0.$$

We have the following:

$$v_n^{(2)}(\eta_{n,t}, \varphi, K) = k^{-1/2} \sum_{j=1}^{v_n} \epsilon_j \sum_{i \in H_j} \left( \varphi(\zeta_i) K \left( \frac{d(\zeta_i, t)}{D_n^-} \right) \right) \mathbb{1}_{\{\kappa_2 F(\zeta_i) \le \lambda_n M_n^{1/2(p-1)}\}}.$$

This is achieved by using the chaining method. In [109], $b_q = b2^{-q}$, $q = 0, \dots, q_n$ was provided, where $q_n$

$$2^{-1} \lambda_n (\log (n))^{-1} \le b_{q_n}^2 \le \lambda_n (\log (n))^{-1}, \quad (110)$$
and we let class $\mathcal{F}\mathfrak{R}_{\theta q}$ of measurable functions of $\mathcal{F}\mathfrak{R}_{\theta}$

$$ \#\mathcal{F}\mathfrak{R}_{\theta q} = N_q := N(b_q, \mathcal{F}\mathfrak{R}_{\theta}, \|\cdot\|_p) \quad \sup_{\eta_{n,t,\theta,\phi_1,K_1} \in \mathcal{F}\mathfrak{R}_{\theta}} \min_{\eta_{n,t,\theta,\phi_2,K_2} \in \mathcal{F}\mathfrak{R}_{\theta q}} \left\| \eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2} \right\|_p \le b_q. $$

There is a map $\pi_q: \mathcal{F}\mathfrak{R}_{\theta} \to \mathcal{F}\mathfrak{R}_{\theta q}$ that takes each $\eta_{n,t,\theta,\phi,K} \in \mathcal{F}\mathfrak{R}_{\theta}$ to its closest function in $\mathcal{F}\mathfrak{R}_{\theta q}$ such that

$$ \left\| \eta_{n,t,\theta,\phi,K} - \pi_q(\eta_{n,t,\theta,\phi,K}) \right\|_p \le b_q. $$

We apply the chaining method

$$ \begin{aligned} & \sup_{\eta_{n,t,\theta,\phi_1,K_1} \eta_{n,t,\theta,\phi_2,K_2} \in \mathcal{F}\mathfrak{R}_{\theta}} v_n^{(2)}(\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}) \\ & \qquad \|\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}\|_p \le b \\ & \le \sup_{\eta_{n,t,\theta,\phi_1,K_1} \eta_{n,t,\theta,\phi_2,K_2} \in \mathcal{F}\mathfrak{R}_{\theta}} v_n^{(2)}(\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}) \\ & \qquad \|\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}\|_p \le b \bar{q}_n \\ & + 2 \sum_{q=1}^{q_n} \sup_{\eta_{n,t,\theta,\phi_1,K_1} \eta_{n,t,\theta,\phi_2,K_2} \in (\mathcal{F}\mathfrak{R}_{\theta})_{q-1}} v_n^{(2)}(\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}) \\ & \qquad \|\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}\|_p \le 3b_q \\ & + \sup_{\eta_{n,t,\theta,\phi_1,K_1} \eta_{n,t,\theta,\phi_2,K_2} \in (\mathcal{F}\mathfrak{R}_{\theta})_0} v_n^{(2)}(\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}). \end{aligned} \quad (111) $$

We let $\delta_q$ be in such a way that

$$ \delta_q = (b_q)^{1/2} \lor (3b_q(8 + c_{p,\beta}^2)^{1/2}(\log N_q)^{1/2}). \quad (112) $$

We let $r$ be chosen so small in such a way that

$$ 2 \sum_{q=1}^{+\infty} \delta_q \le \delta. $$

Therefore, from (111), we readily infer that

$$ \mathbb{P} \left\{ \begin{array}{l} \sup_{\eta_{n,t,\theta,\phi_1,K_1} \eta_{n,t,\theta,\phi_2,K_2} \in \mathcal{F}\mathfrak{R}_{\theta}} v_n^{(2)}(\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}) \ge 3\delta \\ \|\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}\|_p \le b \end{array} \right\} \\ \\ \le \mathbb{P} \left\{ \begin{array}{l} \sup_{\eta_{n,t,\theta,\phi_1,K_1} \eta_{n,t,\theta,\phi_2,K_2} \in \mathcal{F}\mathfrak{R}_{\theta}} v_n^{(2)}(\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}) \ge \delta \\ \|\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}\|_p \le b q_n \end{array} \right\} \\ \\ + 2 \sum_{q=1}^{q_n} \mathbb{P} \left\{ \begin{array}{l} \sup_{\eta_{n,t,\theta,\phi_1,K_1} \eta_{n,t,\theta,\phi_2,K_2} \in (\mathcal{F}\mathfrak{R}_{\theta})_{q-1}} v_n^{(2)}(\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}) \ge \delta_q \\ \|\eta_{n,t,\theta,\phi_1,K_1} - \eta_{n,t,\theta,\phi_2,K_2}\|_p \le 3b_q \end{array} \right\} \end{array} \right\} $$
$$ +\mathbb{P}\left\{ \sup_{\substack{\eta_{n,t,\theta,\varphi_1,K_1},\eta_{n,t,\theta,\varphi_2,K_2} \in (\mathcal{F}\mathfrak{R}_{\theta})_0 \\ \|\eta_{n,t,\theta,\varphi_1,K_1} - \eta_{n,t,\theta,\varphi_2,K_2}\|_{p} \le 2^b}} v_n^{(2)}(\eta_{n,t,\theta,\varphi_1,K_1} - \eta_{n,t,\theta,\varphi_2,K_2}) \ge \delta \right\} =: \mathbb{A} + \mathbb{B} + \mathbb{C}. $$

By the fact that the terms composing $v_n^{(2)}(\eta_{n,t,\theta,\varphi,K})$ are bounded by $a_n\lambda_n M_n^{1/2(p-1)}$, and by applying the Bernstein inequality, we obtain

$$ \mathbb{B} \le 2 \sum_{q=1}^{q_n} \exp \left( 2 \log N_q - \frac{\delta_q^2 k}{nb_q^2 c_{p,\beta}^2 + (4/3)\delta_q a_n \lambda_n n^{p/2(p-1)} \phi(h_K)^{(p-2)/2(p-1)}} \right). $$

By using (110), we have

$$ \delta_q a_n \lambda_n n^{p/2(p-1)} \phi(h_K)^{(p-2)/2(p-1)} = (4/3) \delta_q \lambda_n k (\log(n))^{-1} \le (8/3) n b_q^2 \delta_q \le 8 n b_q^2, $$

that means

$$ \begin{aligned} \mathbb{B} &\le 2 \sum_{q=1}^{q_n} \exp \left( 2 \log N_q - \frac{\delta_q^2}{(8+c_{p,\beta}^2)b_q^2} \right) \le 2 \sum_{q=1}^{q_n} \exp \left( - \frac{\delta_q^2}{2(8+c_{p,\beta}^2)b_q^2} \right) \\ &\le 2 \sum_{q=1}^{\infty} \exp \left( - \frac{2^q}{2(8+c_{p,\beta}^2)b} \right) \to 0 \quad \text{as} \quad b \to 0. \end{aligned} \quad (113) $$

In view of (112), we assume that $\delta < 3$. In a similar way, we have

$$ \mathbb{C} \le 2 \exp \left( 2 \log N_0 - \frac{\delta^2}{(8 + c_{p,\beta}^2)b^2} \right) \to 0 \quad \text{as} \quad b \to 0. $$

