$$ = \mathcal{L} \left( \sum_{(i_1, \dots, i_m) \in I_{S_n}^m} (\xi_{i_1, l_n} - 1) \dots (\xi_{i_m, l_n} - 1) \omega_{\vec{h}} (\mathbf{B}_{i_1}^{**}, \dots, \mathbf{B}_{i_m}^{**}) \right). $$

Hence, as in (33), we have:

$$ \begin{aligned} & P^*(II_n > \epsilon) \\ &= P^{**} \left( \left( \binom{n}{m} \right)^{-1/2} \left\| \sum_{(i_1, \dots, i_m) \in I_{S_n}^m} (\xi_{i_1, l_n} - 1) \dots (\xi_{i_m, l_n} - 1) \omega_{\vec{h}} (\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) \right\|_{\mathcal{F}_{\delta}} > \epsilon \right) \\ &\le C_1 P^{**} \left( \left\| \left( \binom{n}{m} \right)^{-1/2} \sum_{(i_1, \dots, i_m) \in \mathcal{I}_m''} (\xi_{i_1, l_n} - 1) \dots (\xi_{i_m, l_n} - 1) \omega_{\vec{h}} (\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) \right\|_{\mathcal{F}_{\delta}} > C_2 \epsilon \right) \\ &\quad + P^*(S_n^* > K\sqrt{n}), \text{ for any } K > 0, \end{aligned} \quad (40) $$

where

$$ \mathcal{I}_m'' := \left\{ (i_1, \dots, i_m) : 1 \le i_1 < \dots < i_m \le S_n^*, \exists \ell = 1, \dots, m, \right. \\ \left. \qquad S_n^* < i_\ell \le K\sqrt{n}, i_j \ne i_k \text{ for } j \ne k \right\}. $$

Using the same bootstrap procedure defined previously for $I_n$, let

$$ \vec{\mathbf{B}}_i := \{ X_{\mathcal{F}_{i-1}+1}, \dots, X_{\mathcal{F}_i}, 0, 0, \dots \}, $$

for $i = l_n + 1, \dots, \mathbb{E}(l_n)$, and let $\vec{\mathcal{F}}$ be a class of function such that, for every $\vec{\omega}_h \in \vec{\mathcal{F}}$:

$$ \left\{ \begin{array}{ll} \vec{\omega}_h (\vec{\mathbf{B}}_1, \vec{\mathbf{B}}_2, \dots, \vec{\mathbf{B}}_k) = \sum_{i_1=1}^{\infty} \dots \sum_{i_k=1}^{\infty} h(x_{i_1}, \dots, x_{i_k}) \mathbb{1}_{\{x_k \ne 0\}} & \text{if defined} \\ \infty & \text{otherwise.} \end{array} \right. \quad (41) $$

It is classical that $\{\vec{\mathbf{B}}_i\}$ are i.i.d., applying the same bootstrap method of Algorithm 1. This new sample allows us to enlarge and bound (33) by

$$ P^{**} \left( \sup_{\vec{h} \in \vec{\mathcal{H}}} \left| \binom{n}{m}^{-1/2} \sum_{(i_1, \dots, i_m) \in \mathcal{I}_m''} (\xi_{i_1, l_n} - 1) \dots (\xi_{i_m, l_n} - 1) \vec{\omega}_{\vec{h}} (\vec{\mathbf{B}}_{i_1}, \dots, \vec{\mathbf{B}}_{i_m}) \right| > \epsilon \right), \quad (42) $$

where

$$ \vec{h} \in \vec{\mathcal{H}} = \{ \vec{\omega}_{\vec{f}} - \vec{\omega}_{\vec{g}}, \vec{f}, \vec{g} \in \vec{\mathcal{F}} \} $$

corresponding to the class

$$ \mathcal{H} = \{ \omega_f - \omega_g, f, g \in \mathcal{F} \}, $$

with envelope $\vec{F}$ and $F$, respectively. As before, for the class of functions $\mathcal{H}$, consider the bracket $[h^\ell, h^u]$, such that

$$ \mathbb{E}^{**}(h^\ell, h^u) \le \gamma, $$

where $\gamma > 0$ and it is determined later. In this framework, the bracketing entropy number is $N^{**}(\gamma) := N_1(\gamma, \mathcal{H}, \mathbb{D}^{**})$, for

$$ \mathbb{D}^{**} = \left( \mathbb{E}(l_n) - l_n \right)^{-1} \sum_{(i_1, \dots, i_m) \in I_{\mathbb{E}(l_n)-l_n}^m} \xi_{i_1, l_n} \dots \xi_{i_m, l_n} \delta(\vec{\mathbf{B}}_{i_1}, \dots, \vec{\mathbf{B}}_{i_m}). $$

Following the same arguments from Equations (37) through (38), we can find that (42) is
$$
\begin{align}
&\le N^{**}(\gamma) \max_{k \le N^{**}(\gamma)} \mathbb{P}\left(\left|\binom{n}{m}\right|^{-1/2} \sum_{(i_1, \dots, i_m) \in \mathfrak{J}_m''} (\tilde{\xi}_{i_1, l_n} - 1) \dots (\tilde{\xi}_{i_m, l_n} - 1) \right. \nonumber \\
&\qquad \left. \left[ h_k^\ell (\vec{\mathbf{B}}_{i_1}, \dots, \vec{\mathbf{B}}_{i_m}) - \mathbb{E}^{**} (h_k^\ell (\vec{\mathbf{B}}_{i_1}, \dots, \vec{\mathbf{B}}_{i_m})) \right] > \epsilon \right) \nonumber \\
&\le \frac{N^{**}(\gamma)}{\epsilon^2 \binom{n}{m}^{1/2}} c^2 \times 4M_n^2 + \frac{2N^{**}(\gamma)}{\epsilon} \mathbb{E}^{**} \left| (\tilde{\xi}_{1, l_n} - 1) \dots (\tilde{\xi}_{m, l_n} - 1) \tilde{F} \mathbf{1}_{\tilde{F} > M_n} \right| \nonumber \\
&\le \frac{N^{**}(\gamma)}{\epsilon^2 \binom{n}{m}^{1/2}} c^2 \times 4M_n^2 + \frac{4N^{**}(\gamma)}{\epsilon} \nonumber \\
&\qquad \left( \mathbb{E}(l_n) - l_n \right)^{-1} \sum_{(i_1, \dots, i_m) \in I_{\mathbb{E}(l_n)-l_n}^m} (\tilde{\xi}_{i_1, l_n} - 1) \dots (\tilde{\xi}_{i_m, l_n} - 1) \left| \tilde{F} \mathbf{1}_{\tilde{F} > M_n} (\vec{\mathbf{B}}_{i_1}, \dots, \vec{\mathbf{B}}_{i_m}) \right|. \tag{43}
\end{align}
$$

