and

$$I_2(t; H_{n,k}) = (\tilde{r}_{n,2}^{(1)}(\varphi_M, t; H_{n,k}(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi_M, t; H_{n,k}(t))) - r^{(1)}(\varphi, t)(\tilde{r}_{n,1}^{(1)}(1, t; H_{n,k}(t)) - 1)). \quad (8.75)$$

To obtain the desired results, we follow the strategy of [152].

**Lemma 8.10.** Under the assumptions (C.2.1), (C.2.2), (C.3.2), (C.4.2) and (C.8), we have

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

**Lemma 8.11.** Under the assumptions (C.1'), (C.2.1), (C.2.2), (C.3') and (C.5) and if

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

we have

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

**Lemma 8.12.** Under the assumptions (C.1), (C.3.1), and (C.5.1) (or equivalently (C.1'), (C.3') and (C.5.1)), and if

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

then we have for each $t \in \mathcal{X}$:

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

**Lemma 8.13.** 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 $n \to \infty$. Then we have for each $t \in \mathcal{X}$:

$$\tilde{r}_{n,1}^{(1)}(\varphi, t; h_K) \xrightarrow{p.s.} \mathbb{E}(\tilde{r}_{n,1}^{(1)}(\varphi, t; h_K)), \quad (8.76)$$

as $n \to \infty$.

We highlight that this lemma is more general than Lemma 8.5. This result is slightly weaker than the uniform, almost complete convergence with the rate that we obtained in Section 3.14. However, the conditions imposed in this lemma are less restrictive.

## Proof of Theorem 4.2

In this section, we will use the same method as in [43] and earlier [8], that is using the blocking approach which entails breaking down a strictly stationary sequence $(X_1, \dots, X_n)$, into $2\nu_n$, equal-sized blocks, that each one is of length $n - 2\nu_n a_n$, keeping in mind the notation given in the proof of Lemma 8.2. In order to establish the asymptotic equi-continuity of the conditional empirical process

$$\{v_n(\psi | t) = \sqrt{k} (\tilde{r}_n^{(1)}(\psi, t; H_{n,k}(t)) - r^{(1)}(\varphi, t)), \psi \in \mathcal{F}\mathcal{H}, \varphi \in \mathcal{F}\}.$$
Let us introduce, for any $\varphi K \in \mathcal{F}\mathcal{K}$ and $t \in \mathcal{X}$,

$$ \mathcal{W}_n(t, \varphi; H_{n,k}(t)) := \sum_{i=1}^{n} \varphi(Y_i)K\left(\frac{d(X_i, t)}{H_{n,k}(t)}\right) - n\mathbb{E}\left\{\varphi(Y_1)K\left(\frac{d(X_1, t)}{H_{n,k}(t)}\right)\right\}, \quad (8.77) $$

$$ \begin{aligned} v_n(\varphi | t) &= \sqrt{k} (\tilde{r}_n^{(1)}(\varphi, t; H_{n,k}(t)) - r^{(1)}(\varphi, t)) \\ &:= \sqrt{k} (\tilde{r}_n^{(1)}(\psi, t; H_{n,k}(t)) - r^{(1)}(\varphi, t)). \end{aligned} \quad (8.78) $$

Then, we have

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

where for $h_K > 0$, we have

$$ \tilde{r}_{n,2}^{(1)}(\varphi, t, h_K) := \frac{1}{n\phi(h_K(t))} \sum_{i=1}^{n} \varphi(Y_i)\Delta_i(t, h_K), $$

$$ \tilde{r}_{n,1}^{(1)}(1, t, h_K) := \frac{1}{n\phi(h_K(t))} \sum_{i=1}^{n} \Delta_i(t, h_K). $$

We study the asymptotic equi-continuity of each of the previous terms. For a class of functions $\mathcal{G}$, let $\alpha_n(\cdot)$ be an empirical process based on $(X_1, Y_1), \dots, (X_n, Y_n)$ and 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 $t \in \mathcal{X}$, set

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

That implies

$$ \frac{1}{\sqrt{n\phi(h_K)}} \mathcal{W}_n(t, \varphi, h_K) = \frac{1}{\sqrt{\phi(h_K)}} \alpha_n(\eta_{n,t,\varphi,K}). $$

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

$$ \left\{ \sqrt{\frac{n}{k}} \alpha_n(\eta_{n,t,\varphi,K}) : \eta_{n,t,\varphi,K} \in \mathcal{F}\mathcal{K} \right\}, $$
which means, for every $\varepsilon > 0$, that

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

where

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

The idea is to work with the independent block sequence $\{\xi_j = (\zeta_j, \zeta_j)\}_{j=1}^{\infty}$ instead of working on dependent one, which is possible through (A.1), then we have

$$ \mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{n} \left( \varphi(Y_j) K \left( \frac{d(X_j, t)}{H_{n,k}(t)} \right) - \mathbb{P}(\eta_{n,t,\varphi,K}(H_{n,k}(t))) \right) \right\|_{\mathcal{F}\mathcal{H}_{(b,\|\cdot\|_p)}} > \delta \right\} \\ \le 2\mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \sum_{i \in H_j} \left( \varphi(\zeta_i) K \left( \frac{d(\zeta_i, t)}{H_{n,k}(t, \zeta)} \right) - \mathbb{P}(\eta_{n,t,\varphi,K}(H_{n,k}(t, \zeta))) \right) \right\|_{\mathcal{F}\mathcal{H}_{(b,\|\cdot\|_p)}} > \delta' \right\} \\ + 2(v_n - 1)\beta_{a_n}, \qquad (8.79) $$

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

$$ H_{n,k}(t, \zeta) = \min \left\{ h \in \mathbb{R}^{+} : \sum_{i=1}^{n} 1_{B(t,h)}(\zeta_i) = k \right\}, \qquad (8.80) $$

We choose

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

Note that $a_n$ in our setting is equivalent to:

$$ a_n = (\log n)^{-1} (n^{-2} k^p)^{1/2(p-1)}. $$

Making use of the condition (C.5.1), we get $(v_n - 1)\beta_{a_n} \to 0$ as $n \to 0$, then it's just a matter of the right side term of (8.79). Let us begin with, the blocks being independent, we symmetrize using a sequence $\{\epsilon_j\}_{j \in \mathbb{N}^*}$ of i.i.d. Rademacher variables, i.e., r.v's with

$$ \mathbb{P}(\epsilon_j = 1) = \mathbb{P}(\epsilon_j = -1) = 1/2. $$

It should be noted that the sequence $\{\epsilon_j\}_{j \in \mathbb{N}^*}$ is independent of the sequence $\{\xi_i = (\zeta_i, \zeta_i)\}_{i \in \mathbb{N}^*}$, thus it remains to establish, for all $\delta > 0$,

$$ \lim_{b \to 0} \limsup_{n \to \infty} \mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \epsilon_j \sum_{i \in H_j} \left( \varphi(\zeta_i) K \left( \frac{d(\zeta_i, t)}{H_{n,k}(t, \zeta)} \right) \right) \right\|_{\mathcal{F}\mathcal{H}_{(b,\|\cdot\|_p)}} > \delta \right\} = 0. $$
Again, using the fact that that $1_{\{D_n^- \le H_{n,\lambda}(t,\zeta) \le D_n^+\}} \xrightarrow{a.co} 1$ when $\frac{k}{n} \to 0$ and (A.9), it suffices to show that

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

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

$$ k^{-1/2} \sum_{j=1}^{n} \mathbb{E} \left( \kappa_2 F(\zeta_i) 1_{\{F(\zeta_i) \ge \lambda(M_n)^{1/2(p-1)}\}} \right) \quad (8.81) $$

$$ = k^{-1/2} \int_{0}^{\infty} \mathbb{P} \left( \kappa_2 F 1_{\{F \ge \lambda(M_n)^{1/2(p-1)}\}} \ge t \right) dt $$

$$ = k^{-1/2} \int_{0}^{\lambda(M_n)^{1/2(p-1)}} \mathbb{P} \left( F \ge \lambda(M_n)^{1/2(p-1)} \right) dt $$

$$ + k^{-1/2} \int_{\lambda(M_n)^{1/2(p-1)}}^{\infty} \mathbb{P} (F \ge t) dt $$

$$ \xrightarrow{n \to \infty} 0. \qquad (8.82) $$

Hence,

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

Then, it suffices to show

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

We have the following

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

This is done by using the chaining method. [8] gave $b_q = b2^{-q}$, $q = 0, \dots, q_n$ where $q_n$

$$ 2^{-1} \lambda_n (\log(n))^{-1} \le b_{q_n}^2 \le \lambda_n (\log(n))^{-1}, \quad (8.83) $$

and let the class $\mathcal{F}\mathcal{H}_q$ of measurable functions of $\mathcal{F}\mathcal{K}$

$$ \#\mathcal{F}\mathcal{H}_q = N_q := N(b_q, \mathcal{F}\mathcal{K}, \|\cdot\|_p) \quad \sup_{\eta_{n,t,\varphi_1,K_1} \in \mathcal{F}\mathcal{K}} \min_{\eta_{n,t,\varphi_2,K_2} \in \mathcal{F}\mathcal{K}_q} \left\| \eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2} \right\|_p \le b_q. $$

There is a map $\pi_q: \mathcal{F}\mathcal{K} \to \mathcal{F}\mathcal{K}_q$ that takes each $\eta_{n,t,\varphi,K} \in \mathcal{F}\mathcal{K}$ to its closest function in $\mathcal{F}\mathcal{K}_q$ such that

$$ \| \eta_{n,t,\varphi,K} - \pi_q(\eta_{n,t,\varphi,K}) \|_p \le b_q. $$
Applying the chaining method

$$
\begin{align}
& \sup_{
  \begin{subarray}{l}
    \eta_{n,t,\varphi_1,K_1}, \eta_{n,t,\varphi_2,K_2} \in \mathcal{F} \\
    \|\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}\|_p \le b
  \end{subarray}
} \nu_n^{(2)}(\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}) \nonumber \\
& \le \sup_{
  \begin{subarray}{l}
    \eta_{n,t,\varphi_1,K_1}, \eta_{n,t,\varphi_2,K_2} \in \mathcal{F} \\
    \|\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}\|_p \le b q_n
  \end{subarray}
} \nu_n^{(2)}(\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}) \nonumber \\
& \quad + 2 \sum_{q=1}^{q_n} \sup_{
  \begin{subarray}{l}
    \eta_{n,t,\varphi_1,K_1}, \eta_{n,t,\varphi_2,K_2} \in (\mathcal{F}, \mathcal{H})_{q-1} \\
    \|\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}\|_p \le 3bq
  \end{subarray}
} \nu_n^{(2)}(\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}) \nonumber \\
& \quad + \sup_{
  \begin{subarray}{l}
    \eta_{n,t,\varphi_1,K_1}, \eta_{n,t,\varphi_2,K_2} \in (\mathcal{F}, \mathcal{H})_0 \\
    \|\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}\|_p \le 2b
  \end{subarray}
} \nu_n^{(2)}(\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}). \tag{8.84}
\end{align}
$$

Let $\delta_q$ be in such a way that

$$
\delta_q = (bq)^{1/2} \lor (3bq(8 + c_{p,\beta}^2)^{1/2}(\log N_q)^{1/2}). \quad (8.85)
$$

Let $r$ be chosen so small in such a way that

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

Therefore, from (8.84), we readily infer that

$$
\[
\mathbb{P} \left\{
  \begin{array}{@{}l@{}}
    \displaystyle \sup_{
      \begin{subarray}{l}
        \eta_{n,t,\varphi_1,K_1}, \eta_{n,t,\varphi_2,K_2} \in \mathcal{F} \\
        \|\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}\|_p \le b
      \end{subarray}
    } \nu_n^{(2)}(\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}) \ge 3\delta
  \end{array}
\right\} \\
\le \mathbb{P} \left\{
  \begin{array}{@{}l@{}}
    \displaystyle \sup_{
      \begin{subarray}{l}
        \eta_{n,t,\varphi_1,K_1}, \eta_{n,t,\varphi_2,K_2} \in \mathcal{F} \\
        \|\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}\|_p \le b q_n
      \end{subarray}
    } \nu_n^{(2)}(\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}) \ge \delta
  \end{array}
\right\} \\
+ 2 \sum_{q=1}^{q_n} \mathbb{P} \left\{
  \begin{array}{@{}l@{}}
    \displaystyle \sup_{
      \begin{subarray}{l}
        \eta_{n,t,\varphi_1,K_1}, \eta_{n,t,\varphi_2,K_2} \in (\mathcal{F}, \mathcal{H})_{q-1} \\
        \|\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}\|_p \le 3bq
      \end{subarray}
    } \nu_n^{(2)}(\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}) \ge \delta_q
  \end{array}
\right\} \\
+ \mathbb{P} \left\{
  \begin{array}{@{}l@{}}
    \displaystyle \sup_{
      \begin{subarray}{l}
        \eta_{n,t,\varphi_1,K_1}, \eta_{n,t,\varphi_2,K_2} \in (\mathcal{F}, \mathcal{H})_0 \\
        \|\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}\|_p \le 2b
      \end{subarray}
    } \nu_n^{(2)}(\eta_{n,t,\varphi_1,K_1} - \eta_{n,t,\varphi_2,K_2}) \ge \delta
  \end{array}
\right\} =: \mathbf{A} + \mathbf{B} + \mathbf{C}.
\]
$$
By the fact that the terms composing $v_n^{(2)}(\eta_{n,t,\varphi,K})$ are bounded by $a_n\lambda_n M_n^{1/2(p-1)}$, and by applying the
Bernstein's 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 (8.83), 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

$$
\begin{align}
\mathbb{B} &\le 2 \sum_{q=1}^{q_n} \exp \left( 2 \log N_q - \frac{\delta_q^2}{(8 + c_{p,\beta}^2)b_q^2} \right) \le 2 \sum_{q=1}^{q_n} \exp \left( - \frac{\delta_q^2}{2(8 + c_{p,\beta}^2)b_q^2} \right) \nonumber \\
&\le 2 \sum_{q=1}^{\infty} \exp \left( - \frac{2^q}{2(8 + c_{p,\beta}^2)b} \right) \to 0 \quad \text{as} \quad b \to 0. \tag{8.86}
\end{align}
$$

In view of (8.85), 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 (8.83) it suffices to prove, for each $\delta > 0$,

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

Making use of the square root trick (Lemma 5.2 [102]) and see also [137] in similar way as in [8], we
get

$$
\[
\mathbb{P} \left\{ \left\| k^{-1/2} \sum_{j=1}^{v_n} \epsilon_j \sum_{i \in H_j} \left( \varphi(\zeta_i) K \left( \frac{d(\boldsymbol{\zeta}_i, t)}{D_n^{-}} \right) \right) \right\| \times 1_{\{\kappa_2 F(\zeta_i) \le \lambda_n(n/\phi(h_K))^{1/2(p-1)}\}} \right\|_{\mathcal{F}\mathcal{H}_{(\lambda_n^{1/2}(\log(n))^{-1/2}, \|\cdot\|_p)}} \ge 2\delta \}
\]
$$
$$
\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(\boldsymbol{\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)}\}} \right\|_{\mathcal{F}\mathcal{H}_{(\lambda_n^{1/2}(\log(n))^{-1/2}, \|\cdot\|_p)}} \ge 2\delta,
$$
$$
\left\| k^{-1} \sum_{j=1}^{v_n} \left( \sum_{i \in H_j} \varphi(\zeta_i) K \left( \frac{d(\boldsymbol{\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}\mathcal{H}_{(\lambda_n^{1/2}(\log(n))^{-1/2}, \|\cdot\|_p)}}
$$
$$
\le 64 \lambda_n c_{p,\beta}^2 (\log(n))^{-1}
\}
$$
$$ + \mathbb{P} \left\{ \left\| k^{-1} \sum_{j=1}^{\nu_n} \left( \sum_{i \in H_j} \varphi(\zeta_i) K \left( \frac{d(\boldsymbol{\zeta}_i, t)}{D_n^{-}} \right) \mathbb{1}_{\{k_2 F(\zeta_i) \le \lambda_n(n/\phi(h_K))^{1/2}(p-1)\}} \right) \right\|^2 \right\}_{\mathcal{H}_{(\lambda_n^{1/2}(\log(n))^{-1/2}, \|\cdot\|_p)}} \\ > 64 \lambda_n c_{p,\beta}^2 (\log(n))^{-1} \right\} \\ =: \mathbb{P}(A_1) + \mathbb{P}(A_2). $$

Let us introduce the semi-norm

$$ \tilde{d}_{n\phi,2} := \left( k^{-1} \sum_{j=1}^{\nu_n} \sum_{i \in H_j} \left| \eta_{n,t,\varphi_1,K_1}(\boldsymbol{\zeta}_i, \boldsymbol{\zeta}_i) - \eta_{n,t,\varphi_2,K_2}(\boldsymbol{\zeta}_i, \boldsymbol{\zeta}_i) \right|^2 \right)^{1/2}, $$

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 the latter we can bound $\mathbb{P}(A_1)$, (the calculations are detailed in [27]). In the same way, as in [27] and before in [8], as a result of the independence between the blocks and condition (C.4.3), we apply again Lemma 5.2 in [102] and get

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

Therefore, the theorem is proved.