Finally, by (110), it suffices to prove, for each $\delta > 0$,

$$ \lim_{n \to \infty} \mathbb{P} \left\{ \| v_n^{(2)}(\eta_{n,t,\theta,\varphi,K}) \|_{\mathcal{F}\mathfrak{R}_{\theta}(\lambda_n^{1/2}(\log(n))^{-1/2}, \|\cdot\|_p)} \ge \delta \right\} = 0. $$

Making use of the square root trick (Lemma 5.2 [214])—see also [215] in a similar way to that in [109]—we obtain

$$ \begin{aligned} & \mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \epsilon_j \sum_{i \in H_j} \varphi(\zeta_i) K \left( \frac{d\theta(\zeta_i, t)}{D_n^-} \right) \right\| \\ & \times 1_{\{\kappa_2 F(\zeta_i) \le \lambda_n (n/\phi(h_K))^{1/2(p-1)}\}} \|_{\mathcal{F}\mathfrak{R}_{\theta}(\lambda_n^{1/2}(\log(n))^{-1/2}, \|\cdot\|_p)} \ge 2\delta \right\} \\ \le & \mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \epsilon_j \sum_{i \in H_j} \varphi(\zeta_i) K \left( \frac{d\theta(\zeta_i, t)}{D_n^-} \right) \right\| \\ & \times 1_{\{\kappa_2 F(\zeta_i) \le \lambda_n (n/\phi(h_K))^{1/2(p-1)}\}} \|_{\mathcal{F}\mathfrak{R}_{\theta}(\lambda_n^{1/2}(\log(n))^{-1/2}, \|\cdot\|_p)} \ge 2\delta, \right. \\ & \left. \left\| k^{-1} \sum_{j=1}^{v_n} \left( \sum_{i \in H_j} \varphi(\zeta_i) K \left( \frac{d\theta(\zeta_i, t)}{D_n^-} \right) 1_{\{\kappa_2 F(\zeta_i) \le \lambda_n (n/\phi(h_K))^{1/2(p-1)}\}} \right) \right\|^2 \|_{\mathcal{F}\mathfrak{R}_{\theta}(\lambda_n^{1/2}(\log(n))^{-1/2}, \|\cdot\|_p)} \right\} \end{aligned} \quad (114) $$
$$
\left\{
\begin{aligned}
&\le 64\lambda_n c_{p,\beta}^2 (\log(n))^{-1} \\
&+ \mathbb{P} \left\{
\begin{aligned}
&\left\| k^{-1} \sum_{j=1}^{v_n} \left( \sum_{i \in H_j} \varphi(\zeta_i) K \left( \frac{d_{\theta}(\zeta_i, t)}{D_n^{-}} \right) 1_{\{\kappa_2 F(\zeta_i) \le \lambda_n (n/\phi(h_K))^{1/2}(p-1)\}} \right) \right\|^2 \\
&\qquad \cdot \mathcal{F}_{\mathcal{R}_{\theta}^{\lambda_n^{1/2}(\log(n))-1/2, \|\cdot\|_p}} \\
&> 64\lambda_n c_{p,\beta}^2 (\log(n))^{-1}
\end{aligned}
\right\} \\
&:= \mathbb{P}(A_1) + \mathbb{P}(A_2).
\end{aligned}
\right.
$$

Let us introduce the semi-norm

$$
\tilde{d}_{n\phi,2} := \left( k^{-1} \sum_{j=1}^{v_n} \sum_{i \in H_j} \left| \eta_{n,t,\theta,\phi_1,K_1}(\zeta_i, \zeta_j) - \eta_{n,t,\theta,\phi_2,K_2}(\zeta_i, \zeta_j) \right|^2 \right)^{1/2}
$$

and the covering number defined for any class of functions $\mathcal{E}$ by

$$
\tilde{N}_{n\phi,2}(u, \mathcal{E}) := N_{n\phi,2}(u, \mathcal{E}, \tilde{d}_{n\phi,2}).
$$

By utilizing the latter approach, we can bound $\mathbb{P}(A_1)$ (the detailed calculations can be found in [110]). Similarly, as in [110] and earlier in [109], leveraging the independence between the blocks and Condition (C.4.3), we apply Lemma 5.2 from [214] to obtain

$$
\mathbb{P}(A_2) \to 0.
$$

Consequently, the theorem is established. □

8.3.2. Proof of Theorem 14

As previously mentioned, the examination of the weak convergence of the conditional
U-process is grounded in the analysis of two components: the truncated part and the
remainder part.

**Lemma 1.** Let $\mathcal{F}_m\mathcal{R}_\theta^m$ be a uniformly bounded class of measurable canonical functions from $\mathcal{X}^m \times \mathcal{Y}^m \to \mathbb{R}$, where $m \ge 2$. Assume that there exist finite constants **a** and **b** such that the covering number of $\mathcal{F}_m\mathcal{R}_\theta^m$ satisfies

$$
N(\epsilon, \mathcal{F}_m \mathcal{R}_{\theta}^m, \| \cdot \|_{L_2(Q)}) \leq a \epsilon^{-b}, \quad (115)
$$

for every $\epsilon > 0$ and every probability measure $Q$. If the mixing coefficient $\beta$ of the stationary sequence $\{Z_i = (X_i, Y_i)\}_{i \in \mathbb{N}^*}$ fulfills

$$
\beta_s s^r \to 0, \text{ as } s \to \infty \quad (116)
$$

for some $r > 1$, then

$$
\left\| n^{-\frac{3m+1}{2}} \sqrt{k^m} \sum_{i \in I_m^m} G_{\phi, t, \theta, h_{n,k}}(\mathbf{X}_i, \mathbf{Y}_i) \right\|_{\mathcal{F}_m \mathcal{R}_{\theta}^m} \xrightarrow{\mathbb{P}} 0.
$$
### 8.3.3. Proof of Theorem 15

It is well established that the weak convergence of an empirical process can be derived from its finite-dimensional convergence and its asymptotic equicontinuity, provided certain conditions are satisfied. Theorem 14 establishes the finite-dimensional convergence of the conditional U-process {$\mu_n(\varphi, \mathbf{t}, \theta; \mathbf{h}(\mathbf{t}))$}$_{\mathcal{F}_m \otimes \mathcal{F}_\theta^m}$. Thus, the remaining task is to demonstrate its asymptotic equicontinuity. We decompose the U-process $\mu_n(\varphi, \mathbf{t}, \theta)$ into two components: the truncated part and the remainder part.

$$ \mu_n(\varphi, \mathbf{t}, \theta, \mathbf{h}(\mathbf{t})) = \mu_n^{(T)}(\varphi, \mathbf{t}, \theta, \mathbf{h}(\mathbf{t})) + \mu_n^{(R)}(\varphi, \mathbf{t}, \theta, \mathbf{h}(\mathbf{t})). $$

Following the same reasoning as in [155], we also have

$$ \mu_n(\varphi, \mathbf{t}, \theta; \mathbf{D}_n^+) \le \mu_n(\varphi, \mathbf{t}, \theta; \mathbf{h}_{n,k}(t)) \le \mu_n(\varphi, \mathbf{t}, \theta; \mathbf{D}_n^-). \quad (117) $$