Here, we must pay attention to the randomness of $N^{**}$ which depends on $n$. According to Lemma 1 i), we can see that $|\mathbb{E}(l_n) - l_n| \to \infty$ in probability, under the assumption that $l_n < \mathbb{E}(l_n)$. Now, using the same treatment of $l_n$, and for $M_n := n^{1/3}$ (to provide the convergence of $M_n$ to $\infty$), as in [47], this allows the convergence of (43) to zero in probability. Estimating now $N^{**}$ by considering the same $\gamma/2$ brackets $h_1^\ell, \cdots, h_{N(\gamma/2)}^\ell$ and $h_1^u, \cdots, h_{N(\gamma/2)}^u$, we have $N^{**}(\gamma) < N(\gamma/2)$, which does not depend on $n$. Then, $II_n$ is proved. Following the same footsteps, we can prove the case where $l_n > \mathbb{E}(l_n)$. This proves Step 3. $\square$

The end of the previous step yields that we only need to show the stochastic equicontinuity of $\hat{T}_{l_n}^*$, where the number of blocks is replaced by the deterministic quantity $\mathbb{E}(l_n)$. In order to achieve the equicontinuity of this statistic, Lemma 2 shows that it is sufficient to prove that:

$$
\begin{align*}
\mathbb{E} \left\| \sum_{1 \le i_k \le \ell_k, 1 \le k \le m(n)} \epsilon_{i_1}^{(1)} \cdots \epsilon_{i_m}^{(m)} \omega_{\vec{h}} (\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) \right\|_{\mathcal{H}} \\
&\le a(\ell_1, \dots, \ell_m) \left( \prod_{k=1}^{m} \ell_k \right)^{1/2}
\end{align*}
$$

for all $1 \le \ell_1, \dots, \ell_m \le n$. We begin to define the distance:

$$
e_{\ell}^{2}(f,g) \equiv \frac{1}{\prod_{k=1}^{m} \ell_k} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \omega_{\vec{h}}^{2} (\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}),
$$

defined in $L_2$, associated with the Rademacher process

$$
\left\{ \frac{1}{(\prod_{k=1}^{m} \ell_k)^{1/2}} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \epsilon_{i_1}^{(1)} \cdots \epsilon_{i_m}^{(m)} \omega_{\vec{h}} (\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) : \vec{h} \in \mathcal{H} \mid \hat{\mathbf{B}}_1, \dots, \hat{\mathbf{B}}_m \right\}.
$$

Take $\|f\|_\ell^2 \equiv e_\ell^2(f, 0)$ and

$$
r_{\ell}(\delta) \equiv \sup_{f \in \mathcal{F}_{\delta}} \|f\|_{\ell}^{2}.
$$

Using Corollary A1, we have
$$
\begin{align}
&\le N^{**}(\gamma) \max_{k \le N^{**}(\gamma)} \mathbb{P}\left(\left|\binom{n}{m}\right|^{-1/2} \sum_{(i_1, \dots, i_m) \in \mathfrak{J}_m''} (\tilde{\xi}_{i_1, l_n} - 1) \dots (\tilde{\xi}_{i_m, l_n} - 1) \right. \nonumber \\
&\qquad \left. \left[ h_k^\ell (\vec{\mathbf{B}}_{i_1}, \dots, \vec{\mathbf{B}}_{i_m}) - \mathbb{E}^{**} (h_k^\ell (\vec{\mathbf{B}}_{i_1}, \dots, \vec{\mathbf{B}}_{i_m})) \right] > \epsilon \right) \nonumber \\
&\le \frac{N^{**}(\gamma)}{\epsilon^2 \binom{n}{m}^{1/2}} c^2 \times 4M_n^2 + \frac{2N^{**}(\gamma)}{\epsilon} \mathbb{E}^{**} \left| (\tilde{\xi}_{1, l_n} - 1) \dots (\tilde{\xi}_{m, l_n} - 1) \tilde{F} \mathbf{1}_{\tilde{F} > M_n} \right| \nonumber \\
&\le \frac{N^{**}(\gamma)}{\epsilon^2 \binom{n}{m}^{1/2}} c^2 \times 4M_n^2 + \frac{4N^{**}(\gamma)}{\epsilon} \nonumber \\
&\qquad \left( \mathbb{E}(l_n) - l_n \right)^{-1} \sum_{(i_1, \dots, i_m) \in I_{\mathbb{E}(l_n)-l_n}^m} (\tilde{\xi}_{i_1, l_n} - 1) \dots (\tilde{\xi}_{i_m, l_n} - 1) \left| \tilde{F} \mathbf{1}_{\tilde{F} > M_n} (\vec{\mathbf{B}}_{i_1}, \dots, \vec{\mathbf{B}}_{i_m}) \right|. \tag{43}
\end{align}
$$