## Proof of Theorem 4.5

**Lemma 8.14.** Under assumptions of Theorem 4.5, we have

$$ \sqrt{n\tilde{\phi}(\mathbf{h}(t))} (\tilde{r}_n^{*(m)}(\varphi, \mathbf{t}; \mathbf{h}(t)) - \mathbb{E}[u_n(\varphi, \mathbf{t}; \mathbf{h}(t))]) \xrightarrow{d} \mathcal{N}(0, m^2\sigma_{\mathbf{t}}^2(\varphi, \varphi)), \quad (8.88) $$

and, if in addition, condition (C.8) is satisfied, then we have

$$ \sqrt{n^{-m+1}k^m} \{\tilde{r}_n^{*(m)}(\varphi, \mathbf{t}; \mathbf{h}_{n,k}(\mathbf{t})) - \mathbb{E}[u_n(\varphi, \mathbf{t}; \mathbf{h}(\mathbf{t}))]\} \xrightarrow{d} \mathcal{N}(0, m^2\sigma_{\mathbf{t}}^2(\varphi, \varphi)). \quad (8.89) $$

**Lemma 8.15.** Under the assumptions of Theorem 4.5, we have

$$ \sqrt{n\tilde{\phi}(\mathbf{h})} (u_n(\varphi_1, \mathbf{t}, \mathbf{h}) - \mathbb{E}(u_n(\varphi_1, \mathbf{t}, \mathbf{h}), u_n(\varphi_2, \mathbf{t}, \mathbf{h}) - \mathbb{E}(u_n(\varphi_2, \mathbf{t}, \mathbf{h}))) \xrightarrow{d} \mathcal{N}(0, \Sigma), \quad (8.90) $$

where

$$ \Sigma := (m^2\sigma_{\mathbf{t}}^2(\varphi_i, \varphi_j))_{i,j=1,2}. $$
## Proof of Theorem 4.5

As mentioned earlier, the study of the weak convergence of the conditional U-process is based on the study of two parts: the truncated part and the remainder part.

**Lemma 8.16.** Let $\mathcal{F}_m\mathcal{H}^m$ be a uniformly bounded class of measurable canonical functions from $\mathcal{X}^m \times \mathcal{Y}^m \to \mathbb{R}$, $m \ge 2$. Suppose that there are finite constants $a$ and $b$ such that the $\mathcal{F}_m\mathcal{H}^m$ covering number satisfies:

$$N(\epsilon, \mathcal{F}_m\mathcal{H}^m, \|\cdot\|_{L_2(Q)}) \le a\epsilon^{-b}, \qquad (8.91)$$

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

$$\beta_s s^r \to 0, \text{ as } s \to \infty \qquad (8.92)$$

for some $r > 1$, then

$$\left\| n^{\frac{-3m+1}{2}} \sqrt{k^m} \sum_{i \in I_m^k} G_{\varphi, t, \mathbf{h}_{n,k}}(\mathbf{X}_i, \mathbf{Y}_i) \right\|_{\mathcal{F}_m\mathcal{H}^m} \xrightarrow{\mathbb{P}} 0.$$

## Proof of Theorem 4.7

It is known that the weak convergence of an empirical process is obtained from its finite-dimensional convergence and its asymptotic equi-continuity (while respecting certain criteria). Theorem 4.5 gives the finite-dimensional convergence of the conditional U-process $\{\mu_n(\varphi, \mathbf{t}, \mathbf{h}(\mathbf{t}))\}_{\mathcal{F}_m\mathcal{H}^m}$, so what remains to be seen is its asymptotic equi-continuity. We decompose the U-process $\mu_n(\varphi, \mathbf{t})$ into two parts truncated and remainder part:

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

Following the same reasoning to obtain (A.48), we also know that:

$$\mu_n(\varphi, \mathbf{t}; \mathbf{D}_n^+) \le \mu_n(\varphi, \mathbf{t}; \mathbf{h}_{n,k}(t)) \le \mu_n(\varphi, \mathbf{t}; \mathbf{D}_n^-). \qquad (8.93)$$

Therefore, it suffices to prove the weak convergence of $\mu_n(\varphi, \mathbf{t}; \mathbf{D}_n^-)$ and $\mu_n(\varphi, \mathbf{t}; \mathbf{D}_n^+)$ instead of studying $\mu_n(\varphi, \mathbf{t}; \mathbf{h}_{n,k}(t))$. The steps of the proof are similar to [43] while taking into account a multivariate bandwidth $\mathbf{h}(\mathbf{t})$ instead of a univariate bandwidth $h_K$. In this section we only show the proof for $\mu_n^{(T)}(\varphi, \mathbf{t}; \mathbf{D}_n^-)$, the process can be replicated for $\mu_n^{(T)}(\varphi, \mathbf{t}; \mathbf{D}_n^+)$.

As shown earlier, the truncated part $\mu_n^{(T)}(\varphi, \mathbf{t}, \mathbf{h}(\mathbf{t}))$ is decomposed according to the Hoeffding's decomposition:

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

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

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

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

We need to introduce a new function

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

Hence,

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

The linear term of the process is given by

$$
\begin{align*}
m \sqrt{n \tilde{\phi}(\mathbf{D}_n^-)} u_n^{(1)} (\pi_{1,m} G_{\varphi,t}^{(T)}) &= \frac{1}{\sqrt{n \tilde{\phi}(\mathbf{D}_n^-)}} \sum_{i=1}^{n} \{ S_{\varphi,t,\mathbf{D}_n^-}(X_i, Y_i) - \mathbb{E}[S_{\varphi,t,\mathbf{D}_n^-}(X_i, Y_i)] \} \\
&:= \alpha_n(S_{\varphi,t,\mathbf{D}_n^-}).
\end{align*}
$$

Therefore, the linear term of the U-process {$\mu_n(\varphi, \mathbf{t}, \mathbf{D}_n^-)$}$_{\mathcal{F}_m, \mathcal{H}^m}$ is an empirical process indexed by the class of functions $\mathcal{S}$ defined by

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

therefore, its weak convergence may be established in a similar way as in the proof of Theorem 4.5.
It's clear that $\mathcal{S} \subset m\mathcal{G}^{(1)}$. We consider now the nonlinear part, we have to show that

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

This is a consequence of the Lemma 8.16. Note that the choice of the number and size of the blocks
must be made in such a way that the terms I – VI converge to 0. We need to prove that

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

Again, for clarity purposes, we restrict ourselves to $m = 2$. We have

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

We will use blocking arguments and treat the resulting terms. We start by considering the first I'. We
have

$$
\begin{align*}
& \mathbb{P} \left\{ \left\| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^{-})}} \sum_{p \neq q} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \tilde{\phi}(\mathbf{D}_n^{-}) \left\{ G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((X_i, X_j), (Y_i, Y_j)) - \mathbb{E}[G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((X_i, X_j), (Y_i, Y_j))] \right\} \right\|_{\mathcal{F}_2, \mathcal{H}^2} > \delta \right\} \\
& \le \mathbb{P} \left\{ \left\| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^{-})}} \sum_{p \neq q} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \tilde{\phi}(\mathbf{D}_n^{-}) \left\{ G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j)) - \mathbb{E}[G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j))] \right\} \right\|_{\mathcal{F}_2, \mathcal{H}^2} > \delta \right\} \\
& \quad + 2v_n \beta_{b_n}.
\end{align*}
$$

Notice that (4.13) readily implies that $v_n\beta_{b_n} \to 0$ and recall that for all $\varphi \in \mathcal{F}_m$

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

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

$$
\left\| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^{-})}} \sum_{p \neq q} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \tilde{\phi}(\mathbf{D}_n^{-}) \left\{ G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j))\right] \right\} \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\
\le \left| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^{-})}} \sum_{p \neq q} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \left\{ \kappa^2 F(\zeta_i, \zeta_j) \mathbb{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_n^{-}))^{1/2(p-1)}\}} \right. \right. \\
\left. \left. - \mathbb{E}\left[\kappa^2 F(\zeta_i, \zeta_j) \mathbb{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_n^{-}))^{1/2(p-1)}\}}\right] \right\} \right| . \quad (8.94)
$$
We use while maintaining order, Chebyshev's inequality and Hoeffding's trick, then

$$
\begin{align}
& \mathbb{P} \left\{ \left| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^-)}} \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \left\{ \begin{aligned}[t]
    & \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_n^-))^{1/2(p-1)}\}} \\
    & - \mathbb{E}\left[\kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_n^-))^{1/2(p-1)}\}}\right]
    \end{aligned} \right\} \right| > \delta \right\} \nonumber \\
& \le \delta^{-2} n^{-1} \tilde{\phi}^{-1}(\mathbf{D}_n^-) \mathrm{Var} \left( \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \kappa^2 F(\zeta_i, \zeta_j) \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_n^-))^{1/2(p-1)}\}} \right) \nonumber \\
& \le c_2 v_n \delta^{-2} n^{-1} \tilde{\phi}^{-1}(\mathbf{D}_n^-) \mathrm{Var} \left( \sum_{p=1}^{v_n} \sum_{i,j \in H_p^{(U)}} \kappa^2 F(\zeta_i, \zeta_j') \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_n^-))^{1/2(p-1)}\}} \right) \nonumber \\
& \le 2c_2 v_n \delta^{-2} n^{-2} \tilde{\phi}^{-1}(\mathbf{D}_n^-) \mathbb{E}\left[\left(\kappa^2 F(\zeta_1, \zeta_2)\right)^2 \mathbf{1}_{\{\kappa^2 F > \lambda(n/\tilde{\phi}(\mathbf{D}_n^-))^{1/2(p-1)}\}}\right]. \tag{8.95}
\end{align}
$$

Under (C.6), we have for each $\lambda > 0$

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

which tends to 0 as $n \to \infty$. The terms II', V' and VI' are treated in the same way as the first, except that for II', VI' we do not need to apply Hoeffding's trick 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, and for the term IV' we deduce its study from those of I' and III'. Let us consider the term III'. As for the truncated part, we have

$$
\mathbb{P} \left\{ \left| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^-)}} \sum_{p=1}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{q:|q-p| \ge 2}^{v_n} \sum_{j \in T_q^{(U)}} \tilde{\phi}(\mathbf{D}_n^-) \left\{ \begin{aligned}[t]
    & G_{\varphi, \mathbf{t}, \mathbf{D}_n}^{(R)}((X_i, X_j), (Y_i, Y_j)) \\
    & - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \mathbf{D}_n}^{(R)}((X_i, X_j), (Y_i, Y_j))\right]
    \end{aligned} \right\} \right| > \delta \right\} \quad \mathbb{F}_2 \times \mathbb{H}^2 \\
\le \mathbb{P} \left\{ \left| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^-)}} \sum_{p=1}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{q:|q-p| \ge 2}^{v_n} \sum_{j \in T_q^{(U)}} \tilde{\phi}(\mathbf{D}_n^-) \left\{ \begin{aligned}[t]
    & G_{\varphi, \mathbf{t}, \mathbf{D}_n}^{(R)}((\mathcal{S}_i, \mathcal{S}_j), (\zeta_i, \zeta_j)) \\
    & - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \mathbf{D}_n}^{(R)}((\mathcal{S}_i, \mathcal{S}_j), (\zeta_i, \zeta_j))\right]
    \end{aligned} \right\} \right| > \delta \right\}
$$
$$ -\mathbb{E}\left[G_{\varphi, \mathbf{t}, \mathbf{D}_{\bar{n}}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j))\right] \left\|_{\mathcal{F}_2, \mathcal{H}^2} > \delta \right\} \qquad (8.96) \\ + \frac{v_n^2 a_n b_n \beta_{a_n}}{\sqrt{n \tilde{\phi}(\mathbf{D}_{\bar{n}})}} $$

We also have

$$ P\left\{ \left\| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^{-})}} \sum_{i \in H_1^{(U)}} \sum_{q:|q-p| \ge 2} \sum_{j \in T_q^{(U)}} \left\{ G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j))\right] \right\} \right\|_{\mathcal{F}_2, \mathcal{H}^2} > \delta \right\} \\ \le P\left\{ \left\| \frac{1}{\sqrt{n\tilde{\phi}(\mathbf{D}_n^{-})}} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{v_n} \sum_{j \in T_q^{(U)}} \left\{ G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j)) - \mathbb{E}\left[G_{\varphi, \mathbf{t}, \mathbf{D}_n^{-}}^{(R)}((\varsigma_i, \varsigma_j), (\zeta_i, \zeta_j))\right] \right\} \right\|_{\mathcal{F}_2, \mathcal{H}^2} > \delta \right\}. $$

Since the Eq (8.94) is still satisfied, the problem is reduced to

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

we follow the same procedure as in (8.95). The rest has just been shown to be asymptotically negligible, so the process {$\mu_n(\varphi, \mathbf{t}, \mathbf{D}_n^{-})$}$_{\mathcal{F}_m, \mathcal{H}^m}$ converges in law to a Gaussian process which has a version with uniformly bounded and uniformly continuous paths with respect to $|| \cdot ||_2$-norm. By repeating the same steps, this also holds true, for the process {$\mu_n(\varphi, \mathbf{t}, \mathbf{D}_n^{+})$}$_{\mathcal{F}_m, \mathcal{H}^m}$. Consequently, by (8.93) it follows that the process {$\mu_n(\varphi, \mathbf{t}, \mathbf{h}_{n,k}(\mathbf{t}))$}$_{\mathcal{F}_m, \mathcal{H}^m}$ also converges in law to a Gaussian process which has a version with uniformly bounded and uniformly continuous paths with respect to $|| \cdot ||_2$-norm. In a similar way, we treat {$\mu_n(1, \mathbf{t}, \mathbf{h}_{n,k}(\mathbf{t}))$}$_{\mathcal{H}^m}$, and about $\mathbb{E}(u_n(\varphi, \mathbf{t}, \mathbf{h}_{n,k}(\mathbf{t})))$, $u_n(1, \mathbf{t}, \mathbf{h}_{n,k}(\mathbf{t}))$ and $\mathbb{E}(u_n(1, \mathbf{t}, \mathbf{h}_{n,k}(\mathbf{t})))$ the treatment is done as in the proof of Theorem 4.5.

## Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

## Acknowledgments

The authors would like to thank the Editor-in-Chief, an Associate-Editor, and four anonymous referees for their constructive remarks, which resulted in a substantial improvement of the work's
original form and a more sharply focused presentation.

The second author gratefully acknowledges the funding received towards his PhD from the Algerian government PhD fellowship.