Therefore, it suffices to establish the weak convergence of $\mu_n(\varphi, \mathbf{t}, \theta; \mathbf{D}_n^-)$ and $\mu_n(\varphi, \mathbf{t}, \theta; \mathbf{D}_n^+)$ rather than directly studying $\mu_n(\varphi, \mathbf{t}, \theta; \mathbf{h}_{n,k}(t))$. The steps of the proof closely follow those in [112], with the distinction that we now consider a multivariate bandwidth $\mathbf{h}(\mathbf{t})$ instead of a univariate bandwidth $h_K$. In this section, we present the proof for $\mu_n^{(T)}(\varphi, \mathbf{t}, \theta; \mathbf{D}_n^-)$, with the same approach applicable to $\mu_n^{(T)}(\varphi, \mathbf{t}, \theta; \mathbf{D}_n^+)$. As shown earlier, the truncated part $\mu_n^{(T)}(\varphi, \mathbf{t}, \theta, \mathbf{h}(\mathbf{t}))$ is decomposed following Hoeffding's decomposition:

$$ \mu_n^{(T)}(\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-) = \sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)} \left\{ m u_n^{(1)} \left( \pi_{1,m} G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(T)} \right) + \sum_{p=2}^{m} \frac{m!}{(m-p)!} u_n^{(p)} \left( \pi_{p,m} G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(T)} \right) \right\}. $$

We first investigate the linear term $m\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}u_n^{(1)}(\pi_{1,m}G_{\varphi,\mathbf{t},\theta,\mathbf{D}_n^-}^{(T)})$. Notice that

$$ m\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}u_n^{(1)}(\pi_{1,m}G_{\varphi,\mathbf{t},\theta}^{(T)}) = \frac{m\sqrt{\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}}{\sqrt{n}} \sum_{i=1}^{n} \pi_{1,m}G_{\varphi,\mathbf{t},\theta}^{(T)}(\mathbf{X}_i, \mathbf{Y}_i) $$

We can write

$$ \begin{aligned} \pi_{1,m} G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(T)} (x, y) &= \mathbb{E} \left[ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(T)} (x, X_2, \dots, X_m), (y, X_2, \dots, X_m) \right] - \mathbb{E} \left[ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(T)} (\mathbf{X}, \mathbf{Y}) \right] \\ &= \mathbb{E} \left[ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(T)} (\mathbf{X}, \mathbf{Y}) \mid (X_1, Y_1) = (x, y) \right] - \mathbb{E} \left[ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(T)} (\mathbf{X}, \mathbf{Y}) \right]. \end{aligned} $$

We need to introduce a new function

$$ \begin{aligned} S_{\varphi, \mathbf{t}, \theta, \mathbf{h}} : \mathcal{X} \times \mathcal{Y} &\to \mathbb{R} \\ (x, y) &\to m \mathbb{E} \left[ \varphi(y) \tilde{K} \left( \frac{d(\mathbf{t}, \mathbf{x})}{\mathbf{h}(\mathbf{t})} \right) \mid (X_1, Y_1) = (x, y) \right]. \end{aligned} $$

Hence,

$$ m\pi_{1,m} G_{\varphi, \mathbf{t}, \theta}^{(T)}(x, y) = \tilde{\phi}^{-1/m}(\mathbf{D}_n^-) \left( S_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}(x, y) - \mathbb{E}\left[S_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}(x, y)\right] \right). $$

The linear term of the process is given by

$$ \begin{aligned} & m\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}u_n^{(1)}(\pi_{1,m}G_{\varphi,\mathbf{t},\theta}^{(T)}) \\ &= \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}} \sum_{i=1}^{n} \left\{ S_{\varphi,\mathbf{t},\theta,\mathbf{D}_n^-}(X_i, Y_i) - \mathbb{E}\left[S_{\varphi,\mathbf{t},\theta,\mathbf{D}_n^-}(X_i, Y_i)\right] \right\} \\ &:= \alpha_n(S_{\varphi,\mathbf{t},\theta,\mathbf{D}_n^-}). \end{aligned} $$
Thus, the linear term of the U-process $\{\mu_n(\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-)\}_{\mathcal{F}_m, \mathcal{R}_\theta^m}$ is an empirical process indexed by the class of functions $S$, defined as follows:

$$S = \{S_{\varphi, \mathbf{t}, \theta, \mathbf{h}}(\cdot, \cdot) \varphi \in \mathcal{F}_m, \mathbf{t} = (t_1, \dots, t_m) \in \mathcal{X}^m\}.$$

Hence, its weak convergence can be established in a manner similar to the proof of Theorem 14. It is evident that $S \subset m\mathcal{G}^{(1)}$. Now, turning to the nonlinear part, we must demonstrate that

$$\left\| \sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)} u_n^{(p)} \left( \pi_{k,m} G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(T)} \right) \right\|_{\mathcal{F}_m, \mathcal{R}_\theta^m} \xrightarrow{\mathbb{P}} 0, \quad \text{for } 2 \le p \le m.$$

This is a consequence of Lemma 1. It is important to note that the selection of the number and size of the blocks must be performed in such a way that the terms I – VI converge to 0. We need to prove that

$$\mathbb{P}\left\{\left\|\mu_n^{(R)}(\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-)\right\|_{\mathcal{F}_m, \mathcal{R}_\theta^m} > \lambda\right\} \to 0 \text{ as } n \to \infty.$$

Again, we confine our discussion to the case $m = 2$ for clarity. We have

$$
\begin{aligned}
& \mu_n^{(R)}(\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-) \\
&= \sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)} \{u_n^{(R)}(\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-) - \mathbb{E}(u_n^{(R)}(\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-))\} \\
&= \frac{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}}{n(n-1)} \sum_{i \neq j}^{n} \left\{ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j))\right] \right\} \\
&\le \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}} \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \tilde{\phi}(\mathbf{D}_n^-) \left\{ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j))\right] \right\} \\
&\quad + \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}} \sum_{p=1}^{v_n} \sum_{i \neq j, i \in H_p^{(U)}} \tilde{\phi}(\mathbf{D}_n^-) \left\{ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j))\right] \right\} \\
&\quad + 2\frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}} \sum_{p=1}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{q: |q-p| \ge 2} \sum_{j \in T_q^{(U)}} \tilde{\phi}(\mathbf{D}_n^-) \\
&\quad \times \left\{ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j))\right] \right\} \\
&\quad + 2\frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}} \sum_{p=1}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{q: |q-p| \le 1} \sum_{j \in T_q^{(U)}} \tilde{\phi}^{1/m}(\mathbf{D}_n^-) \\
&\quad \times \left\{ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j))\right] \right\} \\
&\quad + \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}} \sum_{p \neq q}^{v_n} \sum_{i \in T_p^{(U)}} \sum_{j \in T_q^{(U)}} \tilde{\phi}(\mathbf{D}_n^-) \left\{ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j))\right] \right\} \\
&\quad + \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_n^-)}} \sum_{p=1}^{v_n} \sum_{i \neq j, i \in T_p^{(U)}} \tilde{\phi}(\mathbf{D}_n^-) \left\{ G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-}^{(R)}((X_i, X_j), (Y_i, Y_j))\right] \right\} \\
&= I' + II' + III' + IV' + V' + VI'.
\end{aligned}
$$

We employ blocking arguments to handle the resulting terms. We begin by examining the first term $I'$. We have
$$
\mathbb{P} \left\{ \left\| \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}}^{-})}} \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}) \left\{ G_{\varphi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}^{-}}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}[G_{\varphi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}^{-}}^{(R)}((X_i, X_j), (Y_i, Y_j))] \right\} \right\|_{\mathcal{F}_2 \mathcal{R}_{\vartheta}^2} > \delta \right\} \\
\le \mathbb{P} \left\{ \left\| \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}}^{-})}} \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}) \left\{ G_{\varphi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}^{-}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) - \mathbb{E}[G_{\varphi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}^{-}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j))] \right\} \right\|_{\mathcal{F}_2 \mathcal{R}_{\vartheta}^2} > \delta \right\} + 2v_n\beta_{b_n}.
$$