Here, we must pay attention to the randomness of $N^{**}$ which depends on $n$. According to Lemma 1 i), we can see that $|\mathbb{E}(l_n) - l_n| \to \infty$ in probability, under the assumption that $l_n < \mathbb{E}(l_n)$. Now, using the same treatment of $l_n$, and for $M_n := n^{1/3}$ (to provide the convergence of $M_n$ to $\infty$), as in [47], this allows the convergence of (43) to zero in probability. Estimating now $N^{**}$ by considering the same $\gamma/2$ brackets $h_1^\ell, \cdots, h_{N(\gamma/2)}^\ell$ and $h_1^u, \cdots, h_{N(\gamma/2)}^u$, we have $N^{**}(\gamma) < N(\gamma/2)$, which does not depend on $n$. Then, $II_n$ is proved. Following the same footsteps, we can prove the case where $l_n > \mathbb{E}(l_n)$. This proves Step 3. $\square$

The end of the previous step yields that we only need to show the stochastic equicontinuity of $\hat{T}_{l_n}^*$, where the number of blocks is replaced by the deterministic quantity $\mathbb{E}(l_n)$. In order to achieve the equicontinuity of this statistic, Lemma 2 shows that it is sufficient to prove that:

$$
\begin{align*}
\mathbb{E} \left\| \sum_{1 \le i_k \le \ell_k, 1 \le k \le m(n)} \epsilon_{i_1}^{(1)} \cdots \epsilon_{i_m}^{(m)} \omega_{\vec{h}} (\hat{\mathbf{B}}_{i_1}^{(1)}, \ldots, \hat{\mathbf{B}}_{i_m}^{(m)}) \right\|_{\mathcal{H}}
\end{align*}
\begin{equation*}
\le a(\ell_1, \ldots, \ell_m) \left( \prod_{k=1}^{m} \ell_k \right)^{1/2}
\end{equation*}
$$

for all $1 \le \ell_1, \dots, \ell_m \le n$. We begin to define the distance:

$$
e_{\ell}^{2}(f,g) \equiv \frac{1}{\prod_{k=1}^{m} \ell_k} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \omega_{\vec{h}}^{2}(\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}),
$$

defined in $L_2$, associated with the Rademacher process

$$
\left\{ \frac{1}{(\prod_{k=1}^{m} \ell_k)^{1/2}} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \varepsilon_{i_1}^{(1)} \cdots \varepsilon_{i_m}^{(m)} \omega_{\vec{h}} (\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) : \vec{h} \in \mathcal{H} \mid \hat{\mathbf{B}}_1, \dots, \hat{\mathbf{B}}_m \right\}.
$$

Take $\|f\|_\ell^2 \equiv e_\ell^2(f, 0)$ and

$$
r_{\ell}(\delta) \equiv \sup_{f \in \mathcal{F}_{\delta}} \|f\|_{\ell}^{2}.
$$

Using Corollary A1, we have
$$
\begin{align}
& \mathbb{E} \left\| \frac{1}{(\prod_{k=1}^m \ell_k)^{1/2}} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \varepsilon_{i_1}^{(1)} \cdots \varepsilon_{i_m}^{(m)} \omega_{\vec{h}} (\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) \right\|_{\mathcal{H}} \nonumber \\
& \le C \int_0^{r_\ell(\delta)} (\log \mathcal{N}(\varepsilon, \mathcal{F}, e_\ell))^{m/2} \, d\varepsilon \nonumber \\
& = C \|F\|_\ell \cdot \int_0^{r_\ell(\delta)/\|F\|_\ell} (\log \mathcal{N}(\varepsilon \|F\|_\ell, \mathcal{F}, e_\ell))^{m/2} \, d\varepsilon \nonumber \\
& \le C \|F\|_\ell \cdot \int_0^{r_\ell(\delta)/\|F\|_\ell} \left( \sup_Q \log \mathcal{N} \left( \varepsilon \|F\|_{L_2(Q), \mathcal{F}, L_2(Q)} \right) \right)^{m/2} \, d\varepsilon. \tag{44}
\end{align}
$$

Assuming that $F \ge 1$, the upper bound in the integral can be replaced by $r_\ell(\delta)$. The following proposition is necessary for the following.

**Proposition 3 ([46]).** Let {$X_i$} be i.i.d. random variables with law $\mathbb{P}$. Let $\mathcal{H}$ be a class of measurable real-valued functions defined on $(\mathcal{X}^m, \mathcal{A}^m)$ with an $\mathbb{P}^m$-integrable envelope such that the following holds: for any fixed $\delta > 0$, $M > 0$, $1 \le k \le m$,

$$
\max_{1 \le j' \le k} \mathbb{E} \left( \frac{\log \mathcal{N} \left( \delta, (\pi_k \mathcal{H})_M, e_{\ell, j'} \right)}{\ell_{j'}} \right)^{1/2} \to 0 \quad (45)
$$

holds for any $\ell_1 \wedge \dots \wedge \ell_k \to \infty$. Here for $\ell = (\ell_1, \dots, \ell_k)$ and $\{X_i\}_{i=1}^\infty$,

$$
e_{\ell,j'}(f,g) \equiv \frac{1}{\ell_{j'}} \sum_{i_{j'}=1}^{\ell_{j'}} \left| \frac{1}{\prod_{j \neq j'} \ell_j} \sum_{1 \le i_j \le \ell_{j'} : j \neq j'} (f-g)(X_{i_1}, \dots, X_{i_{\ell_{j'}}}) \right|
$$

and

$$
(\pi_k \mathcal{H})_M \equiv \{h1_{H_k \le M} : h \in \pi_k \mathcal{H}\},
$$

where $H_k$ is an envelope for $\pi_k \mathcal{H}$. Then,

$$
\sup_{h \in \mathcal{H}} \left| \frac{1}{\prod_{k=1}^{m} \ell_k} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} (h(X_{i_1}, \dots, X_{i_m}) - \mathbb{P}^m h) \right| \to 0
$$

in $L_1$ as $\ell_1 \wedge \dots \wedge \ell_m \to \infty$. The above equation can be replaced by the decoupled version.