## Conflicts of 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.

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A. Proofs of lemmas

*Proof of Lemma 8.2:*

Following the same notation as [43], refer also to [27] and [8], the proof of this lemma is based on the blocking approach, which consists of breaking down a strictly stationary sequence $(X_1, \dots, X_n)$, into $2v_n$, equal-sized blocks, that each one is of length $n - 2v_n a_n$ that is, for $1 \le j \le v_n$,

$$H_j = \{i : 2(j-1)a_n + 1 \le i \le (2j-1)a_n\},$$

$$T_j = \{i : (2j-1)a_n + 1 \le i \le 2ja_n\},$$

$$R = \{i : (2v_n a_n + 1 \le i \le n)\}.$$

The values of $v_n, a_n$ are given in the following. Another important component in this proof is the sequence of independent blocks $(\xi_1, \dots, \xi_n)$ satisfying

$$\mathcal{L}(\xi_1, \dots, \xi_n) = \mathcal{L}(X_1, \dots, X_{a_n}) \times \mathcal{L}(X_{a_n+1}, \dots, X_{2a_n}) \times \dots$$

As in [27, 43], applying the results of [84] on $\beta$-mixing, gives us for any measurable set A

$$|\mathbb{P}\{(\xi_1, \dots, \xi_{a_n}, \xi_{2a_n+1}, \dots, \xi_{3a_n}, \dots, \xi_{2(v_n-1)a_n+1}, \dots, \xi_{2v_n a_n}) \in A\}$$
$$ \begin{aligned} & -\mathbb{P}\{(X_1, \dots, X_{a_n}, X_{2a_n+1}, \dots, X_{3a_n}, \dots, X_{2(\nu_n-1)a_n+1}, \dots, X_{2\nu_n a_n}) \in A\} \\ & \le 2(\nu_n - 1)\beta_{a_n}. \end{aligned} \quad (A.1) $$

Let $t_1, \dots, t_{N_\varepsilon(S_X)}$ be an $\varepsilon$-net of $S_X$ and $\ell(t) = \arg \min_{\ell \in \{1, 2, \dots, N_\varepsilon(S_X)\}} d(t, t_\ell)$. Observing the following decomposition

$$ \begin{aligned} & \sup_{\varphi K \in \mathcal{F}\mathcal{H}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_X} \left| \tilde{r}_{n,2}^{(1)}(\varphi, t, h_K(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t, h_K(t))) \right| \\ & \le \underbrace{\sup_{\varphi K \in \mathcal{F}\mathcal{H}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_X} \left| \tilde{r}_{n,2}^{(1)}(\varphi, t, h_K(t)) - \tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t_\ell)) \right|}_{G_1} \\ & \quad + \underbrace{\sup_{\varphi K \in \mathcal{F}\mathcal{H}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_X} \left| \tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t_\ell)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t_\ell))) \right|}_{G_2} \\ & \quad + \underbrace{\sup_{\varphi K \in \mathcal{F}\mathcal{H}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_X} \left| \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t_\ell))) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t, h_K(t))) \right|}_{G_3}. \end{aligned} \quad (A.2) $$

Let us start with the term $G_2$. We need to prove that there exists $\eta > 0$ and $b_0 > 0$, such that

$$ \sum_{n \ge 1} \mathbb{P} \left\{ \sup_{\varphi K \in \mathcal{F}\mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| \tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t))) \right| > \eta \sqrt{\frac{\psi_{S_X}\left(\frac{\log n}{n}\right)}{n\phi(h_{n,1})}} \right\} < \infty. $$

The key strategy here is to work with independent block sequence $\{\xi_j = (\zeta_j, \zeta_j)\}_{j=1}^\infty$ instead of dependent once, this can be achieved by (A.1). Then we can clearly see that for $1 \le \ell \le N_\varepsilon(S_X)$:

$$ \begin{aligned} & \mathbb{P} \left\{ \sup_{\varphi K \in \mathcal{F}\mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| \tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t))) \right| > \eta \sqrt{\frac{\psi_{S_X}\left(\frac{\log n}{n}\right)}{n\phi(h_{n,1})}} \right\} \\ &= \mathbb{P} \left\{ \sup_{\varphi K \in \mathcal{F}\mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| (n\phi_t(h_K))^{-1} \sum_{i=1}^n \left( \varphi(Y_i)K\left(\frac{d(X_i, t_\ell)}{h_K}\right) - \mathbb{E}\left[\varphi(Y_i)K\left(\frac{d(X_i, t_\ell)}{h_K}\right)\right] \right) \right| \\ &> \eta \sqrt{\frac{\psi_{S_X}\left(\frac{\log n}{n}\right)}{n\phi(h_{n,1})}} \right\} \\ &\le 2\mathbb{P} \left\{ \sup_{\varphi K \in \mathcal{F}\mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| (n\phi_t(h_K))^{-1} \sum_{j=1}^{\nu_n} \sum_{i \in H_j} \left( \varphi(\zeta_i)K\left(\frac{d(\zeta_i, t)}{h_K}\right) - \mathbb{E}\left[\varphi(\zeta_i)K\left(\frac{d(\zeta_i, t)}{h_K}\right)\right] \right) \right| \right\}. \end{aligned} $$
$$ > \eta' \sqrt{\frac{\psi_{S_X}\left(\frac{\log n}{n}\right)}{n\phi(h_{n,1})}} + 2(\nu_n - 1)\beta_{a_n}. \quad (A.3) $$

If we put

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

then, by condition (C.5.1), it follows that $(\nu_n - 1)\beta_{a_n} \to 0$ as $n \to 0$. Hence, only the first term in the right-hand side of (A.3) remains to be dealt with. Set

$$ h_{K,j} = 2^j h_{n,1}, \quad L(n) = \max\{j : h_{K,j} \le 2b_0\} \quad \text{and} \quad \mathcal{H}_{K,j} := [h_{K,j-1}, h_{K,j}]. $$

Then, we have $[h_{n,1}, b_0] \subseteq \bigcup_{j=1}^{L(n)} \mathcal{H}_{K,j}$. The empirical measure on $\{\alpha_n(X_i, Y_i) : i = 1, 2, \dots, n\}$ is defined by:

$$ \alpha_n(g) = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \{g(X_j, Y_j) - \mathbb{E}[g(X_j, Y_j)]\}. $$

For the original sequence, we write

$$ Y_{j,g}(X_{a_n}) = \sum_{i \in H_j} \{g(X_j, Y_j) - \mathbb{E}[g(X_j, Y_j)]\}. $$

For the constructed independent block sequence $\xi$, define

$$ Z_{j,g}(\xi_{a_n}) = \sum_{i \in H_j} \{g(\xi_i) - \mathbb{E}[g(\xi_i)]\} \quad \text{and} \quad \tilde{\alpha}_{\nu_n}(g) = \frac{1}{\sqrt{n}} \sum_{i=1}^{\nu_n} Z_{j,g}. $$

Hence, we have

$$ \tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t))) = \frac{1}{\sqrt{n}\phi_{t_{\ell(t)}}(h_K)} \tilde{\alpha}_{\nu_n}(\varphi K). $$

We consider also the following class of functions for $1 \le j \le L(n)$ and $1 \le \ell \le N_{\varepsilon_n}(S_X)$:

$$ \mathcal{G}_{\varphi K, j}^{(\ell)} = \{ (x, y) \mapsto \varphi(y)K \left( \frac{d(x, t_{\ell})}{h_K} \right) \text{ where } K \in \mathcal{K}, \varphi \in \mathcal{F} \text{ and } h_K \in \mathcal{H}_{K,j} \}, $$

with the envelope function $G_{\varphi K, j}^{(\ell)}$, also keep in mind that $S_X \subseteq \bigcup_{\ell=1}^{N_{\varepsilon}(S_X)} B(t_\ell, \varepsilon)$. Now that we have all the necessary elements and using the fact that we are now dealing with sums of independent blocks, we can use the standard inequalities of independent setting. We have

$$ \mathbb{P} \left\{ \sup_{\varphi K \in \mathcal{F}, \mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| (\sqrt{n}\phi_t(h_K))^{-1} \tilde{\alpha}_{\nu_n}(\varphi K) \right| > \eta' \sqrt{\frac{\psi_{S_X}\left(\frac{\log n}{n}\right)}{n\phi(h_{n,1})}} \right\} $$
$$
\begin{align*}
&\le \sum_{\ell=1}^{N_{\varepsilon_n}(S_X)} \sum_{j=1}^{L(n)} \mathbb{P} \left\{ \frac{1}{\sqrt{n\phi(h_{K,j})}\psi_{S_X}(\varepsilon_n)} \max_{1\le\ell\le N_{\varepsilon_n}(S_X)} \max_{1\le j\le L(n)} \left\| \sqrt{n\tilde{\alpha}_{v_n}(K)} \right\|_{\mathcal{G}_{K,j}^{(\ell)}} \ge \eta' \right\} \\
&\le L(n)N_{\varepsilon_n}(S_X) \max_{1\le\ell\le L(n)} \max_{1\le\ell\le N_{\varepsilon_n}(S_X)} \mathbb{P} \left\{ \left\| \sqrt{n\tilde{\alpha}_{v_n}(\varphi K)} \right\|_{\mathcal{G}_{K,j}^{(\ell)}} > \eta' \sqrt{n\phi(h_{K,j})\psi_{S_X}(\varepsilon_n)} \right\} \\
&\le L(n)N_{\varepsilon_n}(S_X) \max_{1\le\ell\le L(n)} \max_{1\le\ell\le N_{\varepsilon_n}(S_X)} \mathbb{P} \left\{ \max_{1\le p\le n} \left\| \sqrt{p\tilde{\alpha}_{v_p}(\varphi K)} \right\|_{\mathcal{G}_{K,j}^{(\ell)}} > \eta' \sqrt{n\phi(h_{K,j})\psi_{S_X}(\varepsilon_n)} \right\}. \tag{A.4}
\end{align*}
$$

Now, to bound the probability in (A.4), we apply Bernstein's inequality with

$$
\mu_n := \mathbb{E} \left\| \sqrt{n} \alpha_{v_n} (\varphi K) \right\|_{\mathcal{G}_{\varphi K, j}^{(\ell)}},
$$

and

$$
\sigma^2 := \mathbb{E} \left( G_{\varphi K, j}^{(\ell)}(X, Y) \right)^2.
$$

We know that the envelope function $G_{\varphi K,j}^{(\ell)}$, verifies

$$
G_{\varphi K,j}^{(\ell)}(X, Y) \leq M_{K_2} 1_{B(t_l, h_{K,j})}(X).
$$

Hence, by (C.1.1), (C.3.1), (C.4.1) combined with Lemma B.1 and Lemma B.2, we get that

$$
\mu_n = O\left(\sqrt{n\phi(h_{K,j})}\right) \text{ and } \sigma^2 = O\left(\phi(h_{K,j})\right).
$$

Now, we can apply Bernstein’s inequality to the empirical processes, for

$$
z = (\eta'/2) \sqrt{n\phi(h_{K,j})\psi_{S_X}(\epsilon_n)} \quad \text{and} \quad H = \kappa_2 \left(\frac{2}{q!}\right)^{1/(q-2)}, \quad \text{or simply } H = \kappa_2 \text{ if } q=2.
$$

Making use of Lemma B.3, we get

$$
\begin{align}
& \mathbb{P} \left\{ \max_{1 \le p \le n} \left\| \sqrt{p} \tilde{\alpha}_{v_p} (\varphi K) \right\|_{\mathcal{G}_{K,j}^{(\ell)}} \ge \eta' \sqrt{n\phi(h_{K,j})\psi_{S_X}(\varepsilon_n)} \right\} \nonumber \\
& \le \mathbb{P} \left\{ \max_{1 \le p \le n} \left\| \sqrt{p} \tilde{\alpha}_{v_p} (\varphi K) \right\|_{\mathcal{G}_{K,j}^{(\ell)}} \ge \mu_n + z \right\} \tag{A.5}
\end{align}
$$

$$
\begin{align*}
& \le \exp \left( \frac{-(\eta'/2)^2 \sqrt{n} \psi_{S_X}(\epsilon_n)}{8 \kappa_2^2 \sqrt{n} + 4\eta' \kappa_2 \sqrt{\frac{\psi_{S_X}(\epsilon_n)}{\phi(h_{K,j})}}} \right) \\
& \le n^{-C\epsilon_0}, \tag{A.6}
\end{align*}
$$

where $C > 0$. Moreover, from Eq (2.10), and the fact that $L(n) \sim \frac{\log(b_0/h_{n,1})}{\log(2)}$ and by choosing

$$
\epsilon_0^2 \mathbf{C} > 1, \epsilon_n = \log n/n,
$$

we get from (A.6) and (A.4),

$$
\sum_{n \ge 1} \mathbb{P} \left\{ G_2 > \eta \sqrt{\frac{\psi_{S_X} \left( \frac{\log n}{n} \right)}{n \phi(h_{n,1})}} \right\} \le \sum_{n \ge 1} C L(n) N_{\epsilon_n}(S_X) n^{-C \epsilon_0} < \infty.
$$
Now, let us study the term $G_1$. By conditions (C.4.1) and (C.3.1), for some constant $\mathbb{C} > 0$, we have

$$
\begin{aligned}
& \sup_{\varphi K \in \mathcal{F} \mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| \hat{r}_{n,2}^{(1)}(\varphi, t, h_K(t)) - \hat{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t_\ell)) \right| \\
&= \sup_{\varphi K \in \mathcal{F} \mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| \frac{1}{n\phi_t(h_K)} \sum_{i=1}^n \varphi(Y_i) K\left(\frac{d(X_i, t)}{h_K}\right) - \frac{1}{n\phi_{t_{\ell(t)}}(h_K)} \sum_{i=1}^n \varphi(Y_i) K\left(\frac{d(X_i, t_{\ell(t)})}{h_K}\right) \right| \\
&\le M \left| \frac{1}{n\phi_t(h_K)} \sum_{i=1}^n K\left(\frac{d(X_i, t)}{h_K}\right) - \frac{1}{n\phi_{t_{\ell(t)}}(h_K)} \sum_{i=1}^n K\left(\frac{d(X_i, t_{\ell(t)})}{h_K}\right) \right| \\
&\le \sup_{K \in \mathcal{K}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \frac{M\mathbb{C}}{n\phi(h_K)} \sum_{i=1}^n \left| K\left(\frac{d(X_i, t)}{h_K}\right) - K\left(\frac{d(X_i, t_{\ell(t)})}{h_K}\right) \right| 1_{B(t,h_K) \cup B(t_{\ell(t)},h_K)}(X_i).
\end{aligned}
$$

Using the fact that the kernel $K(\cdot)$ is supposedly Lipschitz, we obtain

$$
\begin{aligned}
G_1 &\le \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \frac{1}{n} \sum_{i=1}^n \frac{1}{h_K \phi(h_K)} d(t, t_{\ell(t)}) 1_{B(t,h_K) \cup B(t_{\ell(t)},h_K)}(X_i) \\
&\le \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \frac{1}{n} \sum_{i=1}^n \frac{\varepsilon_n}{h_K \phi(h_K)} 1_{B(t,h_K) \cup B(t_{\ell(t)},h_K)}(X_i) \\
&\le \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \frac{1}{n} \sum_{i=1}^n W_i,
\end{aligned}
$$

where for $1 \le i \le n$:

$$W_i = \frac{\varepsilon_n}{h_K \phi(h_K)} 1_{B(t,h_K) \cup B(t_{\ell(t)},h_K)}(X_i).$$

Taking into account that $h_K \in [h_{n,1}, h_{n,2}]$ and using condition (C.7), we get for $q \ge 2$:

$$\mathbb{E}(W_1)^q \le C_1^q \left( \frac{1}{\sqrt{\phi(h_{n,1})}} \right)^{2(q-1)},$$

where we assume that $C_1$ is the bound of the sequence $\log n/nh_{n,1}$ which converges to 0. Hence, by applying a standard inequality (see Corollary A.8 [91]), the conditions are satisfied here, uniformly on $t \in S_X$ and on $h_K$, on:

$$\frac{1}{n} \sum_{i=1}^{n} W_i,$$

in combination with condition (C.6) and (C.7), we readily obtain

$$G_1 = O_{a,co} \left( \sqrt{\frac{\psi_{S_X} \left( \frac{\log n}{n} \right)}{n \phi(h_{n,1})}} \right). \quad (A.7)$$

Finally, we treat $G_3$. Noticing that

$$G_3 \le \mathbb{E} \left( \sup_{\varphi K \in \mathcal{F} \mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| \hat{r}_{n,2}^{(1)}(\varphi, t, h_K(t)) - \hat{r}_{n,2}^{(1)}(\varphi, t_\ell, h_K(t_\ell)) \right| \right),$$
similar to the procedures of treating $G_1$ once again, it follows directly that

$$G_3 = O_{a.co} \left( \sqrt{\frac{\psi_{S_x} \left( \frac{\log n}{n} \right)}{n\phi(h_{n,1})}} \right). \qquad (A.8)$$

Hence, by combining (A.2) and (A.8), we so obtain (8.2), as sought.

### Proof of (8.3):

Notice that (8.3) is a direct result of (8.2) when the function $\varphi \equiv 1$. This completes the proof of the lemma.