Notice that (48) readily implies that $v_n\beta_{b_n} \to 0$ and recall that for all $\varphi \in \mathcal{F}_m$,

$$
\mathbf{x}, \mathbf{t} \in \mathcal{X}^2, \mathbf{y} \in \mathcal{Y}^2 : \kappa_2^2 \mathbf{1}_{\{d(\mathbf{x}, \mathbf{t}) \le h_K\}} F(\mathbf{y}) \ge \varphi(\mathbf{y}) \tilde{K} \left( \frac{d(\mathbf{x}, \mathbf{t})}{h_K} \right).
$$

By the symmetry of Function $F(\cdot)$, it holds that

$$
\left\| \frac{1}{\sqrt{n \tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}}^{-})}} \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}) \left\{ G_{\varphi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}^{-}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) - \mathbb{E} \left[ G_{\varphi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}^{-}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) \right] \right\} \right\|_{\mathcal{F}_2 \mathcal{R}_{\vartheta}^2} \\
\le \left| \frac{1}{\sqrt{n \tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}}^{-})}} \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \left\{ \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \right. \right. \\
\left. \left. - \mathbb{E} \left[ \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \right] \right\} \right| \quad (118)
$$

Employing Chebyshev's inequality and Hoeffding's trick while maintaining order, we obtain

$$
\mathbb{P} \left\{ \left| \frac{1}{\sqrt{n \tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}}^{-})}} \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \left\{ \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \right. \right. \\
\left. \left. - \mathbb{E} \left[ \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \right] \right\} \right| > \delta \right\} \\
\le \delta^{-2} n^{-1} \tilde{\phi}^{-1/m}(\mathbf{D}_{\bar{n}}^{-}) \mathrm{Var} \left( \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \right) \\
\le c_2 v_n \delta^{-2} n^{-1} \tilde{\phi}^{-1/m}(\mathbf{D}_{\bar{n}}^{-}) \mathrm{Var} \left( \sum_{p=1}^{v_n} \sum_{i,j \in H_p^{(U)}} \kappa^2 F(\zeta_i, \zeta_j') \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \right) \\
\le 2c_2 v_n \delta^{-2} n^{-2} \tilde{\phi}^{-1/m}(\mathbf{D}_{\bar{n}}^{-}) \mathbb{E} \left[ \left( \kappa^2 F(\zeta_1, \zeta_2) \right)^2 \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \right]. \quad (119)
$$
Under (C.6.) we have, for each $\lambda > 0$,

$$
\begin{align*}
& c_2 v_n \delta^{-2} n^{-2} \tilde{\phi}^{-1/m} (\mathbf{D}_{\bar{n}}) \mathbb{E} \left[ \left( \kappa^2 F(\zeta_1, \zeta_2) \right)^2 \mathbb{1}_{\{ \kappa^2 F > \lambda (n/\tilde{\phi}(\mathbf{D}_{\bar{n}}))^{1/2(p-1)} \}} \right] \\
&= c_2 v_n \delta^{-2} n^{-2} \tilde{\phi}^{-1/m} (\mathbf{D}_{\bar{n}}) \int_0^{\infty} \mathbb{P} \left\{ \left( \kappa^2 F(\zeta_1, \zeta_2) \right)^2 \mathbb{1}_{\{ \kappa^2 F > \lambda (n/\tilde{\phi}(\mathbf{D}_{\bar{n}}))^{1/2(p-1)} \}} \ge t \right\} dt \\
&= c_2 v_n \delta^{-2} n^{-2} \tilde{\phi}^{-1/m} (\mathbf{D}_{\bar{n}}) \int_0^{\lambda (n/\tilde{\phi}(\mathbf{D}_{\bar{n}}))^{1/2(p-1)}} \mathbb{P} \left\{ \kappa^2 F > \lambda (n/\tilde{\phi}(\mathbf{D}_{\bar{n}}))^{1/2(p-1)} \right\} dt \\
&\quad + c_2 v_n \delta^{-2} n^{-2} \tilde{\phi}^{-1/m} (\mathbf{D}_{\bar{n}}) \int_{\lambda (n/\tilde{\phi}(\mathbf{D}_{\bar{n}}))^{1/2(p-1)}}^{\infty} \mathbb{P} \left\{ \left( \kappa^2 F \right)^2 > t \right\} dt,
\end{align*}
$$

which tends to 0 as $n \to \infty$. Terms II', V', and VI' are handled in the same manner as the first, with the exception that for II', VI', we do not need to apply Hoeffding's trick. This is because our variables $\{\zeta_i, \zeta_j\}_{i,j \in H_p^{(U)}}$ (or $\{\zeta_i, \zeta_j\}_{i,j \in T_p^{(U)}}$ for VI') are in the same blocks.

For term IV', we deduce its study from those of I' and III'. Let us consider term III'. As for the truncated part, we have

$$
\begin{align}
& \mathbb{P} \left\{ \left\| \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}})}} \sum_{p=1}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{q:|q-p| \ge 2}^{v_n} \sum_{j \in T_q^{(U)}} \tilde{\phi}(\mathbf{D}_{\bar{n}}) \left\{ G_{\phi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}}^{(R)}((X_i, X_j), (Y_i, Y_j)) \right. \right. \nonumber \\
& \qquad \left. - \mathbb{E} \left[ G_{\phi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}}^{(R)}((X_i, X_j), (Y_i, Y_j)) \right] \right\} \right\|_{\mathcal{F}_2 \mathcal{R}_{\theta}^2} > \delta \right\} \nonumber \\
& \le \mathbb{P} \left\{ \left\| \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}})}} \sum_{p=1}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{q:|q-p| \ge 2}^{v_n} \sum_{j \in T_q^{(U)}} \tilde{\phi}(\mathbf{D}_{\bar{n}}) \left\{ G_{\phi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) \right. \right. \nonumber \\
& \qquad \left. - \mathbb{E} \left[ G_{\phi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) \right] \right\} \right\|_{\mathcal{F}_2 \mathcal{R}_{\theta}^2} > \delta \right\} \nonumber \\
& \qquad + \frac{v_n^2 a_n b_n \beta_{a_n}}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}})}}. \tag{120}
\end{align}
$$

We also have

$$
\begin{align*}
& \mathbb{P} \left\{ \left\| \frac{1}{\sqrt{n \tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}})}} \sum_{i \in H_p^{(U)}} \sum_{q:|q-p| \ge 2}^{v_n} \sum_{j \in T_q^{(U)}} \left\{ G_{\phi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) - \mathbb{E} \left[ G_{\phi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) \right] \right\} \right\|_{\mathcal{F}_2 \mathcal{R}_{\theta}^2} > \delta \right\} \\
\le & \mathbb{P} \left\{ \left\| \frac{1}{\sqrt{n \tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}})}} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{v_n} \sum_{j \in T_q^{(U)}} \left\{ G_{\phi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) - \mathbb{E} \left[ G_{\phi, \mathbf{t}, \mathbf{\theta}, \mathbf{D}_{\bar{n}}}^{(R)}((\zeta_i, \zeta_j), (\zeta_i, \zeta_j)) \right] \right\} \right\|_{\mathcal{F}_2 \mathcal{R}_{\theta}^2} > \delta \right\}.
\end{align*}
$$
Since Equation (118) is still satisfied, the problem is reduced to