By this proposition, $\|F\|_\ell \to \mathbb{P} \|F\|_{L_2(P)}$ as $\ell_1 \wedge \dots \wedge \ell_m \to \infty$, therefore, it suffices to
get $r_\ell(\delta) \to_p 0$ as $\ell_1 \wedge \dots \wedge \ell_m \to \infty$ and $\delta \to 0$. It is obvious that all that is left to do now
is to demonstrate that

$$
\sup_{f \in \tilde{\mathcal{F}}_{\delta}} \left| \frac{1}{\prod_{k=1}^{m} \ell_k} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \left( \omega_{\vec{h}}^2 (\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) - \mathbb{P}^m \omega_{\vec{h}}^2 \right) \right| \to_p 0. \quad (46)
$$

Verifying condition (45)

$$
\begin{align*}
& \max_{1 \le j' \le k} \mathbb{E} \left( \frac{\log \mathcal{N} \left( \delta, \mathcal{F}_M^2, e_{\ell, j'} \right)}{\ell_{j'}} \right)^{1/2} \\
& \le (\ell_1 \wedge \dots \wedge \ell_k)^{-1/2} \mathbb{E} \left[ \int_0^\delta \left( \log \mathcal{N} \left( \varepsilon, \mathcal{F}_M^2, e_{\ell, j'} \right) \right)^{m/2} d\varepsilon \right]
\end{align*}
$$
$$
\begin{align}
& \mathbb{E} \left\| \frac{1}{(\prod_{k=1}^m \ell_k)^{1/2}} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \varepsilon_{i_1}^{(1)} \cdots \varepsilon_{i_m}^{(m)} \omega_{\vec{h}} (\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) \right\|_{\mathcal{H}} \nonumber \\
& \le C \int_0^{r_\ell(\delta)} (\log \mathcal{N}(\varepsilon, \mathcal{F}, e_\ell))^{m/2} \, d\varepsilon \nonumber \\
& = C \|F\|_\ell \cdot \int_0^{r_\ell(\delta)/\|F\|_\ell} (\log \mathcal{N}(\varepsilon \|F\|_\ell, \mathcal{F}, e_\ell))^{m/2} \, d\varepsilon \nonumber \\
& \le C \|F\|_\ell \cdot \int_0^{r_\ell(\delta)/\|F\|_\ell} \left( \sup_Q \log \mathcal{N} \left( \varepsilon \|F\|_{L_2(Q), \mathcal{F}, L_2(Q)} \right) \right)^{m/2} \, d\varepsilon. \tag{44}
\end{align}
$$

Assuming that $F \ge 1$, the upper bound in the integral can be replaced by $r_\ell(\delta)$. The following proposition is necessary for the following.

**Proposition 3 ([46]).** Let {$X_i$} be i.i.d. random variables with law $\mathbb{P}$. Let $\mathcal{H}$ be a class of measurable real-valued functions defined on $(\mathcal{X}^m, \mathcal{A}^m)$ with an $\mathbb{P}^m$-integrable envelope such that the following holds: for any fixed $\delta > 0$, $M > 0$, $1 \le k \le m$,

$$
\max_{1 \le j' \le k} \mathbb{E} \left( \frac{\log \mathcal{N} \left( \delta, (\pi_k \mathcal{H})_M, e_{l,j'} \right)}{l_{j'}} \right)^{1/2} \to 0 \quad (45)
$$

holds for any $\ell_1 \wedge \dots \wedge \ell_k \to \infty$. Here for $\ell = (\ell_1, \dots, \ell_k)$ and $\{X_i\}_{i=1}^\infty$,

$$
e_{\ell,j'}(f,g) \equiv \frac{1}{\ell_{j'}} \sum_{i_{j'}=1}^{\ell_{j'}} \left| \frac{1}{\prod_{j \neq j'} \ell_j} \sum_{1 \le i_j \le \ell_{j'} : j \neq j'} (f-g)(X_{i_1}, \dots, X_{i_{\ell_j}}) \right|
$$

and

$$
(\pi_k \mathcal{H})_M \equiv \{h1_{H_k \le M} : h \in \pi_k \mathcal{H}\},
$$

where $H_k$ is an envelope for $\pi_k \mathcal{H}$. Then,

$$
\sup_{h \in \mathcal{H}} \left| \frac{1}{\prod_{k=1}^{m} \ell_k} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} (h(X_{i_1}, \dots, X_{i_m}) - \mathbb{P}^m h) \right| \to 0
$$

in $L_1$ as $\ell_1 \wedge \dots \wedge \ell_m \to \infty$. The above equation can be replaced by the decoupled version.

By this proposition, $\|F\|_\ell \to \mathbb{P} \|F\|_{L_2(P)}$ as $\ell_1 \wedge \dots \wedge \ell_m \to \infty$, therefore, it suffices to get $r_\ell(\delta) \to_p 0$ as $\ell_1 \wedge \dots \wedge \ell_m \to \infty$ and $\delta \to 0$. It is obvious that all that is left to do now is to demonstrate that

$$
\sup_{f \in \mathcal{F}_{\delta}} \left| \frac{1}{\prod_{k=1}^{m} \ell_k} \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \left( \omega_{\vec{h}}^2 (\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) - \mathbb{P}^m \omega_{\vec{h}}^2 \right) \right| \to_p 0. \quad (46)
$$