### Proof of Lemma 8.10

Let us consider the term $I_1(t, H_{n,k}(t))$ such that $1_{\{D_n^- \le H_{n,k}(t) \le D_n^+\}} \xrightarrow{a.co} 1$ when $\frac{k}{n} \to 0$, and making use of condition (C.3.2), we have

$$\hat{r}_{n,2}^{(1)}(\varphi, t; D_n^+) \le \hat{r}_{n,2}^{(1)}(\varphi, t; H_{n,k}(t)) \le \hat{r}_{n,2}^{(1)}(\varphi, t; D_n^-). \qquad (A.9)$$

We have

$$\sqrt{k} |S_1(t, H_{n,k}(t)) - \mathbb{E}(S_1(t, H_{n,k}(t)))| \le \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi, t; H_{n,k}(t)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t; H_{n,k}(t)))| \qquad (A.10)$$

$$+ \sqrt{k} \left| \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t; H_{n,k}(t))) - \hat{r}_{n,2}^{(1)}(\varphi_M, t; H_{n,k}(t)) \right| \qquad (A.11) \\ = I_{11}(t, H_{n,k}(t)) + I_{12}(t, H_{n,k}(t)).$$

The Eq (A.10) gives

$$|I_{11}(t, H_{n,k}(t))| = \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi, t; H_{n,k}(t)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t; H_{n,k}(t)))| \le \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^+))| \le \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-))| \qquad (A.12)$$

$$+ \sqrt{k} \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^+)). \qquad (A.13)$$

Similarly, (A.11) gives us:

$$|I_{12}(t, H_{n,k}(t))| = \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi_M, t; H_{n,k}(t)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t; H_{n,k}(t)))| \le \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-))| + \sqrt{k} \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+)). \qquad (A.14)$$
The following decomposition will be used to treat (A.12)

$$ \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-))| \le \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-) - \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)| \quad (A.15) $$

$$ + \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-))| \quad (A.16) $$

$$ + \sqrt{k} |\mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-))|. \quad (A.17) $$

For some $\xi_n \in (0, 1)$, recall that $D_n^-$ and $D_n^+$ are defined in (8.6) and (8.7) respectively. Moreover, observe that

$$ \begin{aligned} & |\mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-))| \\ & = \frac{1}{n \mathbb{E}(\Delta_1(t, D_n^-))} \left| \mathbb{E} \left( \sum_{i=1}^n \varphi(Y_i) 1_{\{|F(Y_i)|>\delta_n\}} \Delta_i(t, D_n^-) \right) \right| \\ & \le \mathbb{E} \left( |\varphi(Y)| 1_{\{|F(Y)>\delta_n\}} \Delta_1(t, D_n^-) \right) \left( \mathbb{E}(\Delta_1(t, D_n^-)) \right)^{-1}. \end{aligned} $$

Under the assumptions (C.4.2) and (C.8), using Hölder's inequality, for $\alpha_1 = \frac{p}{2}$ and $\alpha_2$ such that $\frac{1}{\alpha_1} + \frac{1}{\alpha_2} = 1$, we can write

$$ \begin{aligned} & \sqrt{k} |\mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-))| \\ & \le \frac{\sqrt{k}}{\phi(D_n^-)} \mathbb{E}^{1/\alpha_1} [|F(Y)_1^\alpha| 1_{\{|F(Y)>\delta_n\}}] \mathbb{E}^{1/\alpha_2} [\Delta_1^{\alpha_2}(t, D_n^-)] \\ & \le \frac{\sqrt{k}}{\phi(D_n^-)} \delta_n^{-1} \mathbb{E}^{1/\alpha_1} (|F(Y)|^p |X) \phi^{1/\alpha_2}(D_n^-) \\ & \le \frac{C \sqrt{k}}{\delta_n \phi^{2/p}(D_n^-)} \\ & \le \frac{C}{\zeta^{\frac{1}{p}} \sqrt{n} \delta_n} \left(\frac{k}{n}\right)^{\frac{1}{2}-\frac{2}{p}} \to 0 \text{ as } n \to \infty. \end{aligned} $$

To obtain the convergence in probability of (A.15), it suffices to use the assumption (C.4.2), (C.8), and Markov's inequality. Indeed, for all $\varepsilon > 0$, we obtain

$$ \begin{aligned} & \mathbb{P} \left( \sqrt{k} |\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-) - \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)| > \sqrt{k} \varepsilon \right) \\ & \le \mathbb{E} \left( |F(Y)| 1_{\{|F(Y)>\delta_n\}} \Delta_1(t, D_n^-) \right) \left( \sqrt{k} \varepsilon \mathbb{E} (\Delta_1(t, D_n^-)) \right)^{-1} \\ & \le \frac{C \phi^{(1-\alpha_2)/\alpha_2}(D_n^-)}{\varepsilon \delta_n \sqrt{k}} \\ & \le \frac{C}{\varepsilon \delta_n} \frac{\phi^{-\frac{2}{p}}(D_n^-)}{\sqrt{k}} \to 0 \end{aligned} $$
for some $\delta_n$ large enough and by condition (C.8). Now, for the second term of (A.14), Lemma 8.2 gives

$$ \left| \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)) \right| \to 0, \text{ a.co. as } n \to \infty. \quad (A.18) $$

Using the fact that $k(\phi^{-1}(\frac{k}{n}))^{2\gamma} \to 0$ and by condition (C.2.1), for $u = u_n = \phi^{-1}(\frac{k}{n})$, (A.13) gives

$$
\begin{align*}
& \sqrt{k} \left| \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^+)) \right| \\
& \le \sqrt{k} \left| \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^-)) - r^{(1)}(\varphi, t) \right| \\
& \phantom{\le} + \sqrt{k} \left| \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi, t, D_n^+)) - r^{(1)}(\varphi, t) \right| \\
& \le \sqrt{k} \left| \mathbb{E} \left( \frac{1}{\mathbb{E}\Delta_1(t, D_n^-)} \Delta_1(t, D_n^-) \varphi(Y) \right) - r^{(1)}(\varphi, t) \right| \\
& \phantom{\le} + \sqrt{k} \left| \mathbb{E} \left( \frac{1}{\mathbb{E}\Delta_1(t, D_n^+)} \Delta_1(t, D_n^+) \varphi(Y) \right) - r^{(1)}(\varphi, t) \right| \tag{A.19} \\
& \le \sqrt{k} \sum_{u=D_n^-, D_n^+} \left| \frac{1}{\mathbb{E}\Delta_1(t, u)} \mathbb{E}' \left( \Delta_1(t, u) 1_{B(t,u)} (r^{(1)}(\varphi, X_1) - r^{(1)}(\varphi, t)) \right) \right| \\
& \le \sqrt{k} (C 1_{B(t,D_n^-)} + C' 1_{B(t,D_n^+)}) d(t, X_1)^\gamma \\
& \le C \sqrt{k} ((D_n^-)^\gamma + (D_n^+)^\gamma) \\
& \le \frac{2C}{\zeta^\gamma} \sqrt{k \left( \phi^{-1} \left( \frac{k}{n} \right) \right)^{2\gamma}} \to 0, \text{ as } n \to \infty.
\end{align*}
$$

On the other side, recall that $\varphi_M(Y) := \varphi(Y)1_{\{F(Y)<M\}}$, then, using the fact that the regression function satisfies the Lipschitz condition and under condition (C.4.1), we have

$$
\begin{align*}
& \sqrt{k} \left| \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)) - \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+)) \right| \\
& \le \sqrt{k} \left| \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-)) - r^{(1)}(\varphi_M, t) \right| \\
& \phantom{\le} + \left| \mathbb{E}(\hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+)) - r^{(1)}(\varphi_M, t) \right| \\
& \le \sqrt{k} \left| \mathbb{E} \left( \frac{1}{\mathbb{E}\Delta_1(t, D_n^-)} \Delta_1(t, D_n^-) \varphi_M(Y) \right) - r^{(1)}(\varphi_M, t) \right| \\
& \phantom{\le} + \sqrt{k} \left| \mathbb{E} \left( \frac{1}{\mathbb{E}\Delta_1(t, D_n^+)} \Delta_1(t, D_n^+) \varphi_M(Y) \right) - r^{(1)}(\varphi_M, t) \right| \tag{A.20} \\
& \le \sqrt{k} \sum_{u=D_n^-, D_n^+} \left| \frac{1}{\mathbb{E}\Delta_1(t, u)} \mathbb{E} \left( \Delta_1(t, u) 1_{B(t,u)} (r^{(1)}(\varphi_M, X_1) - r^{(1)}(\varphi_M, t)) \right) \right| \\
& \le \sqrt{k} (C 1_{B(t,D_n^-)} + C' 1_{B(t,D_n^+)}) d(t, X_1)^\gamma \\
& \le \frac{2C}{\zeta^\gamma} \sqrt{k \left( \phi^{-1} \left( \frac{k}{n} \right) \right)^{2\gamma}} \to 0, \text{ as } n \to \infty.
\end{align*}
$$
For the second term of the right-hand side of the Eq (8.74), we have

$$|S_2(t, H_{n,k}(t))| = |\mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t; H_{n,k}(t)) - r^{(1)}(\varphi, t))| \quad (A.21)$$

$$\leq |\mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t; D_n^{-}) - r^{(1)}(\varphi, t))|, \quad (A.22)$$

making use of the Lemma 8.2, we readily infer

$$\sqrt{k} S_2(t, H_{n,k}(t)) = O\left(\sqrt{\zeta^\gamma k \left(\phi^{-1}\left(\frac{k}{n}\right)\right)^{2\gamma}}\right). \quad (A.23)$$

The proof of $S_2(t, H_{n,k}(t))$ is complete. Hence, the convergence of $I_1(t, H_{n,k}(t))$ to 0 is complete combining the result (8.74) and (A.18), (A.19), (A.20) and (A.23).

### Proof of Lemma 8.3:

In this lemma, we suppose that the class of functions $\mathcal{F}$ is not necessarily bounded but satisfying condition (C.4.2). In order to prove Lemma 8.3, we proceed as follows, for an arbitrary $\lambda > 0$ and each function $\varphi \in \mathcal{F}$:

$$\begin{aligned} \varphi(y) &= \varphi(y)1_{\{F(y) \le \lambda \eta_n^{1/p}\}} + \varphi(y)1_{\{F(y) > \lambda \eta_n^{1/p}\}} \\ &= \varphi^{(T)}(y) + \varphi^{(R)}(y), \end{aligned}$$

where we take $\eta_n = n/(\log n)^2$. So, we write

$$\begin{aligned} \tilde{r}_{n,2}^{(1)}(\varphi, t, h_K(t)) &:= \frac{1}{n \mathbb{E}(\Delta_1(t; h_K(t)))} \sum_{i=1}^{n} \varphi^{(T)}(Y_i) K\left(\frac{d(X_i, t)}{h_K}\right) \\ &\quad + \frac{1}{n \mathbb{E}(\Delta_1(t; h_K(t)))} \sum_{i=1}^{n} \varphi^{(R)}(Y_i) K\left(\frac{d(X_i, t)}{h_K}\right) \\ &= \tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t, h_K(t)) + \tilde{r}_{n,2}^{(1)}(\varphi^{(R)}, t, h_K(t)). \end{aligned}$$

Let us start with the truncated part. Following the same reasoning as in the proof of Lemma 8.2, we have:

$$\begin{aligned} & \sup_{\varphi K \in \mathcal{F}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_X} \left| \tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t, h_K(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t, h_K(t))) \right| \\ & \le \underbrace{\sup_{\varphi K \in \mathcal{F}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_X} \left| \tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t, h_K(t)) - \tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t_{\ell(t)}, h_K(t_{\ell})) \right|}_{G_{1,T}} \\ & \quad + \underbrace{\sup_{\varphi K \in \mathcal{F}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_X} \left| \tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t_{\ell(t)}, h_K(t_{\ell})) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t_{\ell(t)}, h_K(t_{\ell}))) \right|}_{G_{2,T}} \\ & \quad + \underbrace{\sup_{\varphi K \in \mathcal{F}} \sup_{h_{n,1} \le h_K \le h_{n,2}} \sup_{t \in S_X} \left| \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t_{\ell(t)}, h_K(t_{\ell}))) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t, h_K(t))) \right|}_{G_{3,T}}. \end{aligned} \quad (A.24)$$
First, let us start with the term $G_{2,T}$, using the same notation as the proof of Lemma 8.2, we have:

$$ \tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t_{\ell(t)}, h_K(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t_{\ell(t)}, h_K(t))) = \frac{1}{\sqrt{n}\phi_{t_{\ell(t)}}(h_K)} \tilde{\alpha}_{\nu_n}(\varphi^{(T)}K). $$

where $\tilde{\alpha}_{\nu_n}(\varphi^{(T)}K)$ is an empirical process indexed by the class of functions $\mathcal{F}\mathcal{H}\mathcal{I}_{\lambda\eta_n}$, where we define the class of functions of $y \in \mathcal{Y}$, $\mathcal{I}_{\lambda\eta_n}$ by

$$ \mathcal{I}_{\lambda\eta_n} := \{1_{\{F(y) \le \lambda\eta_n^{1/p}\}} : \lambda > 0\}, $$

where $p$ being the moment order given in (C.4.2). We consider also the following class of functions, for $1 \le j \le L(n)$ and $1 \le \ell \le N_{\varepsilon_n}(S_\chi)$,

$$ \mathcal{G}_{\varphi K, j}^{(\ell, T)} = \{ (x, y) \mapsto \varphi(y) K \left( \frac{d(x, t_{\ell})}{h_K} \right) 1_{\{F(y) \le \lambda \eta_n^{1/p}\}} : K \in \mathcal{K}, \varphi \in \mathcal{F} \text{ and } h_K \in \mathcal{H}_{K,j}, \lambda > 0 \}, $$

that the envelope function is denoted by $G_{\varphi K, j}^{(\ell, T)}$, then we have

$$ \mathbb{P} \left\{ \sup_{\varphi K \in \mathcal{F}, \mathcal{H}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_\chi} \left| \tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t_{\ell(t)}, h_K(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi^{(T)}, t_{\ell(t)}, h_K(t))) \right| > \eta \sqrt{\frac{\psi_{S_\chi}\left(\frac{\log n}{n}\right)}{n\phi(h_{n,1})}} \right\} \quad (A.25) $$

$$ \le L(n) N_{\varepsilon_n}(S_\chi) \max_{1 \le j \le L(n)} \max_{1 \le \ell \le N_{\varepsilon_n}(S_\chi)} \mathbb{P} \left\{ \max_{1 \le p \le n} \left\| \sqrt{p} \tilde{\alpha}_{\nu_p}(\varphi^{(T)} K) \right\|_{\mathcal{G}_{\varphi K, j}^{(\ell)}} > \eta' \sqrt{n\phi(h_{K,j})\psi_{S_\chi}(\varepsilon_n)} \right\}. \quad (A.26) $$

We apply Bernstein's inequality to bound (A.26), so, we have to study the asymptotic behavior of the quantities

$$ \mu_{n,T}^{(2)} = \mathbb{E} \left\| \sqrt{n} \tilde{\alpha}_{\nu_n}(\varphi^{(T)} K) \right\|_{\mathcal{G}_{\varphi K, j}^{(\ell,T)}}, $$

and

$$ \mathbb{E} \left( G_{\varphi K, j}^{(\ell, T)} (X, Y) \right)^2. $$

From condition (C.4.2), we remark that

$$ G_{\varphi K, j}^{(\ell, T)}(X, Y) \le F(Y) \kappa_2 \mathbb{1}_{B(t, h_{K,j})}(X) \mathbb{1}_{\{F(y) \le \lambda \eta_n^{1/p}\}}, $$

so thanks to the condition (C.4.2), we get

$$ \mathbb{E} \left( G_{\varphi K, j}^{(\ell, T)} (X, Y) \right)^2 \le \lambda^2 \kappa_2^2 \phi(h_{K,j}) \eta_n^{2/p} = \sigma_{(T)}^2. $$

Furthermore, making use of Lemma B.1 in combination with Lemma B.2 in appendix, and since condition (C.4.3) is satisfied, we infer that

$$ \mu_{n,T}^{(2)} = \mathbb{E} \left\| \sqrt{n} \tilde{\alpha}_{\nu_n}(\varphi^{(T)} K) \right\|_{\mathcal{G}_{\varphi K, j}^{(\ell,T)}} \le C' \kappa_2 \sqrt{n\phi(h_{K,j})\eta_n^{2/p}}, $$
which means that,

$$ \mu_{n,T}^{(2)} = O\left(\sqrt{n\phi(h_{K,j})\eta_n^{2/p}}\right). $$

Now, we can apply Bernstein's inequality for empirical processes with:

$$ z = \eta' \sqrt{n\phi(h_{K,j})\psi_{S_X}(\epsilon_n)}, $$

in Lemma B.3, we obtain, for $\mathcal{C}' > 0$,

$$ \begin{aligned} & \mathbb{P}\left\{\max_{1 \le k \le n} \|\tilde{\alpha}_{v_k}(\varphi^{(T)}K)\|_{\mathcal{G}_{\varphi K,j}^{(l,T)}} \ge \eta' \sqrt{n\phi(h_{K,j})\psi_{S_X}(\epsilon_n)}\right\} \\ & \le \mathbb{P}\left\{\max_{1 \le k \le n} \left\|\sqrt{k}\tilde{\alpha}_{v_k}(\varphi^{(T)}K)\right\|_{\mathcal{G}_{\varphi K,j}^{(l,T)}} \ge \mu_{n,T}^{(2)} + z\right\} \\ & \le \exp\left(-\eta'^2 \frac{n\phi(h_{K,j})\psi_{S_X}(\epsilon_n)}{2n\phi(h_{K,j})\eta_n^{2/p} + \mathcal{C}'\sqrt{n\phi(h_{K,j})\psi_{S_X}(\epsilon_n)}}\right) \\ & \le n^{-\eta_0^2 C'}, \end{aligned} \quad (A.27) $$

where $C' > 0$. From Eq (2.10), and fact that $L(n) \sim \frac{\log(b_0/h_{n,1})}{\log(2)}$ and by choosing

$$ \epsilon_n = \frac{\log n}{n}, $$

we get

$$ \sum_{n \ge 1} \mathbb{P} \left\{ G_{2,T} > \eta \sqrt{\frac{\psi_{S_X} \left( \frac{\log n}{n} \right)}{n\phi(h_{n,1})}} \right\} \le \sum_{n \ge 1} C L(n) N_{\epsilon_n}(S_X) n^{-C' \eta_0} < \infty. $$