$$ \mathbb{P} \left\{ \left| \frac{1}{\sqrt{n\tilde{\phi}^{1/m}(\mathbf{D}_{\bar{n}}^{-})}} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{v_n} \sum_{j \in T_q^{(U)}} \left\{ \begin{aligned} & \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \\ & - \mathbb{E}\left[\kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}}\right] \end{aligned} \right\} \right| > \delta \right\} \\ \le \delta^{-2} n^{-1/m} \tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}) \mathrm{Var} \left( \sum_{i \in H_1^{(U)}} \sum_{q=3}^{v_n} \sum_{j \in T_q^{(U)}} \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_{\bar{n}}^{-}))^{1/2(p-1)}\}} \right), $$

and we follow the same procedure as in (119). The remaining terms have been shown to be asymptotically negligible. Consequently, process $\{\mu_n(\varphi, \mathbf{t}, \theta, \mathbf{D}_n^-)\}_{\mathcal{F}_m, \mathcal{R}_\theta^m}$ converges in distribution to a Gaussian process, which admits a version with uniformly bounded and uniformly continuous paths with respect to the $\|\cdot\|_2$-norm. By applying the same reasoning, the same result holds for process $\{\mu_n(\varphi, \mathbf{t}, \theta, \mathbf{D}_n^+)\}_{\mathcal{F}_m, \mathcal{R}_\theta^m}$. Thus, according to (117), it follows that process $\{\mu_n(\varphi, \mathbf{t}, \theta, \mathbf{h}_{n,k}(\mathbf{t}))\}_{\mathcal{F}_m, \mathcal{R}_\theta^m}$ also converges in distribution to a Gaussian process, which has a version with uniformly bounded and uniformly continuous paths with respect to the $\|\cdot\|_2$-norm. Similarly, process $\{\mu_n(1, \mathbf{t}, \theta, \mathbf{h}_{n,k}(\mathbf{t}))\}_{\mathcal{F}_m, \mathcal{R}_\theta^m}$ is treated in the same manner, and for $\mathbb{E}(u_n(\varphi, \mathbf{t}, \theta, \mathbf{h}_{n,k}(\mathbf{t}))), u_n(1, \mathbf{t}, \theta, \mathbf{h}_{n,k}(\mathbf{t})))$, and $\mathbb{E}(u_n(1, \mathbf{t}, \theta, \mathbf{h}_{n,k}(\mathbf{t}))),$ the analysis follows the same steps as in the proof of Theorem 14. $\square$

**Funding:** This research received no external funding.

**Data Availability Statement:** No new data were created or analyzed in this study.

**Acknowledgments:** The author extends his sincere gratitude to the Editor-in-Chief, the Associate Editor, and the three referees for their invaluable feedback and for pointing out a number of oversights in the version initially submitted. Their insightful comments have greatly refined and focused the original work, resulting in a markedly improved presentation.

**Conflicts of Interest:** The author declares that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix A. Technical Lemmas

### Appendix A.1. General Lemma

Lemma A1 is presented in a broad context, as illustrated in [51], making it applicable beyond the specific scenario under consideration. This represents an extension of the result established in [118]. In a more general sense, this technical tool proves invaluable for addressing random bandwidths. Consider a sequence of random vectors, $(A_i, B_i)_{n \ge 1}$, taking values in $(\Omega^m \times \Omega'^m, \mathcal{A} \times \mathcal{B})$, a general space. Let $S_\Omega$ be a fixed subset of $\Omega$. Additionally, define $G: \mathbb{R} \times (S_\Omega \times \Omega) \to \mathbb{R}^+$ as a function such that, for every $t \in S_\Omega$, $G(\cdot, (t, \cdot))$ is measurable. Furthermore, for all $x, x' \in \mathbb{R}$, the following condition holds:
$$ (L_0) \quad \text{If } x \le x', \text{ then } G(\langle x, \theta \rangle, z) \le G(\langle x', \theta \rangle, z), \text{ for all } z \in S_\Omega \times \Omega, \theta \in \theta. $$

Define the pointwise measurable class of functions for $1 \le m \le n$:

$$ \mathcal{G}^m := \left\{ (\langle x_1, \theta_1 \rangle, \dots, \langle x_m, \theta_m \rangle) \mapsto \prod_{i=1}^{m} G(h_i, (\langle x_i, \theta_i \rangle, t_i)), (h_1, \dots, h_m) \in \mathbb{R}_+^m \setminus \{0\}, (t_1, \dots, t_m) \in S_\Omega^m \right\}. $$

Next, consider a sequence $(\mathbf{D}_{n,k}(\mathbf{t}, \theta))_{n \in \mathbb{N}}$ consisting of random real vectors, where for all $\mathbf{t} = (t_1, \dots, t_m) \in S_\Omega^m$ and $\theta \in \theta^m$, $\mathbf{D}_{n,k}(\mathbf{t}, \theta) = (D_{n,k}(\langle x_1, \theta_1 \rangle), \dots, D_{n,k}(\langle x_m, \theta_m \rangle))$. Let
$\varphi : \Omega'^m \to \mathbb{R}$ be a measurable function belonging to a class of functions $\mathcal{F}_m$. Define $\mathfrak{M}^{(m)} : \mathcal{F}_m \times S_{\Omega}^m \times \theta^m \to \mathbb{R}$ as a non-random function satisfying

$$ \sup_{\varphi \in \mathcal{F}_m} \sup_{t \in S_{\Omega}^m} \sup_{\theta \in \theta^m} |\mathfrak{M}^{(m)}(\varphi, t, \theta)| < \infty. $$

For all $\mathbf{t} \in S_{\Omega}^m$, $\theta \in \theta^m$, $\varphi \in \mathcal{F}_m$, and $n \in \mathbb{N} \setminus \{0\}$, define

$$ \mathfrak{M}_n^{(m)}(\varphi, \mathbf{t}, \theta; \mathbf{h}) = \frac{\sum_{i \in I(m,n)} \varphi(B_{i_1}, \dots, B_{i_m}) \prod_{j=1}^{m} G(H_{j'}, (\langle t_j, \theta_j \rangle, A_{i,j}))}{\sum_{i \in I(m,n)} \prod_{j=1}^{m} G(H_{j'}, (\langle t_j, \theta_j \rangle, A_{i,j}))}. $$

**Lemma A1.** Let $(\mathcal{U}_n)_{n \in \mathbb{N}}$ be a decreasing positive sequence such that $\lim_{n \to \infty} \mathcal{U}_n = 0$. If, for all increasing sequence $\xi_n \in (0,1)$ with $\xi_n - 1 = O(\mathcal{U}_n)$, there exist two sequences of r.r.v. $(D_{n,k}^{-}(\xi_n, \mathbf{t}, \theta))_{n \in \mathbb{N}}$ and $(D_{n,k}^{+}(\xi_n, \mathbf{t}, \theta))_{n \in \mathbb{N}}$ such that

$$ (L_1) \qquad \forall n \in \mathbb{N} \text{ and } \mathbf{t} \in S_{\Omega}^{m}, D_{n,k}^{-}(\xi_n, \langle t_j, \theta_j \rangle) \le D_{n,k}^{+}(\xi_n, \langle t_j, \theta_j \rangle), \forall j = 1, \dots, m, $$

$$ (L_2) \qquad \prod_{j=1}^{m} 1 \left\{ D_{n,k}^{-}(\xi_n, \langle t_j, \theta_j \rangle) \le D_{n,k}^{+}(\langle t_j, \theta_j \rangle) \le D_{n,k}^{+}(\xi_n, \langle t_j, \theta_j \rangle) \right\} \to 1, \quad \text{a.co, } \forall \mathbf{t} \in S_{\Omega}^{m}, $$