Verifying condition (45)

$$
\begin{align*}
& \max_{1 \le j' \le k} \mathbb{E} \left( \frac{\log \mathcal{N} \left( \delta, \mathcal{F}_M^2, e_{\ell,j'} \right)}{\ell_{j'}} \right)^{1/2} \\
& \le (\ell_1 \wedge \dots \wedge \ell_k)^{-1/2} \mathbb{E} \left[ \int_0^\delta \left( \log \mathcal{N} \left( \varepsilon, \mathcal{F}_M^2, e_{\ell,j'} \right) \right)^{m/2} d\varepsilon \right]
\end{align*}
$$
$$ \begin{aligned} & \le \left(\frac{\delta}{\sqrt{2M}}\right)^{-1} (\ell_1 \wedge \dots \wedge \ell_k)^{-1/2} \mathbb{E} \left[ \int_0^{\delta/\sqrt{2M}} \left( \log \mathcal{N}(\varepsilon, \mathcal{F}_M^2, e_{\ell,j'}) \right)^{m/2} d\varepsilon \right] \\ & \le (\delta/2M)^{-1} (\ell_1 \wedge \dots \wedge \ell_k)^{-1/2} \int_0^1 \left( \sup_Q \log \mathcal{N}(\varepsilon \|F\|_{L_2(Q)}, \mathcal{F}, L_2(Q)) \right)^{m/2} d\varepsilon \\ & \times \|F\|_{L_2(\mathbb{P}^m)} \to 0. \end{aligned} \quad (47) $$

The shift from the second to the third line is true because

$$ \mathcal{N}(\delta, \mathcal{F}_M^2, L_2(Q)) \le \mathcal{N}(\delta/\sqrt{2M}, \mathcal{F}_M, L_2(Q)). $$

As the condition is verified, as well as $\ell_1 \wedge \dots \wedge \ell_m \to \infty$, (46) follows directly using the previous proposition. Hence, there exists some sequence $\{a_\ell\}$, in a way that $a_\ell \to 0$ for any sequence $\{\delta_\ell\}$ with $\delta_\ell \to 0$ both under $\ell_1 \wedge \dots \wedge \ell_m \to \infty$, such that:

$$ \mathbb{E} \left\| \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \varepsilon_{i_1}^{(1)} \cdots \varepsilon_{i_m}^{(m)} \omega_{\tilde{h}}(\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) \right\|_{\mathcal{H}} \le a_\ell \left( \prod_{k=1}^{m} \ell_k \right)^{1/2}. \quad (48) $$

An application of Lemma 2 proves that

$$ n^{-m/2} \mathbb{E} \left\| \sum_{1 \le i_1, \dots, i_m \le n} (\zeta_{i_1} - 1) \cdots (\zeta_{i_m} - 1) \omega_{\tilde{h}}(\hat{\mathbf{B}}_{i_1}, \dots, \hat{\mathbf{B}}_{i_m}) \right\|_{\mathcal{F}_{\delta_n}} \to 0, \quad n \to \infty. $$

This completes the proof for the asymptotic equicontinuity.

**Author Contributions:** I.S. and S.B.: conceptualization, methodology, investigation, writing—original draft, writing—review and editing. All authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.

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

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the Special Issue Editor of the Special Issue on "Current Developments in Theoretical and Applied Statistics", Christophe Chesneau for the invitation. The authors are indebted to the Editor-in-Chief and the three referees for their very generous comments and suggestions on the first version of our article, which helped us to improve the content, presentation and layout of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

## Appendix A

This appendix contains supplementary information that is an essential part of providing a more comprehensive understanding of the paper. We also refer to [46] for more details.

**Proof of Proposition 1.** Let $\mathbf{B}_0 = \{X_1, \dots, X_{\mathcal{F}_0}\}$ and $\mathbf{B}_{l_n}^{(n)} = \{X_{\mathcal{F}_{l_n-1}+1}, \dots, X_n\}$ the possibly empty non-regenerative blocks of observations. Note that, for $l_n \le 2$, the demonstration can be viewed directly in [59], under the assumptions (C1), (C2) and (C3), we can see that

$$ \mathbb{P}_v(l_n \le 2) = O(n^{-2}). $$

Otherwise, for $l_n > 2$, we can write $W_n(h)$ as follows:

$$ W_n(h) = (I) + (II), $$
$$ \begin{aligned} & \le \left(\frac{\delta}{\sqrt{2M}}\right)^{-1} (\ell_1 \wedge \dots \wedge \ell_k)^{-1/2} \mathbb{E} \left[ \int_0^{\delta/\sqrt{2M}} \left( \log \mathcal{N}(\varepsilon, \mathcal{F}_M^2, e_{\ell,j'}) \right)^{m/2} d\varepsilon \right] \\ & \le (\delta/2M)^{-1} (\ell_1 \wedge \dots \wedge \ell_k)^{-1/2} \int_0^1 \left( \sup_Q \log \mathcal{N}(\varepsilon \|F\|_{L_2(Q)}, \mathcal{F}, L_2(Q)) \right)^{m/2} d\varepsilon \\ & \times \|F\|_{L_2(\mathbb{P}^m)} \to 0. \end{aligned} \quad (47) $$

The shift from the second to the third line is true because

$$ \mathcal{N}(\delta, \mathcal{F}_M^2, L_2(Q)) \le \mathcal{N}(\delta/\sqrt{2M}, \mathcal{F}_M, L_2(Q)). $$

As the condition is verified, as well as $\ell_1 \wedge \dots \wedge \ell_m \to \infty$, (46) follows directly using the previous proposition. Hence, there exists some sequence $\{a_\ell\}$, in a way that $a_\ell \to 0$ for any sequence $\{\delta_\ell\}$ with $\delta_\ell \to 0$ both under $\ell_1 \wedge \dots \wedge \ell_m \to \infty$, such that:

$$ \mathbb{E} \left\| \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \varepsilon_{i_1}^{(1)} \cdots \varepsilon_{i_m}^{(m)} \omega_{\tilde{h}}(\hat{\mathbf{B}}_{i_1}^{(1)}, \dots, \hat{\mathbf{B}}_{i_m}^{(m)}) \right\|_{\mathcal{H}} \le a_\ell \left( \prod_{k=1}^{m} \ell_k \right)^{1/2}. \quad (48) $$

An application of Lemma 2 proves that

$$ n^{-m/2} \mathbb{E} \left\| \sum_{1 \le i_1, \dots, i_m \le n} (\zeta_{i_1} - 1) \cdots (\zeta_{i_m} - 1) \omega_{\tilde{h}}(\hat{\mathbf{B}}_{i_1}, \dots, \hat{\mathbf{B}}_{i_m}) \right\|_{\mathcal{F}_{\delta_n}} \to 0, \quad n \to \infty. $$

This completes the proof for the asymptotic equicontinuity.