Next, we prove that the result is valid for the term $G_{1,T}$. By conditions (C.4.2) and (C.3.1), for some constant $\mathcal{C}' > 0$, we have

$$ \begin{aligned} & \left| \hat{r}_{n,2}^{(1)}(\varphi, t, h_K(t)) - \hat{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t_{\ell})) \right| \\ &= \left| \frac{1}{n\phi_t(h_K)} \sum_{i=1}^n \varphi^{(T)}(Y_i) K\left(\frac{d(X_i, t)}{h_K}\right) - \frac{1}{n\phi_{t_{\ell(t)}}(h_K)} \sum_{i=1}^n \varphi^{(T)}(Y_i) K\left(\frac{d(X_i, t_{\ell(t)})}{h_K}\right) \right| \\ &\le \frac{\mathcal{C}'}{n\phi(h_K)} \left| \sum_{i=1}^n F^{(T)}(Y_i) K\left(\frac{d(X_i, t)}{h_K}\right) - \sum_{i=1}^n F^{(T)}(Y_i) K\left(\frac{d(X_i, t_{\ell(t)})}{h_K}\right) \right|, \end{aligned} $$

where

$$ F^{(T)}(y) := F(y) \mathbb{1}_{\{F(y) \le \lambda \eta_n^{1/p}\}}. $$
Observe that

$$
\begin{align*}
& \sup_{\varphi^{(T)} K \in \mathcal{F} \mathcal{H} \mathcal{I}_{\lambda, \eta_n}} \sup_{h_{n,1} \le h_K \le b_0} \sup_{t \in S_X} \left| \hat{r}_{n,2}^{(1)}(\varphi, t, h_K(t)) - \hat{r}_{n,2}^{(1)}(\varphi, t_{\ell(t)}, h_K(t_{\ell})) \right| \\
& \le \sup_{\varphi^{(T)} K \in \mathcal{F} \mathcal{H} \mathcal{I}_{\lambda, \eta_n}} \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \sup_{n} \frac{1}{n\phi(h_K)} \sum_{i=1}^{n} \left| F^{(T)}(Y_i) \right| \left| K\left(\frac{d(X_i, t)}{h_K}\right) - K\left(\frac{d(X_i, t_{\ell(t)})}{h_K}\right) \right| \mathbb{1}_{B(t, h_K) \cup B(t_{\ell(t)}, h_K)}(X_i) \\
& := |\mathcal{G}_{1,T}|.
\end{align*}
$$

Furthermore, following the same steps as the proof of $G_1$ in the previous lemma’s proof, we have:

$$
\begin{align*}
|\mathcal{G}_{1,T}| & \le \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n} \sum_{i=1}^n \frac{1}{h_K \phi(h_K)} |F^{(T)}(Y_i)| d(t, t_{\ell(t)}) \mathbb{1}_{B(t, h_K) \cup B(t_{\ell(t)}, h_K)}(X_i) \\
& \le \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n} \sum_{i=1}^n \frac{\varepsilon_n}{h_K \phi(h_K)} |F^{(T)}(Y_i)| \mathbb{1}_{B(t, h_K) \cup B(t_{\ell(t)}, h_K)}(X_i) \\
& = \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n} \sum_{i=1}^n W_{i,T},
\end{align*}
$$

where, for $1 \le i \le n$,

$$
W_{i,T} = \frac{\varepsilon_n}{h_K \phi(h_K)} |F^{(T)}(Y_i)| \mathbb{1}_{B(t,h_K) \cup B(t_{\ell(t)},h_K)}(X_i).
$$

We get uniformly on $t \in S_X$ and on $h_K$:

$$
\sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} W_{1,T} \le \sup_{t \in S_X} \frac{\varepsilon_n}{h_{n,1} \phi(h_{n,1})} |F^{(T)}(Y_1)|.
$$

So, for $2 \le q \le p$, we obtain

$$
\begin{align}
\mathbb{E} (W_{1,T})^q &\le \left( \frac{\varepsilon_n}{h_{n,1} \phi(h_{n,1})} \right)^q \mathbb{E} (|F^{(T)}(Y_1)|^q) \nonumber \\
&\le \sup_{t \in S_X} \left( \frac{\varepsilon_n}{h_{n,1} \phi(h_{n,1})} \right)^q \mathbb{E} (|F^{(T)}(Y_1)|^q |X=t) \tag{A.28}
\end{align}
$$

$$
\le \left( \frac{\varepsilon_n}{h_{n,1} \phi(h_{n,1})} \right)^q \theta_p^{q/p} \varepsilon_n^q \left( \frac{1}{\phi(h_{n,1})} \right)^{q-1}. \quad (A.29)
$$

The transition from (A.28) to (A.29) is done by using Jensen's inequality used for the concave function $z^a$, for $0 < a \le 1$. Moreover, from condition (C.7), we deduce that the quantity $\varepsilon_n/h_{n,1}\phi(h_{n,1})$ is bounded, then for $q \ge 2$

$$
\sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \mathbb{E} (W_{1,T})^q \le \mathcal{C} \theta_p^{q/p} \left( \frac{1}{\sqrt{\phi(h_{n,1})}} \right)^{2(q-1)},
$$

where $\mathcal{C} > 0$. Hence, by applying a standard inequality (see Corollary A.8 [91]) that the conditions are satisfied here, uniformly on $t \in S_X$, we get

$$
G_{1,T} = O_{a,co} \left( \sqrt{\frac{\psi_{S_X} \left( \frac{\log n}{n} \right)}{n \phi(h_{n,1})}} \right).
$$
Finally, all that is left is to evaluate the term $G_{3,T}$. We have

$$
\begin{aligned}
& \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)} (\varphi^{(T)}, t_{\ell(t)}, h_K(t_\ell)) \right) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)} (\varphi^{(T)}, t, h_K(t)) \right) \right| \\
&= \left| \mathbb{E} \left[ \hat{r}_{n,2}^{(1)} (\varphi^{(T)}, t_{\ell(t)}, h_K(t_\ell)) - \hat{r}_{n,2}^{(1)} (\varphi^{(T)}, t, h_K(t)) \right] \right| \\
&\le \frac{1}{n\phi(h_K)} \left| \sum_{i=1}^{n} \mathbb{E} \left( F^{(T)}(Y_i) K \left( \frac{d(X_i, t)}{h_K} \right) - F^{(T)}(Y_i) K \left( \frac{d(X_i, t_{\ell(t)})}{h_K} \right) \right) \right|,
\end{aligned}
$$

which means that

$$
\begin{aligned}
& \sup_{\varphi^{(T)} K \in \mathcal{F} \mathcal{H} \mathcal{I}_{\lambda\eta_n}} \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)} (\varphi^{(T)}, t_{\ell(t)}, h_K(t_\ell)) \right) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)} (\varphi^{(T)}, t, h_K(t)) \right) \right| \\
&\le \sup_{\varphi^{(T)} K \in \mathcal{F} \mathcal{H} \mathcal{I}_{\lambda\eta_n}} \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n\phi(h_K)} \sum_{i=1}^{n} \mathbb{E} \left( \left| F^{(T)}(Y_i) \right| \left| K \left( \frac{d(X_i, t)}{h_K} \right) - K \left( \frac{d(X_i, t_{\ell(t)})}{h_K} \right) \right| 1_{B(t,h_K) \cup B(t_{\ell(t)},h_K)}(X_i) \right) \\
&=: \left| \mathcal{G}_{3,T}(\varphi K, t; h_K) \right|.
\end{aligned}
$$

Let's take a look at $|\mathcal{G}_{1,T}(\varphi K, t; h_K)|$ and following the same steps, by the fact that $K(\cdot)$ is a Lipschitz function, we get

$$
\begin{aligned}
|\mathcal{G}_{3,T}(\varphi K, t; h_K)| & \le \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n} \sum_{i=1}^n \frac{1}{h_K \phi(h_K)} \mathbb{E} \left( |F^{(T)}(Y_i)| d(t, t_{\ell(t)}) 1_{B(t,h_K) \cup B(t_{\ell(t)},h_K)}(X_i) \right) \\
& \le \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n} \sum_{i=1}^n \frac{\varepsilon_n}{h_K \phi(h_K)} \mathbb{E} \left( |F^{(T)}(Y_i)| 1_{B(t,h_K) \cup B(t_{\ell(t)},h_K)}(X_i) \right) \\
& \le \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n} \sum_{i=1}^n \frac{\varepsilon_n}{h_K \phi(h_K)} \mathbb{E} \left( |F^{(T)}(Y_i)| \right) \\
& \le \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n} \sum_{i=1}^n \frac{\varepsilon_n}{h_K \phi(h_K)} \mathbb{E} \left( \mathbb{E} (F^{(T)}(Y_i) | X = t) \right) \\
& = \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \frac{1}{n} \sum_{i=1}^n W_{i,T}^{(3)},
\end{aligned}
\quad (A.30)
$$

where, for $1 \le i \le n$,

$$
W_{i,T}^{(3)} = \frac{\varepsilon_n}{h_K \phi(h_K)} \mathbb{E} \left( |F^{(T)}(Y_i)| \right).
$$

We get for $q \ge 2$ and uniformly on $t \in S_X$ and on $h_K$:

$$
\sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \mathbb{E} (Z_{1,T}^{(31)})^q \le \sup_{t \in S_X} \left( \frac{\varepsilon_n}{h_{n,1} \phi(h_{n,1})} \right)^q \mathbb{E} \left( \mathbb{E} \left( |F^{(T)}(Y_1)|^q |X=t\right) \right) \quad (A.31)
$$

$$
\le \left( \frac{\varepsilon_n}{h_{n,1} \phi(h_{n,1})} \right)^q \theta_p^{q/p} \varepsilon_n^q \left( \frac{1}{h_{n,1}^q \phi(h_{n,1})} \right)^{q-1}. \quad (A.32)
$$
The transition from (A.31) to (A.32) is done by using Minkowski's and Jensen's inequalities. Then, following the same way as in treating $|\mathcal{G}_{1,T}(\varphi K, t; h_K)|$, we get

$$ \sup_{\varphi^{(T)} K \in \mathcal{F} \mathcal{H} \mathcal{J}_{\eta\eta}} \sup_{t \in S_X} \sup_{h_{n,1} \le h_K \le b_0} \left| \mathcal{G}_{3,T}(\varphi K, t; h_K) \right| = O_{a,co} \left( \sqrt{\frac{\psi_{S_X} \left( \frac{\log n}{n} \right)}{n\phi(h_{n,1})}} \right). $$

Therefore, the proof is done for the truncated part. Regarding the remainder term, the idea consists of proving its asymptotic negligibility, that is

$$ \sup_{h_{n,1} \le h_K \le b_n} \sup_{t \in S_X} \sup_{\varphi K \in \mathcal{F} \mathcal{H}} \left| \widehat{r}_{n,2}^{(1)}(\varphi^{(R)}, t, h_K(t)) - \mathbb{E}(\widehat{r}_{n,2}^{(1)}(\varphi^{(R)}, t, h_K(t))) \right| = o_{a,co}(1), $$

which could be derived directly from the proof of the remainder part of the U-statistics developed in the sequel.

### Proof of Lemma 8.11

We recall that

$$ \begin{aligned} I_2(t, u) &= \widehat{r}_{n,2}^{(1)}(\varphi_M, t, u) - \mathbb{E}(\widehat{r}_{n,2}^{(1)}(\varphi_M, t, u)) - r^{(1)}(\varphi, t)(\widehat{r}_{n,1}^{(1)}(1, t, u) - 1) \\ &= [\widehat{r}_{n,2}^{(1)}(\varphi_M, t, u) - r^{(1)}(\varphi, t)\widehat{r}_{n,1}^{(1)}(1, t, u)] - \mathbb{E}(\widehat{r}_{n,2}^{(1)}(\varphi_M, t, u) - r^{(1)}(\varphi, t)\widehat{r}_{n,1}^{(1)}(1, t, u)). \end{aligned} $$

Therefore, the proof of this lemma is based on the following decomposition:

$$ \begin{aligned} & \sqrt{k} I_2(\varphi, t, H_{n,k}(t)) \\ &= \sqrt{k} (\widehat{r}_{n,2}^{(1)}(\varphi_M, t, H_{n,k}(t)) - r^{(1)}(\varphi, t) \widehat{r}_{n,1}^{(1)}(1, t, H_{n,k}(t)) - \mathbb{E}(\widehat{r}_{n,2}^{(1)}(\varphi_M, t, H_{n,k}(t)) - r^{(1)}(\varphi, t) \widehat{r}_{n,1}^{(1)}(1, t, H_{n,k}(t))))) \\ &= \sqrt{k} [\widehat{r}_{n,2}^{(1)}(\varphi_M, t, H_{n,k}(t)) - \widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+)] \\ &\quad + \sqrt{k} [\widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) - r^{(1)}(\varphi, t) \widehat{r}_{n,1}^{(1)}(1, t, D_n^+) - \mathbb{E}(\widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) - r^{(1)}(\varphi, t) \widehat{r}_{n,1}^{(1)}(1, t, D_n^+))] \\ &\quad + \sqrt{k} [r^{(1)}(\varphi, t) (\widehat{r}_{n,1}^{(1)}(1, t, D_n^+) - \mathbb{E}(\widehat{r}_{n,1}^{(1)}(1, t, D_n^+)))] \\ &\quad + \sqrt{k} [r^{(1)}(\varphi, t) (\mathbb{E}(\widehat{r}_{n,1}^{(1)}(1, t, H_{n,k}(t)) - \widehat{r}_{n,1}^{(1)}(1, t, H_{n,k}(t)))] \\ &\quad + \sqrt{k} [\mathbb{E}(\widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) - \mathbb{E}(\widehat{r}_{n,2}^{(1)}(\varphi_M, t, H_{n,k}(t)))] \\ &= J_1(x) + J_2(x) + J_3(x) + J_4(x) + J_5(x). \end{aligned} \quad (A.33) $$

By the same arguments as those involved in the proof of $I_1(\varphi, t, H_{n,k}(t))$. We get

$$ \begin{aligned} |J_1(x)| &= \sqrt{k} |\widehat{r}_{n,2}^{(1)}(\varphi_M, t, H_{n,k}(t)) - \widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+)| \\ &\le \sqrt{k} |\widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) - \widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+)| \\ &\le \sqrt{k} |\widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) - \mathbb{E}(\widehat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-))| \end{aligned} $$
$$ + \sqrt{k} \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right) - \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) \right| + \sqrt{k} \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) \right) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right) \right|, \quad (A.34) $$

and

$$ \begin{aligned} |J_5(x)| &= \sqrt{k} \left( \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, H_{n,k}(t)) \right) \right) \\ &\le \sqrt{k} \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right) - \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right| \\ &\quad + \sqrt{k} \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, H_{n,k}(t)) \right) \right| \\ &\le 2 \sqrt{k} \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right) - \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right|. \end{aligned} \quad (A.35) $$

By combining (A.34) and (A.35) and by similar arguments used to treat the $I_{11}(\varphi, t, H_{n,k}(t))$ and $I_{12}(\varphi, t, H_{n,k}(t))$, we get

$$ \begin{aligned} |J_1(x) + J_5(x)| &\le \sqrt{k} \left| \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) \right) \right| \\ &\quad + 3 \sqrt{k} \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right) - \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right| \\ &\quad + \sqrt{k} \left| \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^-) \right) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) \right) \right|. \end{aligned} $$

We then obtain that $|J_1(x) + J_5(x)|$ tending to zero as $n$ tends to infinity. We evaluate the term $J_2(x)$ on the right side of (A.33). Let us introduce the following sum

$$ J_2(x) = \sum_{i=1}^{n} \mathbb{Z}_{ni}, $$

where

$$ \mathbb{Z}_{ni} = \frac{\sqrt{k}}{\sqrt{n} \mathbb{E}(\Delta_1(t, D_n^+))} \left( (\varphi_M(Y_i) - r^{(1)}(\varphi_M, t)) \Delta_i(t, D_n^+) - \mathbb{E}((\varphi_M(Y_i) - r^{(1)}(\varphi_M, t)) \Delta_i(t, D_n^+)) \right), $$

and

$$ J_2(x) = \sqrt{k} \left[ \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) - r^{(1)}(\varphi, t) \hat{r}_{n,1}^{(1)}(1, t, D_n^+) - \mathbb{E} \left( \hat{r}_{n,2}^{(1)}(\varphi_M, t, D_n^+) - r^{(1)}(\varphi, t) \hat{r}_{n,1}^{(1)}(1, t, D_n^+) \right) \right]. $$

Thus, the claimed result now is

$$ J_2(x) \to \mathcal{N}(0, \sigma^2(t)). $$

The asymptotic normality of $J_2(x)$ was proved in Lemma 8.8 by choosing the bandwidth parameter as $u = D_n^+$. For $J_3(x)$ and $J_4(x)$, we obtain by using Lemma 8.2 and the fact that $\mathbb{E}(\hat{r}_{n,1}^{(1)}(1, t, u)) = 1$ with $u = D_n^-$ or $D_n^+$.

$$ \begin{aligned} |J_3(x) + J_4(x)| = \sqrt{k} \left( \left| r^{(1)}(\varphi, t) \left[ \left( \hat{r}_{n,1}^{(1)}(1, t, D_n^+) - \mathbb{E} \left( \hat{r}_{n,1}^{(1)}(1, t, D_n^+) \right) \right) \right] \right| \right. \\ \left. + \sqrt{k} \left( \left| r^{(1)}(\varphi, t) \left[ \mathbb{E} \left( \hat{r}_{n,1}^{(1)}(1, t, H_{n,k}(t)) \right) - \hat{r}_{n,1}^{(1)}(1, t, H_{n,k}(t)) \right] \right| \right) \end{aligned} $$
$$ \begin{aligned} & \le \sqrt{k} |r^{(1)}(\varphi, t)(\tilde{r}_{n,1}^{(1)}(1, t, D_n^+) - 1)| + \sqrt{k} |r^{(1)}(\varphi, t)(1 - \tilde{r}_{n,1}^{(1)}(1, t, D_n^+))| \\ & \le 2\sqrt{k} |r^{(1)}(\varphi, t)(\tilde{r}_{n,1}^{(1)}(1, t, D_n^+) - 1)|. \end{aligned} $$

Using Lemma 8.5, we get

$$ \sqrt{k} |\tilde{r}_{n,1}^{(1)}(1, t, D_n^+) - 1| \to 0 \text{ as } n \to \infty. $$

Consequently, we have

$$ |J_3(x) + J_4(x)| \to 0. $$

Hence, the proof is complete.