$$ (L_3) \qquad \sup_{\tilde{\mathcal{G}} \in \mathcal{G}^m} \sup_{\mathbf{t} \in S_{\Omega}^m} \left| \frac{\sum_{i \in I(m,n)} \prod_{j=1}^{m} G(D_{n,k}^{-}(\xi_n, \langle t_j, \theta_j \rangle), (\langle t_j, \theta_j \rangle, A_{i,j}))}{\sum_{i \in I(m,n)} \prod_{j=1}^{m} G(D_{n,k}^{+}(\xi_n, \langle t_j, \theta_j \rangle), (\langle t_j, \theta_j \rangle, A_{i,j}))} - \xi_n \right| = O_{a \cdot c o}(\mathcal{U}_n), $$

$$ (L_4) \qquad \sup_{\varphi \in \mathcal{F}_m} \sup_{\mathcal{G}^m} \sup_{\mathbf{t} \in S_{\Omega}^m} \left| \mathfrak{M}_n^{(m)}(\varphi, \mathbf{t}, \theta; D_{n,k}^{-}(\xi_n, \mathbf{t}, \theta)) - \mathfrak{M}^{(m)}(\varphi, \mathbf{t}, \theta) \right| = O_{a \cdot c o}(\mathcal{U}_n), $$

$$ (L_5) \qquad \sup_{\varphi \in \mathcal{F}_m} \sup_{\mathcal{G}^m} \sup_{\mathbf{t} \in S_{\Omega}^m} \left| \mathfrak{M}_n^{(m)}(\varphi, \mathbf{t}, \theta; D_{n,k}^{+}(\xi_n, \mathbf{t}, \theta)) - \mathfrak{M}^{(m)}(\varphi, \mathbf{t}, \theta) \right| = O_{a \cdot c o}(\mathcal{U}_n). $$

Then, as $n \to \infty$,

$$ \sup_{\varphi \in \mathcal{F}_m} \sup_{\mathcal{G}^m} \sup_{\mathbf{t} \in S_{\Omega}^m} \left| \mathfrak{M}_n^{(m)}(\varphi, \mathbf{t}, \theta; D_{n,k}(\mathbf{t}, \theta)) - \mathfrak{M}^{(m)}(\varphi, \mathbf{t}, \theta) \right| = O_{a \cdot c o}(\mathcal{U}_n). \quad (A1) $$

Refer to [51] for proof of this lemma.

**Lemma A2.** Under Assumptions (C.1.1), (C.3.1), (C.4.1), (C.5.1), (C.5.2'), (C.6.) and (C.7.), obtain, as $n \to \infty$,

$$ \sup_{\varphi \in \mathcal{F}} \sup_{\mathbf{h}_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_{\mathcal{H}}^m} \sup_{\theta \in \theta} \left| \hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle, h_K(t)) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle, h_K(t)) \right) \right| \\ = O_{a \cdot c o} \left( \sqrt{\frac{\psi_{S_{\mathcal{H}}} \left( \frac{\log n}{n} \right) + \psi_{\theta} \left( \frac{\log n}{n} \right)}{n \phi(h_{n,1})}} \right), \quad (A2) $$
and

$$
\begin{align}
& \sup_{\varphi K \in \mathcal{F}\mathcal{R}_{\theta}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_H} \sup_{\theta \in \theta} \left| \hat{r}_{n,1}^{(1)}(\varphi, \langle t, \theta \rangle, h_K(t)) - 1 \right| \nonumber \\
& = O_{a,co} \left( \sqrt{\frac{\psi_{S_H} \left( \frac{\log n}{n} \right) + \psi_{\theta} \left( \frac{\log n}{n} \right)}{n\phi(h_{n,1})}} \right). \tag{A3}
\end{align}
$$

This lemma establishes the consistency rates for the stochastic component when the function class $\mathcal{F}$ is restricted in scope. The following lemma extends this result to the case where the function class is unbounded.

**Lemma A3.** Under Assumptions (C.1.1), (C.3.1), (C.4.2), (C.5.1), (C.5.2'), (C.6.) and (C.7.), obtain, as $n \to \infty$,

$$
\begin{align}
& \sup_{\varphi K \in \mathcal{F}\mathcal{R}_{\theta}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_H} \left| \hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle, h_K(t)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle, h_K(t))) \right| \nonumber \\
& = O_{a,co} \left( \sqrt{\frac{\psi_{S_H} \left( \frac{\log n}{n} \right) + \psi_{\theta} \left( \frac{\log n}{n} \right)}{n\phi(h_{n,1})}} \right). \tag{A4}
\end{align}
$$

**Lemma A4.** Under Conditions (C.1.), (C.3.1), and (C.5.1) (or equivalently (C.1'.), (C.3'.) and (C.5.1)), and if $n\phi(h_K) \to \infty$ as $n \to \infty$, obtain, for each $t \in \mathcal{H}$,

$$
\hat{r}_{n,1}^{(1)}(1, \langle t, \theta \rangle; h_K) \xrightarrow{p.s.} 1. \qquad (A5)
$$

Before presenting the next result, it is necessary to introduce the following notation:

$$
\mu_{n,1}(t, \theta; h_K) := \mathbb{E} \left[ (\varphi(Y_1) - r^{(1)}(\varphi, t)) \Delta_1(t, \theta; h_K) \right], \quad (A6)
$$

$$
Z_{n,i}(h_K) := (\varphi(Y_i) - r^{(1)}(\varphi, t; h_K)) \Delta_i(t, \theta; h_K) - \mu_{n,1}(\langle t, \theta \rangle; h_K). \quad (A7)
$$

Set

$$
Q_n(t, \theta; h_K) = \frac{1}{n \mathbb{E}[\Delta_1(t, \theta; h_K)]} \sum_{i=1}^{n} Z_{n,i}(h_K).
$$

and

$$
\sigma_{n,0}^2(t, \theta; h_K) := \frac{1}{\mathbb{E}^2[\Delta_1(t, \theta; h_K)]} \mathrm{Var}[Z_{n,1}(h_K)].
$$

**Lemma A5.** Under Conditions (C.1.), (C.2.), (C.3.1) and (C.5.1) (with $p = \infty$ and $\delta > 1$ for condition (C.5.1)), obtain, for $n\phi(h_K) \xrightarrow{n\to\infty} \infty$, $t \in \mathcal{X}$ and positive constants $C_4, C_5$,

$$
C_4 \frac{g_2(t, \theta, \varphi)}{f_1(t)} \le \phi(h_K) \sigma_{n,0}^2(t, \theta) \le C_5 \frac{g_2(t, \theta, \varphi)}{f_1(t)}, \quad (A8)
$$

whenever $f_1(t) > 0$. Obtain

$$
\frac{1}{n \mathbb{E}^2[\Delta_1(t, \theta; h_K)]} \sum_{i=1}^{n} \sum_{\substack{i=1 \\ i \neq j}}^{n} \mathrm{Cov}[Z_{n,i}, Z_{n,j}] = o(\sigma_{n,0}^2(t, \theta)). \quad (A9)
$$

$$
\mathrm{Var}[Q_n(t, \theta)] = \sigma_{n,0}^2(t, \theta)(1 + o(1))/n. \quad (A10)
$$
**Lemma A6.** Under Conditions (C.1'.), (C.2.), (C.3'.) and (C.5.1) (with $p = \infty$ and $\delta > 1$ for condition (C.5.1)), obtain, for $n\phi(h_K) \to \infty$ as $n \to \infty$ and $t \in \mathcal{H}$,

$$ \phi(h_K)\sigma_{n,0}^2(t, \theta) \to \frac{\kappa_1' g_2(t, \theta, \varphi)}{\kappa_2' f_1(t)}, \qquad (A11) $$

whenever $f_1(t) > 0$, and $\kappa_1'$, $\kappa_2'$ are constants specified previously.