**Author Contributions:** I.S. and S.B.: conceptualization, methodology, investigation, writing—original draft, writing—review and editing. All authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.

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

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the Special Issue Editor of the Special Issue on "Current Developments in Theoretical and Applied Statistics", Christophe Chesneau for the invitation. The authors are indebted to the Editor-in-Chief and the three referees for their very generous comments and suggestions on the first version of our article, which helped us to improve the content, presentation and layout of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

## Appendix A

This appendix contains supplementary information that is an essential part of providing a more comprehensive understanding of the paper. We also refer to [46] for more details.

**Proof of Proposition 1.** Let $\mathbf{B}_0 = \{X_1, \dots, X_{\mathcal{F}_0}\}$ and $\mathbf{B}_{l_n}^{(n)} = \{X_{\mathcal{F}_{l_n-1}+1}, \dots, X_n\}$ the possibly empty non-regenerative blocks of observations. Note that, for $l_n \le 2$, the demonstration can be viewed directly in [59], under the assumptions (C1), (C2) and (C3), we can see that

$$ \mathbb{P}_v(l_n \le 2) = O(n^{-2}). $$

Otherwise, for $l_n > 2$, we can write $W_n(h)$ as follows:

$$ W_n(h) = (I) + (II), $$
where

$$ (I) = \frac{1}{\binom{n}{m}} \left[ \sum_{1 \le i_1 < \dots < i_{m-1} \le l_n - 1} \omega_{\tilde{h}}(\mathbf{B}_0, \mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1}}) \right. \\ \left. + \sum_{1 \le i_1 < \dots < i_{m-1} \le l_n - 1} \omega_{\tilde{h}}(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1}}, \mathbf{B}_{l_n}) \right], $$

$$ (II) = \frac{1}{\binom{n}{m}} \left[ \sum_{j=2}^{m} \sum_{0 \le k < i_1 < \dots < i_{m-1-j} \le l_n} \omega_{\tilde{h}}(\mathbf{B}_{k}, \dots, \mathbf{B}_{k'}, \mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1-j}}) \right. \\ \left. - \sum_{j=2}^{m} \sum_{1 \le k < i_1 < \dots < i_{m-1-j} \le n} \tilde{h}(X_k, \dots, X_k', X_{i_1}, \dots, X_{i_m}) \right] \\ = \frac{1}{\binom{n}{m}} \left[ \sum_{(i_m^n)^c} \omega_{\tilde{h}}(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) - \sum_{(I_m^n)^c} \tilde{h}(X_{i_1}, \dots, X_{i_m}) \right], $$

where

$$ (I_m^s)^c = \{(i_1, \dots, i_m) : i_j \in \mathbb{N}, 1 \le i_j \le n; \text{ at least there are } j \text{ and } k \text{ such that } i_j = i_k\}, $$

the complement of index set, with cardinal equal to $\binom{s+m-1}{m} - \binom{s}{m} := \overline{\binom{s}{m}}$. To prove the convergence of $W_n(h)$ to zero in probability, we must fulfill the convergence of (I) and (II) to zero in probability.

$$ \mathfrak{A} = \overline{\binom{n}{m}}^{-1} \sum_{\binom{l_m^n-1}{m}} \omega_{\tilde{h}}(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) \\ \xrightarrow{n \to \infty} \alpha^{-m} \left[ \mathbb{E}(\omega_h(\mathbf{B}_1, \underbrace{\mathbf{B}_{k'}, \dots, \mathbf{B}_{k_u}}_{\text{u times}} \mathbf{B}_{k+u}, \dots, \mathbf{B}_m)) - \mathbb{E}_A((\tau)^u) (\mathbb{E}_A(\tau))^{m-u} \mu(h) \right], $$

where $1 \le k \le m$ and $1 \le u \le k$. We apply the SLLN for Harris Markov chains to find the convergence of

$$ \mathfrak{B} = \overline{\binom{n}{m}}^{-1} \sum_{(I_m^n)^c} \tilde{h}(X_{i_1}, \dots, X_{i_m}), $$

to

$$ \int \dots \int h(x_1, \underbrace{x_k, \dots, x_k}_{\text{u times}}, x_{k+u}, \dots, x_m) d\mu(x_1) d\mu^u(x_1) d\mu(x_{k+u}) \dots d\mu(x_m) - \mu(h). $$

Using the conditions, all terms in $\mathfrak{A}$ and $\mathfrak{B}$ are finite and we can prove the convergence of (II) to zero. Now, for (I), applying the SLLN and by Lemma 3.2 in [47] part i), we can see that

$$ \mathbb{P}_v \left( \lim_{n \to +\infty} \frac{l_n}{n} \to \alpha^{-1} = (\mathbb{E}_A(\tau))^{-1} \right) = 1. \qquad (A1) $$