### Proof of Lemma 8.12:

For the proof of this Lemma, it suffices to use the result of [58]) in inequality (A.9), Tchebychev's inequality and Lemma 8.2. For $\varepsilon > 0$, we readily infer

$$ \begin{aligned} & \mathbb{P}(|\tilde{r}_{n,1}^{(1)}(1, t, H_{n,k}(t)) - 1| > \delta) \\ & \le \mathbb{P}(|\tilde{r}_{n,1}^{(1)}(1, t, D_n^-) - \mathbb{E}(\tilde{r}_{n,1}^{(1)}(1, t, D_n^-))| > \varepsilon) \le \frac{\mathrm{Var}(\tilde{r}_{n,1}^{(1)}(1, t, D_n^-))}{\varepsilon^2}. \end{aligned} $$

Making use of the fact that

$$ \mathrm{Var}(\tilde{r}_{n,1}^{(1)}(1, t, D_n^-)) = O\left(\frac{1}{n\phi(x, D_n^-)}\right). $$

We finally obtain

$$ \tilde{r}_{n,1}^{(1)}(1, t, H_{n,k}(t)) - 1 \to 0, \text{ in probability as } n \to \infty $$

Hence, the proof is complete.

### Proof of Lemma 8.13:

We have

$$ \begin{aligned} & \tilde{r}_{n,2}^{(1)}(\varphi, t, h_K(t)) - \mathbb{E}(\tilde{r}_{n,2}^{(1)}(\varphi, t, h_K(t))) \\ &= \frac{1}{n\mathbb{E}[\Delta_1(t, h_K(t))]} \sum_{i=1}^{n} \{\varphi(Y_i)\Delta_i(t; h_K(t)) - \mathbb{E}[\varphi(Y_i)\Delta_i(t; h_K(t))]\} \\ &= \frac{1}{n\mathbb{E}[\Delta_1(t, h_K(t))]} \sum_{i=1}^{n} Z_i(t, h_K(t)), \end{aligned} $$

where

$$ Z_i(t, h_K(t)) = \varphi(Y_i)\Delta_i(t; h_K(t)) - \mathbb{E}[\varphi(Y_i)\Delta_i(t; h_K(t))]. $$

Taking into account condition (C.4.2) and using Hölder's inequality, for $2 < q < p$ such that $\frac{1}{p} + \frac{1}{q} = 1$, we can write for all $i \ne j$:

$$ \mathbb{E}[\varphi(Y_i)\Delta_i(t; h_K(t))\varphi(Y_j)\Delta_j(t; h_K(t))] $$
$$
\begin{align*}
&= \mathbb{E} \left[ \Delta_i(t; h_K(t)) \Delta_j(t; h_K(t)) \right] \mathbb{E} \left[ \varphi(Y_i) \varphi(Y_j) \mid (X_i, X_j) \right] \\
&\le \mathbb{E} \left[ \Delta_i(t; h_K(t)) \Delta_j(t; h_K(t)) \right] \mathbb{E}^{1/p} [F^p(Y) \mid X] \mathbb{E}^{1/q} [F^q(Y) \mid X] \\
&\le C \mathbb{E} \left[ \Delta_i(t; h_K(t)) \Delta_j(t; h_K(t)) \right] \\
&\le C \sup_{i \ne j} \mathbb{P} \left\{ (X_i, X_j) \in B(t, h_K(t)) \times B(t, h_K(t)) \right\} \\
&\le C \Psi(h_K(t)) f_2(t), \tag{A.36}
\end{align*}
$$

the later inequality is due to condition (C.1.2). Next, we have

$$
\begin{align*}
& \mathrm{Var} \left( \frac{1}{n \mathbb{E}[\Delta_1(t, h_K(t))]} \sum_{i=1}^{n} Z_i(t, h_K(t)) \right) \\
&= \frac{1}{n^2 \mathbb{E}^2(\Delta_1(t, h_K(t)))} \sum_{i=1}^{n} \sum_{j=1}^{n} \mathrm{Cov} (\varphi(Y_i) \Delta_i(t; h_K(t)), \varphi(Y_j) \Delta_j(t; h_K(t))) \\
&= \frac{1}{n \mathbb{E}^2(\Delta_1(t, h_K(t)))} \mathrm{Var} (\varphi(Y_1) \Delta_1(t; h_K(t))) \\
&\quad + \frac{1}{n^2 \mathbb{E}^2(\Delta_1(t, h_K(t)))} \sum_{i=1}^{n} \sum_{j=1}^{n} \mathrm{Cov} (\varphi(Y_i) \Delta_i(t; h_K(t)), \varphi(Y_j) \Delta_j(t; h_K(t))) \\
&=: V_1 + V_2.
\end{align*}
$$

Let us start with the term $V_1$, considering the conditions (C.1.1) and (C.3.1), it follows that

$$
\kappa_1^j C_1 f_1(t) \phi(h_K(t)) \le \mathbb{E} (\Delta_1^j(t, h_K(t))) \le \kappa_2^j C_2 f_1(t) \phi(h_K(t)), \quad (A.37)
$$

and we have

$$
\begin{align*}
\mathrm{Var}(\varphi(Y_1)\Delta_1(t; h_K(t))) &= \mathbb{E}[\varphi^2(Y_1)\Delta_1^2(t, h_K(t))] - \mathbb{E}^2[\varphi(Y_1)\Delta_1(t, h_K(t))] \\
&\le \mathbb{E}[\varphi^2(Y_1)\Delta_1^2(t, h_K(t))] \\
&\le \mathbb{E}[\mathbb{E}[F^2(Y) \mid X] \Delta_1^2(t, h_K(t))] \\
&\le \mathbb{E}^{2/p} [F^p(Y) \mid X] \mathbb{E}^{2/q} [\Delta_1^q(t, h_K(t))] \\
&\le C^{2/p} C_2^{2/q} \kappa_2^2 \phi^2(h_K(t)) f_1^2(t).
\end{align*}
$$

Hence, combining the later inequality with (A.37) gives us:

$$
\frac{\text{const1}}{nf_1(t)\phi(h_K(t))} \le V_1 \le \frac{\text{const2}}{nf_1(t)\phi(h_K(t))}, \quad (A.38)
$$

whenever $f_1(t) > 0$ and const1 < const2. Next, we consider $V_2$. We have

$$
V_2 = \frac{1}{n^2 \mathbb{E}^2 (\Delta_1(t, h_K(t)))} \left\{ \sum_{i=1}^{n} \sum_{j=1}^{n} \mathrm{Cov} (\varphi(Y_i) \Delta_i(t; h_K(t)), \varphi(Y_j) \Delta_j(t; h_K(t))) \right\}
$$
Once more by using (A.37) and by a simple calculation (reduction of the double sum), we get:

$$V_{2,2} \le \frac{\text{const}}{n\omega_n^\delta f_1^{2(1-1/p)}(t) \{\phi(h_K)\}^{2(1-1/p)}} \sum_{k=\omega_n+1}^{\infty} [k^\delta \beta(k)]^{1-2/p}.$$

This latter, with the boundary on $V_1$ implies:

$$\frac{V_{2,2}}{V_1} \le \frac{\text{const}}{\omega_n^\delta (\log(w_n))^{\delta(1-1/p)} f_1^{1-2/p}(t) \{\phi(h_K)\}^{1-2/p}} \sum_{k=\omega_n+1}^{\infty} k^\delta (\log(k))^{\delta(1-1/p)} (\beta(k))^{1-2/p}. \quad (A.40)$$

Choosing $\omega_n = \{\phi(h_K)\}^{-(1-2/p)/\delta}$ and making use of condition (C.5.1), we get that:

$$\frac{V_{2,2}}{V_1} \le \frac{\text{const}}{\left[-\frac{1}{\delta}\left(1-\frac{2}{p}\right)\log(\phi(h_K))\right]^{\delta(1-1/p)} f_1^{1-2/p}(t)} \sum_{k=\omega_n+1}^{\infty} k^\delta (\log(k))^{\delta(1-1/p)} (\beta(k))^{1-2/p} \\ \xrightarrow{n \to \infty} 0. \qquad (A.41)$$

By the same value chosen for $\omega_n$:

$$\frac{V_{2,1}}{V_1} \le \frac{\text{const} f_2(t) \Psi(h_K)}{f_1(t) \{\phi(h_K)\}^2} \omega_n \phi(h_K) + \{\phi(h_K)\}^{1-(1-2/p)/\delta},$$

by the fact that $\Psi(h_K)/\phi^2(h_K)$ is assumed to be bounded and $\phi(h_K)\omega_n \to 0$, then the first term of the last inequality tends to 0 as $n \to \infty$. Furthermore, the second term also tends to 0, since $1 - (1 - 2/p)/\delta < 1$. Hence, the proof is complete.

### Proof of Lemma 8.14:

Based on (4.8), we have

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

Remark that the first term of the last sum is an empirical process indexed by $m\mathcal{G}^{(1)}$, so from Section 4.1, we have

$$\sqrt{n\tilde{\phi}(\mathbf{h}(\mathbf{t}))}mu_n^{(1)}(\pi_{1,m}G_{\varphi,\mathbf{t},\mathbf{h}}) \xrightarrow{d} \mathcal{N}(0, m^2\sigma_t^2(\varphi, \varphi)). \quad (A.42)$$

So, it is enough to show, for $2 \le p \le m$, that

$$\sqrt{n\tilde{\phi}(\mathbf{h}(\mathbf{t}))}u_n^{(p)}(\pi_{p,m}G_{\varphi,\mathbf{t},\mathbf{h}}) \xrightarrow{\mathbb{P}} 0. \qquad (A.43)$$

To avoid ambiguity, we consider the case where $m = 2$ (the other cases are treated in the same way). Then, we have

$$u_n^{(2)}(\pi_{2,m} G_{\varphi, \mathbf{t}, \mathbf{h}}) = \frac{1}{n(n-1)} \sum_{1 \le i < j \le n} \pi_{2,m} G_{\varphi, \mathbf{t}, \mathbf{h}}((X_i, Y_i), (X_j, Y_j)).$$
The projections being $\mathbb{P}$-canonical, then to show (A.43) it suffices to show that

$$ \mathbb{E}[u_n^{(2)}(\pi_{2,m} G_{\varphi,t})]^2 \xrightarrow{\mathbb{P}} 0. $$

We have

$$ \mathbb{E}[u_n^{(2)}(\pi_{2,m} G_{\varphi,t})]^2 = \frac{1}{n^2(n-1)^2} \sum_{1 \le i_1 < i_2 \le n} \sum_{1 \le j_1 < j_2 \le n} \mathbb{E}((i_1, i_2); (j_1, j_2)), $$

where

$$ \mathbb{E}((i_1, i_2); (j_1, j_2)) = \mathbb{E}[\pi_{2,m} G_{\varphi,t}((X_{i_1}, Y_{i_1}), (X_{i_2}, Y_{i_2})) \times \pi_{2,m} G_{\varphi,t}((X_{j_1}, Y_{j_1}), (X_{j_2}, Y_{j_2}))]. $$

We first remark that

$$ \begin{aligned} & \mathbb{E}[n^2(n-1)^2 u_n^{(2)}(\pi_{2,m} G_{\varphi,t})^2] \\ &= \sum_{1 \le i_1 < i_2 \le n} \sum_{1 \le j_1 < j_2 \le n} \mathbb{E}((i_1, i_2), (j_1, j_2)) \\ &\le \left| \sum_{1 \le i_1 < i_2 < j_1 < j_2 \le n} \mathbb{E}((i_1, i_2), (j_1, j_2)) \right| + \left| \sum_{1 \le i_1 < j_1 < i_2 < j_2 \le n} \mathbb{E}((i_1, i_2), (j_1, j_2)) \right| \\ &\quad + \left| \sum_{1 \le i_1 < j < j_2 < i_2 \le n} \mathbb{E}((i_1, i_2), (j_1, j_2)) \right| + \left| \sum_{1 \le i_1 < i_2 < j_2 \le n} \mathbb{E}((i_1, i_2), (i_2, j_2)) \right| \\ &\quad + \left| \sum_{i_1, i_2=1}^n \mathbb{E}((i_1, i_2), (i_1, i_2)) \right|. \end{aligned} $$

Following similar reasoning to obtain (A.41), the following bound suffices to our need

$$ \begin{aligned} & \left| \sum_{1 \le i_1 < i_2 < j_1 < j_2 \le n} \mathbb{E}((i_1, i_2), (j_1, j_2)) \right| \\ &\le \left| \sum_{\substack{1 \le i_1 < i_2 < j_1 < j_2 \le n \\ i_2 - i_1 \ge j_2 - j_1}} \mathbb{E}((i_1, i_2), (j_1, j_2)) \right| + \left| \sum_{\substack{1 \le i_1 < i_2 < j_1 < j_2 \le n \\ j_2 - j_1 > i_2 - i_1}} \mathbb{E}((i_1, i_2), (j_1, j_2)) \right| \\ &\le \text{const} \frac{n^2}{\tilde{\phi}(\mathbf{h})^4} \sum_{s=1}^{\infty} s^{\delta} (\log(s))^{\delta(1-1/p)} (\beta(s))^{1-2/p}. \end{aligned} $$

Treating the other terms as in Lemma 2 of [203] (refer for detailed proof in the conditional setting to [13, Lemma 3]) we get

$$ \mathbb{E}[u_n^{(2)}(\pi_{2,m} G_{\varphi,t})]^2 = O(n\tilde{\phi}(\mathbf{h}))^{-2} \to 0. $$

Hence, the proof of (8.88) is complete. The statement (8.89) is a direct consequence of (8.88) in connection with condition (C.8).
### Proof of Lemma 8.15

For any two constants the sum of the product of these constants by the components of the vector (8.90) is a centered U-statistic, i.e., for $C^{(1)}, C^{(2)} \in \mathbb{R}$,

$$C^{(1)}[u_n(\varphi_1, \mathbf{t}, \mathbf{h}) - \mathbb{E}(u_n(\varphi_1, \mathbf{t}, \mathbf{h}))] + C^{(2)}[u_n(\varphi_2, \mathbf{t}, \mathbf{h}) - \mathbb{E}(u_n(\varphi_2, \mathbf{t}, \mathbf{h}))] = u_n(\varphi', \mathbf{t}, \mathbf{h}) - \mathbb{E}(u_n(\varphi', \mathbf{t}, \mathbf{h})),$$

where

$$\varphi' = C^{(1)}\varphi_1 + C^{(2)}\varphi_2.$$

So, by the Cramér-Wold theorem the proof of the present lemma is directly deduced from Lemma 8.14. What remains to be done is to complete the proof of the theorem in question based on the two previous lemmas. Indeed,

$$\hat{\tau}_n^{(m)}(\varphi, \mathbf{t}; \mathbf{h}) - \mathbb{E}(u_n(\varphi, \mathbf{t}, \mathbf{h})) = \frac{u_n(\varphi, \mathbf{t}, \mathbf{h}) - u_n(1, \mathbf{t}, \mathbf{h})\mathbb{E}(u_n(\varphi, \mathbf{t}, \mathbf{h}))}{u_n(1, \mathbf{t}, \mathbf{h})},$$

and because

$$u_n(\varphi, \mathbf{t}, \mathbf{h}) \xrightarrow{\mathbb{P}} 1,$$

then all we have to do is prove that

$$\sqrt{n\tilde{\phi}(\mathbf{h})} \{u_n(\varphi, \mathbf{t}, \mathbf{h}) - u_n(1, \mathbf{t}, \mathbf{h})\mathbb{E}(u_n(\varphi, \mathbf{t}, \mathbf{h}))\} \xrightarrow{d} \mathcal{N}(0, \rho^2). \quad (A.44)$$

We have

$$\sqrt{n\tilde{\phi}(\mathbf{h})} \{u_n(\varphi, \mathbf{t}, \mathbf{h}) - u_n(1, \mathbf{t}, \mathbf{h})\mathbb{E}(u_n(\varphi, \mathbf{t}, \mathbf{h}))\} = (1, \mathbb{E}(u_n(\varphi, \mathbf{t}, \mathbf{h}))) \begin{pmatrix} \sqrt{n\tilde{\phi}(\mathbf{h})} \{u_n(\varphi, \mathbf{t}, \mathbf{h}) - \mathbb{E}(u_n(\varphi, \mathbf{t}, \mathbf{h}))\} \\ \sqrt{n\tilde{\phi}(\mathbf{h})} \{u_n(1, \mathbf{t}, \mathbf{h}) - 1\} \end{pmatrix},$$

the latter, the fact that $r^{(m)}(\varphi, \mathbf{t})$ is continuous on $\mathbf{t}$ and $\mathbb{E}(u_n(\varphi, \mathbf{t}, \mathbf{h})) \to r^{(m)}(\varphi, \mathbf{t})$ leads to the desired result.