$$ \frac{1}{n\mathbb{E}^2[\Delta_1(t, \theta; h_K)]} \sum_{i=1}^{n} \sum_{i=1}^{n} \mathrm{Cov}[Z_{n,i}, Z_{n,j}] = o(\sigma_{n,0}^2(t)), \qquad (A12) $$

$$ n\phi(h_K)\mathrm{Var}[Q_n(t, \theta; h_K)] \to \frac{\kappa_1' g_2(t, \theta, \varphi)}{\kappa_2' f_1(t)}. \qquad (A13) $$

To simplify the notation and for convenience, denote

$$ \sigma^2(t, \theta) := (\kappa_1' g_2(t, \theta, \varphi)) / (\kappa_2' f_1(t)). $$

**Lemma A7.** Under Conditions (C.1'.), (C.2.), (C.3'.), (C.5.1) and (C.5.2), if $n\phi(h_K) \to \infty$ as $n \to \infty$, obtain, for $t \in \mathcal{H}$, as $n \to \infty$,

$$ (n\phi(h_K))^{1/2} Q_n(t, \theta, h_K) \to \mathcal{N}(0, \sigma^2(t, \theta)). \qquad (A14) $$

**Lemma A8.** Under Conditions (C.1.), (C.2.), (C.3.1) and (C.5.1), and if

$$ (n\phi(h_K)/\log \log n) \underset{n\to\infty}{\longrightarrow} \infty, $$

we obtain, as $n \to \infty$,

$$ \left( \frac{n\phi(h_K)}{\log \log n} \right)^{1/2} \left[ \hat{r}_n^{(1)}(\varphi, \langle t, \theta \rangle, \theta) - r^{(1)}(\varphi, \langle t, \theta \rangle, \theta) - B_n(\langle t, \theta \rangle) \right] \xrightarrow{\mathbb{P}} 0. \qquad (A15) $$

The detailed proofs of the aforementioned lemmas can be found in [111,112,155].

The following result concerning the bias term is all that remains to be established. This lemma can be derived using a similar approach to that in [111], where more comprehensive details are provided.

**Lemma A9.** Under Condition (C.2.1), obtain, as $n \to \infty$,

$$ \sup_{\varphi K \in \mathcal{F}, \mathcal{K}_{\theta}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_{\mathcal{H}}} \left| \mathbb{E} \left[ \hat{r}_{n,2}^{(1)}(\varphi, \langle t, \theta \rangle; h_K(t)) \right] - r^{(1)}(\varphi, \langle t, \theta \rangle) \right| = O(h_{n,2}^{\gamma}). \qquad (A16) $$

**Lemma A10.** Under assumptions of Theorem 14, obtain

$$ \sqrt{n\tilde{\phi}^{1/m}(\mathbf{h}(t))} \left( \hat{r}_n^{*(m)}(\varphi, \mathbf{t}, \theta; \mathbf{h}(t)) - \mathbb{E}[u_n(\varphi, \mathbf{t}, \theta; \mathbf{h}(t))] \right) \xrightarrow{d} \mathcal{N}(0, m^2\sigma_{\mathbf{t},\theta}^2(\varphi, \varphi)), \qquad (A17) $$

and, if, in addition, Condition (C.8) is satisfied, then obtain

$$ \sqrt{k} \left\{ \hat{r}_n^{*(m)}(\varphi, \mathbf{t}, \theta; \mathbf{h}_{n,k}(\mathbf{t})) - \mathbb{E}[u_n(\varphi, \mathbf{t}, \theta; \mathbf{h}(\mathbf{t}))] \right\} \xrightarrow{d} \mathcal{N}(0, m^2 \sigma_{\mathbf{t},\theta}^2(\varphi, \varphi)). \qquad (A18) $$

**Lemma A11.** Under the assumptions of Theorem 14, obtain

$$ \sqrt{n\tilde{\phi}^{1/m}(\mathbf{h})} (u_n(\varphi_1, \mathbf{t}, \theta, \mathbf{h}) - \mathbb{E}(u_n(\varphi_1, \mathbf{t}, \theta, \mathbf{h}), u_n(\varphi_2, \mathbf{t}, \theta, \mathbf{h}) - \mathbb{E}(u_n(\varphi_2, \mathbf{t}, \theta, \mathbf{h}))) \xrightarrow{d} \mathcal{N}(0, \Sigma), \qquad (A19) $$
where

$$ \Sigma := (m^2 \sigma_{\mathbf{t},\theta}^2 (\varphi_i, \varphi_j))_{i,j=1,2}. $$

The proofs of the preceding lemmas can be found in [112,155]. Similarly, the following lemmas follow the same steps as in [155] and are thus omitted for brevity.

**Lemma A12.** Under Assumptions (C.2.1), (C.2.2), (C.3.2), (C.4.2) and (C.8), obtain

$$ \sqrt{k}I_1(\langle t, \theta \rangle, H_{n,k}(t)) \xrightarrow{\mathbb{P}} 0, \quad \text{as } n \to \infty $$

**Lemma A13.** Under Assumptions (C.1'), (C.2.1), (C.2.2), (C.3') and (C.5.) and if

$$ k \left( \phi^{-1} \left( \frac{k}{n} \right) \right)^{2\gamma} \to 0 \quad \text{as } n \to \infty, $$

obtain

$$ \sqrt{k}I_2(\varphi, \langle t, \theta \rangle, H_{n,k}(t, \theta)) \xrightarrow{\mathcal{N}} (0, \sigma^2(t, \theta)), \quad \text{as } n \to \infty. $$

**Lemma A14.** Under Assumptions (C.1), (C.3.1), and (C.5.1) (or equivalently (C.1'), (C.3') and (C.5.1)), and if

$$ k \left( \phi^{-1} \left( \frac{k}{n} \right) \right)^{2\gamma} \to 0 \quad \text{as } n \to \infty, $$

obtain, for each $t \in \mathcal{H}$:

$$ \hat{r}_{n,1}^{(1)}(1, \langle t, \theta \rangle, H_{n,k}(t, \theta)) - 1 \xrightarrow{\mathbb{P}} 0 \quad \text{as } n \to \infty. $$

**Lemma A15.** Under Conditions (C.1.), (C.3.1), and (C.5.1) (or equivalently (C.1'), (C.3') and (C.5.1)), and if $n\phi(h_K) \to \infty$ as $n \to \infty$, obtain, for each $t \in \mathcal{H}$,

$$ \hat{r}_{n,1}^{(1)}(\varphi, \langle t, \theta \rangle; h_K) \xrightarrow{\text{p.s.}} \mathbb{E}(\hat{r}_{n,1}^{(1)}(\varphi, \langle t, \theta \rangle; h_K)), \quad (A20) $$

as $n \to \infty$.

It is important to highlight that this lemma is broader in scope than Lemma A4. While the result obtained here is somewhat weaker than the uniform almost sure convergence rate established in Section 7, the conditions imposed in this lemma are less stringent, making it more generally applicable.