We have

$$ n^{-2m} \mathbb{E}_v \left[ \left( \sum_{1 \le i_1 < \dots < i_{m-1} \le l_n - 1} \omega_{\tilde{h}}(\mathbf{B}_0, \mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1}}) \right)^2 \right] $$
where

$$ (I) = \frac{1}{\binom{n}{m}} \left[ \sum_{1 \le i_1 < \dots < i_{m-1} \le l_n - 1} \omega_{\tilde{h}}(\mathbf{B}_0, \mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1}}) \right. \\ \left. + \sum_{1 \le i_1 < \dots < i_{m-1} \le l_n - 1} \omega_{\tilde{h}}(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1}}, \mathbf{B}_{l_n}) \right], $$

$$ (II) = \frac{1}{\binom{n}{m}} \left[ \sum_{j=2}^{m} \sum_{0 \le k < i_1 < \dots < i_{m-1-j} \le l_n} \omega_{\tilde{h}}(\mathbf{B}_{k}, \dots, \mathbf{B}_{k'}, \mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1-j}}) \right. \\ \left. - \sum_{j=2}^{m} \sum_{1 \le k < i_1 < \dots < i_{m-1-j} \le n} \tilde{h}(X_k, \dots, X_{k'}, X_{i_1}, \dots, X_{i_m}) \right] \\ = \frac{1}{\binom{n}{m}} \left[ \sum_{(i_m^n)^c} \omega_{\tilde{h}}(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) - \sum_{(I_m^n)^c} \tilde{h}(X_{i_1}, \dots, X_{i_m}) \right], $$

where

$$ (I_m^s)^c = \{(i_1, \dots, i_m) : i_j \in \mathbb{N}, 1 \le i_j \le n; \text{ at least there are } j \text{ and } k \text{ such that } i_j = i_k\}, $$

the complement of index set, with cardinal equal to $\binom{s+m-1}{m} - \binom{s}{m} := \overline{\binom{s}{m}}$. To prove the convergence of $W_n(h)$ to zero in probability, we must fulfill the convergence of (I) and (II) to zero in probability.

$$ \mathfrak{A} = \overline{\binom{n}{m}}^{-1} \sum_{\binom{l_m^n-1}{m}} \omega_{\tilde{h}}(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) \\ \xrightarrow{n \to \infty} \alpha^{-m} \left[ \mathbb{E}(\omega_h(\mathbf{B}_1, \underbrace{\mathbf{B}_{k'}, \dots, \mathbf{B}_{k_{k+u}}, \dots, \mathbf{B}_m}_{\text{u times}})) - \mathbb{E}_A((\tau)^u) (\mathbb{E}_A(\tau))^{m-u} \mu(h) \right], $$

where $1 \le k \le m$ and $1 \le u \le k$. We apply the SLLN for Harris Markov chains to find the convergence of

$$ \mathfrak{B} = \overline{\binom{n}{m}}^{-1} \sum_{(I_m^n)^c} \tilde{h}(X_{i_1}, \dots, X_{i_m}), $$

to

$$ \int \dots \int h(x_1, \underbrace{x_k, \dots, x_k}_{\text{u times}}, x_{k+u}, \dots, x_m) d\mu(x_1) d\mu^u(x_1) d\mu(x_{k+u}) \dots d\mu(x_m) - \mu(h). $$

Using the conditions, all terms in $\mathfrak{A}$ and $\mathfrak{B}$ are finite and we can prove the convergence of (II) to zero. Now, for (I), applying the SLLN and by Lemma 3.2 in [47] part i), we can see that

$$ \mathbb{P}_v \left( \lim_{n \to +\infty} \frac{l_n}{n} \to \alpha^{-1} = (\mathbb{E}_A(\tau))^{-1} \right) = 1. \qquad (A1) $$

We have

$$ n^{-2m} \mathbb{E}_v \left[ \left( \sum_{1 \le i_1 < \dots < i_{m-1} \le l_n - 1} \omega_{\tilde{h}}(\mathbf{B}_0, \mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1}}) \right)^2 \right] $$
$$ \leq 2\alpha^{-2m} \left\{ \mathbb{E}_v \left[ \left( \omega_{|h|} (\mathbf{B}_0, \mathbf{B}_1, \dots, \mathbf{B}_{m-1}) \right)^2 \right] + \mathbb{E}_v (\tau_0) (\mathbb{E}_A (\tau)^{m-1} \mu(h))^2 \right\} < \infty. $$

We obtain, in turn, that

$$ \begin{aligned} & n^{-2m} \mathbb{E}_v \left[ \left( \sum_{1 \le i_1 < \dots < i_{m-1} \le l_n} \omega_{\tilde{h}} (\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1}}, \mathbf{B}_{l_n}) \right)^2 \right] \\ & \le 2\alpha^{-2m} \left\{ \mathbb{E}_v \left[ \left( \omega_{|h|} (\mathbf{B}_1, \dots, \mathbf{B}_{m-1}, \mathbf{B}_{l_n}) \right)^2 \right] + (\mathbb{E}_A (\tau))^m \mu(|h|)^2 \right\} < \infty. \end{aligned} $$

Hence, (I) also converges to zero a.s under $\mathbb{P}_v$ as $n \to \infty$. $\square$

**Proof of Theorem 1.** In what follows, let $L = l_n - 1$ denote the number of blocks observed. We find that

$$ R_L(h) = S_L(h) + D_L(h), $$

where

$$ \begin{aligned} S_L(h) &= \frac{m}{L} \sum_{i=1}^{L} \tilde{h}^{(1)}(\mathbf{B}_i), \\ D_L(h) &= \sum_{j=2}^{m} \binom{m}{j} \binom{L}{j}^{-1} \sum_{1 \le i_1 < \dots < i_j \le L} \tilde{h}^{(j)}(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_j}), \end{aligned} $$

where $\tilde{h}^{(c)}(\cdot)$ represents the conditional expectation of $\omega_{\tilde{h}}(\cdot)$ given the $c$ of the coordinates, for all $\mathbf{B}_c \in \mathbb{T}$. The U-statistics $D_L(h)$ is obtained by truncating the Hoeffding decomposition after the first term $S_L(h)$. Then, we just need to show that:

1. $L^{1/2}S_L(h)$ converges weakly to a Gaussian process $G_{\mathbb{P}}$ on $l^\infty(\mathcal{F})$,

2. $\|L^{-m+1/2}D_L(h)\|_{\mathcal{F}} \to 0$.