### Proof of Lemma 8.16:

For the clarity of the exposition, we present the proof for $m = 2$; this case already contains the main idea. As in the proof of Theorem 4.2, we divide the sequence $\{(\mathbf{X}_i, \mathbf{Y}_i)\}$ into $\nu_n$ alternate blocks, here the sizes $a_n, b_n$ are different satisfying

$$b_n \ll a_n, \quad (\nu_n - 1)(a_n + b_n) < n \le \nu_n(a_n + b_n), \quad (A.45)$$

and set, for $1 \le j \le \nu_n - 1$:

$$H_j^{(U)} = \{i : (j-1)(a_n + b_n) + 1 \le i \le (j-1)(a_n + b_n) + a_n\},$$

$$T_j^{(U)} = \{i : (j-1)(a_n + b_n) + a_n + 1 \le i \le (j-1)(a_n + b_n) + a_n + b_n\},$$

$$H_{\nu_n}^{(U)} = \{i : (\nu_n - 1)(a_n + b_n) + 1 \le i \le n \wedge (\nu_n - 1)(a_n + b_n) + a_n\},$$
$$T_{\nu_n}^{(U)} = \{i : (\nu_n - 1)(a_n + b_n) + a_n + 1 \le i \le n\}.$$

Note that the notation $b_n$ used here and in the proof of Theorem 4.7 denotes the size of the alternative blocks. However, in the proof of Theorem 4.2, it denotes the radius of the nets of the class of functions. Then, we have

$$ \begin{aligned} & \sum_{i \neq j}^{n} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \\ &= \sum_{p \neq q}^{\nu_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \\ &\quad + \sum_{p=1}^{\nu_n} \sum_{i \neq j, i \in H_p^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \\ &\quad + 2 \sum_{p=1}^{\nu_n} \sum_{i \in H_p^{(U)}} \sum_{q: |q-p| \ge 2} \sum_{j \in T_q^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \\ &\quad + 2 \sum_{p=1}^{\nu_n} \sum_{i \in H_p^{(U)}} \sum_{q: |q-p| \le 1} \sum_{j \in T_q^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \\ &\quad + \sum_{p \neq q}^{\nu_n} \sum_{i \in T_p^{(U)}} \sum_{j \in T_q^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \\ &\quad + \sum_{p=1}^{\nu_n} \sum_{i \neq j, i \in T_p^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \\ &=: \text{I} + \text{II} + \text{III} + \text{IV} + \text{V} + \text{VI}. \end{aligned} \quad (A.46) $$

We have to treat each of the terms I–VI. The treatment of V and VI is readily achieved through similar techniques used to investigate I and II, which we omit.

The same type of block but not the same block (I):

Suppose that the sequence of independent blocks {$\xi_i = (\varsigma_i, \zeta_i)$}$_{i \in \mathbb{N}^*}$ is of size $a_n$. An application of (A.1), shows that

$$ \begin{aligned} & \mathbb{P} \left\{ \left\| n^{-1/2} k^{-1} \sum_{p \neq q}^{\nu_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \right\|_{\mathcal{F}_2 \times \mathcal{H}^2} > \delta \right\} \\ & \le \mathbb{P} \left\{ \left\| n^{-1/2} k^{-1} \sum_{p \neq q}^{\nu_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\varsigma_i, t_1)}{H_{n,k}(t_1, \varsigma)} \right) K \left( \frac{d(\varsigma_j, t_2)}{H_{n,k}(t_2, \varsigma)} \right) \right\|_{\mathcal{F}_2 \times \mathcal{H}^2} > \delta \right\} \\ & \quad + 2 \nu_n \beta_{b_n}. \end{aligned} $$
We keep the choice of $b_n$ and $v_n$ such that :

$$v_n b_n^r \le 1, \qquad (A.47)$$

which implies that $2v_n\beta_{b_n} \to 0$ as $n \to \infty$, so the term to consider is the second summand. For some $(\xi_{n,1}, \xi_{n,2}) \in (0, 1)^2$, we choose $\mathbf{D}_n^- = (D_{n,1}^-, D_{n,2}^-)$ and $\mathbf{D}_n^+ = (D_{n,1}^+, D_{n,2}^+)$ such that for $j = 1, 2$

$$\phi(D_{n,j}^-) = \sqrt{\xi_{n,j}^-} \frac{k}{n}, \quad \phi(D_{n,j}^+) = \frac{k}{\sqrt{\xi_{n,j}^- n}} \quad \text{and} \quad \xi_n := \xi_{n,1} \xi_{n,2} \to 1,$$

then using the results of [44], we have $1_{\{D_{n,j}^- \le H_{n,k}(t_j) \le D_{n,j}^+\}} \xrightarrow{a.co} 1$ when $\frac{k}{n} \to 0$ and making use of condition (C.3.2), we have :

$$u_n(\varphi, \mathbf{t}; \mathbf{D}_n^+) \le u_n(\varphi, \mathbf{t}; \mathbf{h}_{n,k}(t)) \le u_n(\varphi, \mathbf{t}; \mathbf{D}_n^-), \qquad (A.48)$$

where $\mathbf{D}_n^- = (D_{n,1}^-, D_{n,2}^-)$ and $\mathbf{D}_n^+ = (D_{n,1}^+, D_{n,2}^+)$. This implies that

$$\mathbb{P} \left\{ \left\| n^{-1/2} k^{-1} \sum_{p \ne q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{H_{n,k}(t_1, \zeta)} \right) K \left( \frac{d(\zeta_j, t_2)}{H_{n,k}(t_2, \zeta)} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} > \delta \right\} \\ \le \mathbb{P} \left\{ \left\| n^{-1/2} k^{-1} \sum_{p \ne q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^-} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^-} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} > \delta \right\}.$$

Now, combining Lemma A.1 of [67] with Proposition B.8 in the Appendix, we obtain

$$\begin{aligned} & \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{p \ne q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^-} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^-} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ & \le c_2 \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{p \ne q}^{v_n} \epsilon_p \epsilon_q \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^-} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^-} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ & \le c_2 \mathbb{E} \int_0^{D_{nh}^{(U_1)}} N(u, \mathcal{F}_2, \mathcal{H}^2, \tilde{d}_{nhK,2}^{(1)}) du, \end{aligned} \qquad (A.49)$$

where $D_{nh}^{(U_1)}$ is the diameter of $\mathcal{F}_2\mathcal{H}^2$ according to the distance $\tilde{d}_{nhK,2}^{(1)}$, which are defined respectively by

$$D_{nh}^{(U_1)} := \left\| \mathbb{E}_{\epsilon} \left\| n^{-1/2} k^{-1} \sum_{p \ne q}^{v_n} \epsilon_p \epsilon_q \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^-} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^-} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \right\|_{\mathcal{F}_2, \mathcal{H}^2},$$

and

$$\tilde{d}_{nhK,2}^{(1)} (\varphi_1 \tilde{K}_1, \varphi_2 \tilde{K}_2)$$
$$ := \mathbb{E}_{\epsilon} \left| n^{-1/2} k^{-1} \sum_{p \neq q}^{v_n} \epsilon_p \epsilon_q \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \left[ \varphi_1(\zeta_i, \zeta_j) K_1 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_1 \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) - \varphi_2(\zeta_i, \zeta_j) K_2 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_2 \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) \right] \right| $$

Let consider another semi-norm $\tilde{d}_{nhK,2}^{(2)}$

$$ \tilde{d}_{nh,2}^{(2)} (\varphi_1 \tilde{K}_1, \varphi_2 \tilde{K}_2) = \frac{1}{n \tilde{\phi}(\mathbf{D}_n^{-})} \left[ \sum_{i \neq j}^{v_n} \left( \varphi_1(\zeta_i, \zeta_j) K_1 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_1 \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) - \varphi_2(\zeta_i, \zeta_j) K_2 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_2 \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) \right]^{1/2} $$

One can see that we have

$$ \tilde{d}_{nhK,2}^{(1)} (\varphi_1 K_1, \varphi_2 K_2) \leq a_n n^{1/2} k^{-3} \tilde{d}_{nh,2}^{(2)} (\varphi_1 K_1, \varphi_2 K_2). $$

We readily infer that

$$ \begin{aligned} \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{p \neq q}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{H_{n,k}(t_1, \zeta)} \right) K \left( \frac{d(\zeta_j, t_2)}{H_{n,k}(t_2, \zeta)} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ \leq c_2 \mathbb{E} \int_0^{D_{nhK}^{(U_1)}} N(u a_n^{-1} n^{-1/2} k^3, \mathcal{F}_2 \mathcal{H}^2, \tilde{d}_{nhK,2}^{(2)}) du \\ \leq c_2 a_n n^{1/2} k^{-3} \mathbb{P} \{ D_{nhK}^{(U_1)} a_n^{-1} n^{1/2} k^{-3} \geq \lambda_n \} + c_2 a_n n^{-3/2} k^{-1} \int_0^{\lambda_n} \log u^{-1} du, \end{aligned} $$

where $\lambda_n \to 0$. Notice that as $\lambda \to 0$, we have

$$ \left( \int_0^\lambda \log u^{-1} du \right) (\lambda \log \lambda^{-1})^{-1} \to 0, $$

where $a_n$ and $\lambda_n$ are chosen in such a way that the following relation will be fulfilled

$$ a_n \lambda_n n^{1/2} k^{-3} \log \lambda_n^{-1} \to 0. \qquad (A.50) $$

Making use of the triangle inequality, in combination with Hoeffding's trick, for instance, see [8, page 62], we obtain readily that

$$ a_n n^{1/2} k^{-3} \mathbb{P} \{ D_{nhK}^{(U_1)} \ge \lambda_n a_n n^{1/2} k^{-3} \} \\ \le \lambda_n^{-2} a_n^{-1} n^{-3/2} k \mathbb{E} \left\| \sum_{p \ne q}^{v_n} \left[ \sum_{i \in H_p^{(U)}} \sum_{j \in H_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) \right]^2 \right\|_{\mathcal{F}_2, \mathcal{H}^2} $$
$$ \le c_2 v_n \lambda_n^{-2} a_n^{-1} n^{-1/2} k \mathbb{E} \left\| \sum_{p=1}^{v_n} \left[ \sum_{i,j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j') K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j', t_2)}{D_{n,2}^{-}} \right) \right]^2 \right\|_{\mathcal{F}_2 \mathcal{H}^2}, \quad (A.51) $$

where $\{\xi'_i = (\zeta'_i, \zeta'_j)\}_{i \in \mathbb{N}^*}$ are independent copies of $\{\xi_i = (\zeta_i, \zeta_j)\}_{i \in \mathbb{N}^*}$. By imposing

$$ \lambda_n^{-2} a_n^{1-r} n^{-1/2} k \to 0, \quad (A.52) $$

we readily infer that

$$ \left\| v_n \lambda_n^{-2} a_n^{-1} n^{-1/2} k \mathbb{E} \sum_{p=1}^{v_n} \left[ \sum_{i,j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j') K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j', t_2)}{D_{n,2}^{-}} \right) \right]^2 \right\|_{\mathcal{F}_2 \mathcal{H}^2} \\ \le O(\lambda_n^{-2} a_n^{1-r} n^{-1/2} k). $$

By symmetrizing the expression in the expression in (A.51) and applying again the Proposition B.8 in the Appendix, we get

$$ v_n \lambda_n^{-2} a_n^{-1} n^{-1/2} k \mathbb{E} \left\| \sum_{p=1}^{v_n} \left[ \sum_{i,j \in H_p^{(U)}} \epsilon_p \varphi(\zeta_i, \zeta_j') K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j', t_2)}{D_{n,2}^{-}} \right) \right]^2 \right\|_{\mathcal{F}_2 \mathcal{H}^2} \\ \le c_2 \mathbb{E} \left( \int_0^{D_{nhK}^{(U_2)}} \left( \log N(u, \mathcal{F}_2 \mathcal{H}^2, \tilde{d}_{nhK,2}) \right)^{1/2} \right), \quad (A.53) $$

where

$$ D_{nhK}^{(U_2)} = \left\| \mathbb{E}_{\epsilon} \left| v_n \lambda_n^{-2} a_n^{-1} n^{-1/2} k \sum_{p=1}^{v_n} \epsilon_p \left[ \sum_{i,j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j') K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j', t_2)}{D_{n,2}^{-}} \right) \right] \right\|_{\mathcal{F}_2 \mathcal{H}^2}, $$

and for $\varphi_1 \tilde{K}_1, \varphi_2 \tilde{K}_2 \in \mathcal{F}_2 \mathcal{H}^2$

$$ \tilde{d}_{nhK,2}(\varphi_1 \tilde{K}_1, \varphi_2 \tilde{K}_2) = \mathbb{E}_{\epsilon} \left| v_n \lambda_n^{-2} a_n^{-1} n^{-1/2} k \sum_{p=1}^{v_n} \epsilon_p \left[ \left( \sum_{i,j \in H_p^{(U)}} \varphi_1(\zeta_i, \zeta_j') K_1 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_1 \left( \frac{d(\zeta_j', t_2)}{D_{n,2}^{-}} \right) \right)^2 - \left( \sum_{i,j \in H_p^{(U)}} \varphi_2(\zeta_i, \zeta_j') K_2 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_2 \left( \frac{d(\zeta_j', t_2)}{D_{n,2}^{-}} \right) \right)^2 \right] \right| $$

The fact that

$$ \mathbb{E}_{\epsilon} \left| v_n \lambda_n^{-2} a_n^{-1} n^{-1/2} k \sum_{p=1}^{v_n} \epsilon_p \left( \sum_{i,j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j') K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j', t_2)}{D_{n,2}^{-}} \right) \right) \right|^2 $$
$$ \leq a_n^{3/2} \lambda_n^{-2} n^{-1/2} \left[ \nu_n^{-1} a_n^{-2} k^2 \sum_{p=1}^{\nu_n} \sum_{i,j \in H_p^{(U)}} \left( \varphi(\zeta_i, \zeta'_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta'_j, t_2)}{D_{n,2}^{-}} \right) \right)^4 \right]^{1/2}, $$

so,

$$ a_n^{3/2} \lambda_n^{-2} n^{-1/2} k^2 \to 0, \qquad (A.54) $$

we have the convergence of (A.53) to zero. For the choice of $a_n$, $b_n$, and $v_n$, it should be noted that all the values satisfying (A.45), (A.47), (A.50), (A.52) and (A.54) are accepted.