## Appendix B. Auxiliary Results

This appendix provides supplementary information that is essential for a more thorough understanding of the paper. In what follows, we let $X, X_1, \dots, X_n$ be i.i.d. random variables defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$, taking values in a measurable space $(\Psi, \mathcal{B})$, and we let $\mathcal{F}$ be a $\mathbb{P}$-measurable class of functions with an envelope function $F$, such that

$$ \mathbb{E}(F^2(X)) < \infty. $$

We further assume that $\mathcal{F}$ satisfies the following condition:

* For any sequence of i.i.d. $\mathcal{X}$-valued random variables $Z_1, Z_2, \dots$, the following inequality holds:

$$ \mathbb{E} \left\| \sum_{i=1}^{k} \{ g(Z_i) - \mathbb{E}g(Z_1) \} \right\|_{\mathcal{G}} \leq C_1 \sqrt{k} \|G(Z_1)\|_{2}, \quad 1 \leq k \leq n, $$
where $C_1 > 1$ is a constant that depends only on $\mathcal{G}$.

**Lemma A16 (Theorem 2.14.1 [181]).** For an empirical process $\alpha_n(f)$ indexed by the class of functions $\mathcal{F}$ with notation

$$ \|\alpha_n(f)\|_{\mathcal{F}} := \sup_{f \in \mathcal{F}} |\alpha_n(f)| $$

and $J(\delta, \mathcal{F})$ meaning

$$ \sup_{Q} \int_{0}^{\delta} \sqrt{1 + \log N(\epsilon, \mathcal{F}, \| \cdot \|_{L_{2}(Q)})} d\epsilon, $$

we have, for $p > 1$,

$$ \|\|\alpha_n(f)\|_{\mathcal{F}}\|_p \leq C J(\delta, \mathcal{F}) \|F\|_{p \lor 2}. $$

**Lemma A17 (Theorem 3.1 [107]).** We let $\mathcal{F}$ be a pointwise measurable function class satisfying the above assumptions. If we suppose that the empirical process $\alpha_n(f)$ satisfies

$$ \mathbb{E}\|\alpha_n(f)\|_{\mathcal{F}} \leq C\|F\|_2, \qquad (A21) $$

then, for any measurable subset $B \in \mathcal{B}$,

$$ \mathbb{E}\|\alpha_n(f1_B)\|_{\mathcal{F}} \leq 2C\|F1_B\|_2. $$

From Theorem 3.2 of [107], it follows that a VC-type class of functions satisfies, always, Condition (A21).

**Lemma A18 (Bernstein-type inequality Fact 4.2 [107]).** We assume that for some $H > 0$ and $p \geq 2$ the r.v.'s $X, X_1, \dots, X_n$ satisfy

$$ \mathbb{E}(F^p(X)) \leq (p!/2)\sigma^2 H^{p-2}, \quad \text{where} \quad \sigma^2 \geq \mathbb{E}(F^2(X)); $$

then, for $\mu'_n = \mathbb{E}(\|\sqrt{n}\alpha_n(f)\|_{\mathcal{F}})$, we have, for any $z > 0$,

$$ \mathbb{P}\left\{ \max_{1 \leq k \leq n} \left\| \sqrt{k\alpha_k(f)} \right\|_{\mathcal{F}} \geq \mu'_n + z \right\} \leq \exp\left( \frac{-z^2}{2n\sigma^2 + 2zH} \right) \\ \leq \exp\left( \frac{-z^2}{4n\sigma^2} \right) \lor \exp\left( \frac{-z}{4H} \right). $$

**Lemma A19 (Proposition 4. [153]).** We let $X_1, \dots, X_m$ be i.i.d. random variables with values in some measurable space $(S, \mathcal{S})$. We let $\mathcal{F}$ be a class of symmetric functions $f$ from $S^m$ satisfying some measurability conditions. We suppose that there exists a finite constant $c_0$ such that for each $x_j \in S$, we have

$$ \mathbb{E}\left\{ \left\| n^{1-m} \sum_{i=1}^{m} \zeta_{i_1} \zeta_{i_2} f(x_{i_1}, \dots, x_{i_m}) \right\|_{\mathcal{F}} \right\} \leq c_0, $$

and that there is a finite constant $b$ such that $\sup_{f \in \mathcal{F}} |f(X)| \leq b$, almost surely. Then, for each $u > 0$,

$$ \mathbb{P}\left\{ n^{1/2} \left\| \sum_{j=2}^{m} \binom{m}{j} u_n(\pi_{j,m} f) \right\|_{\mathcal{F}} \geq u \right\} \leq 2 \exp \left( - \frac{un^{1/2}}{2^{m+5}m^{m+1}bc_0} \right), $$

where variables $\zeta_i$ are Rademacher variables.
**Lemma A20** (Lemma 6.1 in [108], p. 186). We let $X_1, \dots, X_m$ be independent Bernoulli random variables with $\mathbb{P}(X_i=1) = p$, for all $i=1, \dots, n$. We set $U = X_1 + \dots + X_m$ and $\mu = pn$. Then, for any $\omega \ge 1$, we have

$$ \mathbb{P}(U \ge (1+\omega)\mu) \le \exp\left\{\frac{-\mu\min(\omega^2, \omega)}{4}\right\}, $$

and if $\omega \in (0,1)$, we have

$$ \mathbb{P}\{U \le (1-\omega)\mu\} \le \exp\{-\mu(\omega^2/2)\}. $$

**Lemma A21** ([150]). We let $\{X_n\}_{n \in \mathbb{Z}}$ be a data sequence, along with kernel function $h(\cdot)$, satisfying Assumptions (A1), (A2), and (A3). We then deetrmine that there exist absolute constants $C_4, C_5 > 0$ only depending on $\gamma$ and $r$, such that, for any $x \ge 0$ and $T$ sufficiently large,

$$ \mathbb{P}\left(\left|u_n^{(m)}(h) - \theta(h)\right| \ge C_4 M / \sqrt{n} + x\right) \le 2 \exp\left(-\frac{C_5 x^2 n}{M^2 + Mx(\log n)(\log \log 4n)}\right). $$

**Proposition A1** ([216]). We let $\{Z_{n,i}, 1 \le i \le n, n \ge 1\}$ be a strong mixing non-stationary sequence of random variables and let $V$ be a $\sigma(Z_{n,i}, 1 \le i \le j)$-measurable and $W$ be a $\sigma(Z_{n,i}, i \ge j + m)$-measurable. If $\mathbb{E}|V|^p < \infty$ and $\mathbb{E}|W|^q < \infty$, then

$$ \mathrm{Cov}(V, W) \le [\alpha(m)]^{1/r} \{\mathbb{E}|V|^p\}^{1/p} \{\mathbb{E}|W|^q\}^{1/q}, $$

where $p, q, r > 0$:

$$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1. $$

**Proposition A2** (Proposition 3.6 of [10]). We let $\{X_t : t \in T\}$ be a process satisfying, for $m \ge 1$,

$$ (\mathbb{E}\|\mathbf{X}_t - \mathbf{X}_s\|^p)^{1/p} \le \left(\frac{p-1}{q-1}\right)^{m/2} (\mathbb{E}\|\mathbf{X}_t - \mathbf{X}_s\|^q)^{1/q}, \quad 1 < q < p < \infty, $$

and the semi-metric $\rho(s, t) = (\mathbb{E}\|\mathbf{X}_t - \mathbf{X}_s\|^2)^{1/2}$. There exists a constant $K = K(m)$ such that

$$ \mathbb{E} \sup_{s,t \in T} \|\mathbf{X}_t - \mathbf{X}_s\| \le K \int_0^D [\log N(\epsilon, T, \rho)]^{m/2} d\epsilon, $$

where $D$ is the $\rho$-diameter of $T$.

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