The same blocks (II):

Remark that we have

$$ \mathbb{P} \left\{ \left\| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \sum_{i \neq j, j \in H_p^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \right\|_{\mathcal{F}_2 \mathcal{H}^2} > \lambda \right\} \\ \le 2 \nu_n \beta_{b_n} + \mathbb{P} \left\{ \left\| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \sum_{i \neq j, j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2 \mathcal{H}^2} > \lambda \right\}. $$

In a similar way as in the preceding proof, it suffices to prove that

$$ \mathbb{E} \left( \left\| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \sum_{i \neq j, j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,2}^{-}} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2 \mathcal{H}^2} \to 0. $$

Because of computation by [138, p. 53] and the fact that the classes functions are uniformly bounded, we obtain uniformly in $\mathcal{F}_2 \mathcal{H}^2$

$$ \mathbb{E} \left( \sum_{i \neq j, j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) \right) = O(a_n). $$

This implies that we have to prove that

$$ \mathbb{E} \left( \left\| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \sum_{i \neq j, j \in H_p^{(U)}} \left[ \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) - \mathbb{E} \left( \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_2)}{D_{n,2}^{-}} \right) \right) \right\|_{\mathcal{F}_2 \mathcal{H}^2} \right) \to 0. \qquad (A.55) $$

Like for empirical processes, to prove (A.55), it suffices to symmetrize and show that

$$ \mathbb{E} \left( \left\| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \sum_{i \neq j, j \in H_p^{(U)}} \epsilon_p \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2 \mathcal{H}^2} \to 0. $$
In a similar way as in (A.49), we infer that

$$ \mathbb{E} \left( \left\| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \sum_{i \neq j, i, j \in H_p^{(U)}} \epsilon_p \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \right) \\ \le \mathbb{E} \left( \int_0^{D_{nhK}^{(U_3)}} \left( \log N(u, \mathcal{F}_2 \mathcal{H}^2, \tilde{d}_{nhK,2}^{(3)}) \right)^{1/2} du \right), $$

where

$$ D_{nhK}^{(U_3)} = \left\| \mathbb{E}_{\epsilon} \left| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \epsilon_p \sum_{i \neq j, i, j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right| \right\|_{\mathcal{F}_2, \mathcal{H}^2}, $$

and the semi-metric $\tilde{d}_{nhK,2}^{(3)}$ is defined by

$$ \tilde{d}_{nhK,2}^{(3)}(\varphi_1 \tilde{K}_1, \varphi_2 \tilde{K}_2) = \mathbb{E}_{\epsilon} \left| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \epsilon_p \sum_{i \neq j, i, j \in H_p^{(U)}} \left( \varphi_1(\zeta_i, \zeta_j) K_1 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_1 \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) - \varphi_2(\zeta_i, \zeta_j) K_2 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_2 \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right) \right|. $$

Since we are trading uniformly bounded classes of functions, we infer that

$$ \mathbb{E}_{\epsilon} \left| n^{-1/2} k^{-1} \sum_{p=1}^{\nu_n} \epsilon_p \sum_{i \neq j, i, j \in H_p^{(U)}} \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right| \\ \le a_n^{3/2} n^{-1/2} k^{-1} \left[ \frac{1}{\nu_n a_n^2} \sum_{p=1}^{\nu_n} \sum_{i \neq j, i, j \in H_p^{(U)}} \left( \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right)^2 \right]^{1/2} \\ \le O(a_n^{3/2} n^{-1/2} k^{-1}). $$

Since $a_n^{3/2}(n)^{-1/2}k^{-1} \to 0$, $D_{nhK}^{(U_3)} \to 0$, we obtain $\Pi \to 0$ as $n \to \infty$.

## Different types of blocks (III)

An application of (A.1), shows that

$$ \sum_{p=1}^{\nu_n} \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{i \in H_p^{(U)}} \sum_{q: |q-p| \ge 2} \sum_{j \in T_q^{(U)}} \varphi(Y_i, Y_j) K\left(\frac{d(X_i, t_1)}{h_K}\right) K\left(\frac{d(X_j, t_2)}{h_K}\right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ \le \sum_{p=1}^{\nu_n} \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{i \in H_p^{(U)}} \sum_{q: |q-p| \ge 2} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} $$
$$+n^{-1/2}k^{-1}\nu_n^2 a_n b_n \beta_{a_n},$$

by the last choice of the parameters $a_n, b_n, \nu_n$ and the condition (8.92) imposed on the $\beta$-coefficients, we have

$$n^{-1/2}k^{-1}\nu_n^2 a_n b_n \beta_{a_n} \to 0.$$

For $p=1$ and $p=\nu_n$, since we have independent exchangeable blocks, we infer that

$$ \begin{aligned} & \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q:|q-p| \ge 2} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ &= \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{i \in H_{\nu_n}^{(U)}} \sum_{q:|q-p| \ge 2} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ &= \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2}. \end{aligned} $$

For $2 \le p \le \nu_n - 1$, we obtain

$$ \begin{aligned} & \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{i \in H_p^{(U)}} \sum_{q:|q-p| \ge 2} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ &= \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=4}^{\nu_n} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ &\le \mathbb{E} \left\| n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2}, \end{aligned} $$

therefore, it suffices to treat the convergence

$$ \mathbb{E} \left\| \nu_n n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \to 0. $$

By similar arguments as in [8], the usual symmetrization gives

$$ \begin{aligned} & \mathbb{E} \left\| \nu_n n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \\ &\le 2 \mathbb{E} \left\| \nu_n n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \epsilon_q \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2, \mathcal{H}^2} \end{aligned} $$
$$
\begin{align}
&= 2\mathbb{E}\left\{ \left\| \nu_n n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \epsilon_q \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right\|_{\mathcal{F}_2\mathcal{H}^2} \mathbb{1}_{\{D_{nhK}^{(U4)} \le \gamma_n\}} \right\} \nonumber \\
&\quad + 2\mathbb{E}\left\{ \left\| \nu_n n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \epsilon_q \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right\|_{\mathcal{F}_2\mathcal{H}^2} \mathbb{1}_{\{D_{nhK}^{(U4)} > \gamma_n\}} \right\} \nonumber \\
&= 2\mathrm{III}_1 + 2\mathrm{III}_2, \tag{A.56}
\end{align}
$$

where

$$
D_{nhK}^{(U4)} = \left\| \nu_n n^{-1/2} k^{-1} \left[ \sum_{q=3}^{\nu_n} \left( \sum_{j \in T_q^{(U)}} \sum_{i \in H_1^{(U)}} \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right)^2 \right]^{1/2} \right\|_{\mathcal{F}_2 \mathcal{H}^2} .
$$

In a similar way as in (A.49), we infer that

$$
\mathrm{III}_1 \le c_2 \int_0^{\gamma_n} \left( \log N \left( u, \mathcal{F}_2 \mathcal{H}^2, \bar{d}_{nhK,2}^{(4)} \right) \right)^{1/2} du, \quad (A.57)
$$

where

$$
\begin{align*}
& \bar{d}_{nhK,2}^{(4)}(\varphi_1 \tilde{K}_1, \varphi_2 \tilde{K}_2) \\
&= \mathbb{E}_{\epsilon} \left| \nu_n n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \epsilon_q \left[ \varphi_1(\zeta_i, \zeta_j) K_1 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_1 \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right. \right. \\
& \quad \left. - \varphi_2(\zeta_i, \zeta_j) K_2 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_2 \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right] \left. \right| .
\end{align*}
$$

Since we have

$$
\mathbb{E}_{\epsilon} \left| \nu_n n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \epsilon_q \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right| \\
\le a_n^{-1/2} b_n n^{1/2} k \left[ \left( \frac{1}{a_n b_n \nu_n n^{-4} k^4} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \left[ \varphi(\zeta_i, \zeta_j) K\left(\frac{d(\zeta_i, t_1)}{D_{n,1}^{-}}\right) K\left(\frac{d(\zeta_j, t_1)}{D_{n,2}^{-}}\right) \right]^2 \right)^{1/2} \right]
$$

and by considering the semi-metric

$$
\begin{align*}
& \bar{d}_{nhK,2}^{(5)}(\varphi_1 \tilde{K}_1, \varphi_2 \tilde{K}_2) \\
&= \left( \frac{1}{a_n b_n \nu_n \tilde{\phi}^4(\mathbf{D}_n^-)} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{\nu_n} \sum_{j \in T_q^{(U)}} \left[ \varphi_1(\zeta_i, \zeta_j) K_1 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^-} \right) K_1 \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^-} \right) \right] \right)^{1/2}.
\end{align*}
$$
$$ - \varphi_2(\zeta_i, \zeta_j) K_2 \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K_2 \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right)^2^{1/2} . $$

We show that the expression in (A.57) is bounded as follows

$$ v_n^{1/2} b_n n^{-1/2} \tilde{\phi}(\mathbf{D}_n^{-}) \int_0^{v_n^{-1/2} b_n^{-1} n^{1/2} h^{-1} \gamma_n} \left( \log N \left( u, \mathcal{F}_2 \mathcal{H}^2, \tilde{d}_{nhK,2}^{(5)} \right) \right)^{1/2} du, $$

by choosing $\gamma_n = n^{-\alpha}$ for some $\alpha > (17r - 26)/60r$, we get the convergence to zero of the previous quantity. To bound the second term in the right-hand side of (A.56), we remark that

$$ \text{III}_2 \tag{A.58} $$

$$ = \mathbb{E} \left\{ \left\| v_n n^{-1/2} k^{-1} \sum_{i \in H_1^{(U)}} \sum_{q=3}^{v_n} \sum_{j \in T_q^{(U)}} \epsilon_q \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right\|_{\mathcal{F}_2 \mathcal{H}^2} \mathbb{1}_{\{D_{nhK}^{(U4)} > \gamma_n\}} \right\} $$

$$ \le a_n^{-1} b_n n^{1/2} \tilde{\phi}^{-1}(\mathbf{D}_n^{-}) \mathbb{P} \left\{ \left\| v_n^2 n^{-3} \tilde{\phi}^{-2}(\mathbf{D}_n^{-}) \sum_{q=3}^{v_n} \left( \sum_{j \in T_q^{(U)}} \sum_{i \in H_1^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right)^2 \right\|_{\mathcal{F}_2 \mathcal{H}^2} \right. \\ \left. \ge \gamma_n^2 \right\}. \tag{A.59} $$

We now apply the square root trick to the last expression conditionally on $H_1^U$. We denote by $\mathbb{E}_T$ the expectation with respect to $\sigma\{(\zeta_j, \zeta_i) : j \in T_q, q \ge 3\}$ and we get by (C.6), for $2r/(r-1) < s < \infty$, (in the notation in Lemma 5.2 of [102])

$$ M_n = v_n^{1/2} \mathbb{E}_T \left( \sum_{j \in T_q^{(U)}} \sum_{i \in H_1^{(U)}} \varphi(\zeta_i, \zeta_j) K \left( \frac{d(\zeta_i, t_1)}{D_{n,1}^{-}} \right) K \left( \frac{d(\zeta_j, t_1)}{D_{n,2}^{-}} \right) \right)^2, \\ t = \gamma_n^2 a_n^{5/2} n^{1/2} \tilde{\phi}^{-1}(\mathbf{D}_n^{-}), \quad \rho = \lambda = 2^{-4} \gamma_n a_n^{5/4} n^{1/4} \tilde{\phi}^{-1/2}(\mathbf{D}_n^{-}), \quad m = \exp(\gamma_n^2 n \tilde{\phi}^{-2}(\mathbf{D}_n^{-}) b_n^{-2}). $$

Since we need $t > 8M_n$, and $m \to \infty$, by similar arguments as in [8] page 69, we get the convergence of (A.57) and (A.59) to zero.

## Different types of blocks (IV)

We have:

$$ \left\| n^{-1/2} k^{-1} \sum_{p=1}^{v_n} \sum_{i \in H_p^{(U)}} \sum_{q:|q-p| \le 1}^{v_n} \sum_{j \in T_q^{(U)}} \varphi(Y_i, Y_j) K \left( \frac{d(X_i, t_1)}{H_{n,k}(t_1)} \right) K \left( \frac{d(X_j, t_2)}{H_{n,k}(t_2)} \right) \right\|_{\mathcal{F}_2 \mathcal{H}^2} \\ \le c_2 v_n a_n b_n n^{-1/2} k^{-1} \to 0. $$

Hence, the proof of the lemma is complete.
## B. Auxiliary results

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

In the sequel, we define $X, X_1, \dots, X_n$ to be i.i.d. random variables defined on the probability space $(\Omega, \mathcal{A}, \mathcal{P})$ and taking values in some measurable space $(\Psi, \mathcal{B})$, and $\mathcal{F}$ to be a $\mathcal{P}$-measurable class of measurable functions with envelope function $F$, such that:

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

We further assume that $\mathcal{F}$ has the following property:

* For any sequence of i.i.d. $X$-valued random variables $Z_1, Z_2, \dots$ it holds that

$$ \mathbb{E} \left\| \sum_{i=1}^{k} \{g(Z_i) - \mathbb{E}g(Z_1)\} \right\|_{\mathcal{G}} \le C_1 \sqrt{k} \|G(Z_1)\|_2, \quad 1 \le k \le n, $$

where $C_1 > 1$ is a constant depending on $\mathcal{G}$ only.

**Lemma B.1 (Theorem 2.14.1 [193]).** For an empirical process $\alpha_n(f)$ indexed by the class of functions $\mathcal{F}$ with the 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 $p > 1$,

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

**Lemma B.2 (Theorem 3.1 [78]).** Let $\mathcal{F}$ be a pointwise measurable function class satisfying the above assumptions. If we suppose that the empirical process $\alpha_n(f)$ satisfies:

$$ \mathbb{E} \|\alpha_n(f)\|_{\mathcal{F}} \le C \|F\|_2, \qquad (B.1) $$

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

$$ \mathbb{E} \|\alpha_n(f 1_B)\|_{\mathcal{F}} \le 2C \|F 1_B\|_2. $$

From Theorem 3.2 of [78] it follows that a VC-type class of functions satisfies, always, the condition (B.1).

**Lemma B.3 (Bernstein type inequality Fact 4.2 [78]).** Assume that for some $H > 0$ and $p \ge 2$ the rvs $X, X_1, \dots, X_n$ satisfy:
$$ \mathbb{E}(F^p(X)) \leq (p!/2)\sigma^2 H^{p-2}, \text{ where } \sigma^2 \geq \mathbb{E}(F^2(X)), $$

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

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

**Lemma B.4 (Proposition 4. [9]).** Let $X_1, \dots, X_m$ be i.i.d. random variables with values in some measurable space $(S, \mathcal{S})$. Let $\mathcal{F}$ be a class of symmetric functions $f$ from $S^m$ satisfying some measurability conditions. Suppose that there exists a finite constant $c_0$ such that for each $x_j \in S$ we have:

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

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

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

where the variables $\zeta_i$ are a Rademacher variables.

**Lemma B.5 (Lemma 6.1 in [124], p.186).** Let $X_1, \dots, X_m$ be independent Bernoulli random variables with $\mathbb{P}(X_i=1) = p$, for all $i=1, \dots, n$. 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 B.6.** Let $\{X_n\}_{n \in \mathbb{Z}}$ be a data sequence, along with the kernel function $h(\cdot)$, satisfying Assumptions (A1)–(A3). We then have, 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,

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

**Proposition B.7.** Let $\{Z_{n,i}, 1 \le i \le n, n \ge 1\}$ 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:
$$ \mathrm{Cov}(V, W) \leq [\alpha(m)]^{1/r} \{\mathbb{E}|V|^p\}^{1/p} \{\mathbb{E}|W|^q\}^{1/q}, $$

where $p, q, r > 0$:

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

**Proposition B.8 (Proposition 3.6 of [10]).** Let $\{\mathbf{X}_t : t \in T\}$ 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:

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

where $D$ being the $\rho$-diameter of $\mathbf{T}$.

**Remark B.9.** In a similar way as in [35], Theorem 4.2 can be used to investigate the following problems.

1) (Expectile regression). For $p \in (0, 1)$, let $\psi(T - \theta) = (p - \{T - \theta \le 0\})|T - \theta|$, then the zero of $m(\psi, \cdot)$ with respect to $\theta$ leads to quantities called expectiles by [155]. Expectiles, as defined by [155], may be introduced either as a generalization of the mean or as an alternative to quantiles. Indeed, classical regression provides us with a high sensitivity to extreme values, allowing for more reactive risk management. Quantile regression, on the other hand, provides the ability to acquire exhaustive information on the effect of the explanatory variable on the response variable by examining its conditional distribution, refer to [3, 4, 150, 151] for further details on expectiles in functional data setting.

2) (Quantile regression). For $p \in (0, 1)$, let $\psi(T - \theta) = p - \{T - \theta \le 0\}$. Then the zero of $m(\psi, \cdot)$ with respect to $\theta$ is the conditional $p$-th quantile, initially introduced in [129] in the real and linear framework, for more general setting, refer to [35].

3) (Conditional winsorized mean). As in [117], if we consider $\psi(T - \theta) = -k, T - \theta, k$ if $T - \theta < -k, |T - \theta| \le k$, or $T - \theta > k$, then the zero of $m(\psi, \cdot)$ with respect to $\theta$ will be the conditional winsorized mean. Notably, this parameter was not considered in the literature on nonparametric functional data analysis involving wavelet estimators. Our paper offers asymptotic results for the conditional winsorized mean when the covariates are functions.

<figure><img src="image_4.png" /></figure>

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