For $\mathbb{P}_L(h_1) := \frac{1}{L} \sum_{i=1}^L \tilde{h}_1(\mathbf{B}_i)$, introduce

$$ Z_L(h) = \sqrt{L}(\mathbb{P}_L(h_1) - \mathbb{P}(h_1)) = \frac{1}{\sqrt{L}} \sum_{i=1}^{L} (\mathbb{P}_L(\mathbb{P}^{m-1}(h)) - \mathbb{P}^m(h)). $$

Using (A1), we can replace the random variable $L = l_n - 1$ with the deterministic quantity $\tilde{L}$ and we write

$$ Z_L(h) = \frac{1}{\sqrt{\tilde{L}}} \sum_{i=1}^{\tilde{L}} (\mathbb{P}_L(\mathbb{P}^{m-1}(h)) - \mathbb{P}^m(h)) + o_{\mathbb{P}}, $$

where $\tilde{L} = 1 + \lfloor \frac{n}{\mathbb{E}_A(\tau)} \rfloor$. In order to establish the weak convergence for the empirical process $Z_{\tilde{L}}(h)$, it is sufficient and necessary to prove the finite dimensional convergence and the stochastic equicontinuity. For the finite multidimensional convergence, we have to prove that $(Z_L(h_{i_1}), \dots, Z_L(h_{i_k}))$ converges weakly to $(G(h_{i_1}), \dots, G(h_{i_k}))$, for every fixed finite collection of functions

$$ \{h_{i_1}, \dots, h_{i_k}\} \subset \mathcal{F}. $$

In order to fix this, it is enough to show that for every fixed $a_1, \dots, a_k \in \mathbb{R}$,

$$ \sum_{j=1}^{k} a_j Z_L(h_{i_j}) \to N(0, \sigma^2), \text{ in distribution,} $$
$$ \leq 2\alpha^{-2m} \left\{ \mathbb{E}_{\nu} \left[ \left( \omega_{|h|} (\mathbf{B}_0, \mathbf{B}_1, \dots, \mathbf{B}_{m-1}) \right)^2 \right] + \mathbb{E}_{\nu}(\tau_0) (\mathbb{E}_A(\tau)^{m-1} \mu(h))^2 \right\} < \infty. $$

We obtain, in turn, that

$$ \begin{aligned} & n^{-2m} \mathbb{E}_{\nu} \left[ \left( \sum_{1 \le i_1 < \dots < i_{m-1} \le l_n} \omega_{\tilde{h}} (\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_{m-1}}, \mathbf{B}_{l_n}) \right)^2 \right] \\ & \le 2\alpha^{-2m} \left\{ \mathbb{E}_{\nu} \left[ \left( \omega_{|h|} (\mathbf{B}_1, \dots, \mathbf{B}_{m-1}, \mathbf{B}_{l_n}) \right)^2 \right] + (\mathbb{E}_A(\tau))^m \mu(|h|)^2 \right\} < \infty. \end{aligned} $$

Hence, (I) also converges to zero a.s under $P_{\nu}$ as $n \to \infty$. $\square$

**Proof of Theorem 1.** In what follows, let $L = l_n - 1$ denote the number of blocks observed. We find that

$$ R_L(h) = S_L(h) + D_L(h), $$

where

$$ S_L(h) = \frac{m}{L} \sum_{i=1}^{L} \tilde{h}^{(1)}(\mathbf{B}_i), $$

$$ D_L(h) = \sum_{j=2}^{m} \binom{m}{j} \binom{L}{j}^{-1} \sum_{1 \le i_1 < \dots < i_j \le L} \tilde{h}^{(j)}(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_j}), $$

where $\tilde{h}^{(c)}(\cdot)$ represents the conditional expectation of $\omega_{\tilde{h}}(\cdot)$ given the $c$ of the coordinates, for all $\mathbf{B}_c \in \mathbb{T}$. The U-statistics $D_L(h)$ is obtained by truncating the Hoeffding decomposition after the first term $S_L(h)$. Then, we just need to show that:

1. $L^{1/2}S_L(h)$ converges weakly to a Gaussian process $G_{\mathbb{T}}$ on $l^{\infty}(\mathcal{F})$,

2. $\|L^{-m+1/2}D_L(h)\|_{\mathcal{F}} \to 0$.

For $\mathbb{P}_L(h_1) := \frac{1}{L} \sum_{i=1}^L \tilde{h}_1(\mathbf{B}_i)$, introduce

$$ Z_L(h) = \sqrt{L}(\mathbb{P}_L(h_1) - \mathbb{P}(h_1)) = \frac{1}{\sqrt{L}} \sum_{i=1}^{L} (\mathbb{P}_L(\mathbb{P}^{m-1}(h)) - \mathbb{P}^m(h)). $$

Using (A1), we can replace the random variable $L = l_n - 1$ with the deterministic quantity $\tilde{L}$ and we write

$$ Z_L(h) = \frac{1}{\sqrt{\tilde{L}}} \sum_{i=1}^{\tilde{L}} (\mathbb{P}_L(\mathbb{P}^{m-1}(h)) - \mathbb{P}^m(h)) + o_{\mathbb{P}}, $$

where $\tilde{L} = 1 + \lfloor \frac{n}{\mathbb{E}_A(\tau)} \rfloor$. In order to establish the weak convergence for the empirical process $Z_{\tilde{L}}(h)$, it is sufficient and necessary to prove the finite dimensional convergence and the stochastic equicontinuity. For the finite multidimensional convergence, we have to prove that $(Z_{\tilde{L}}(h_{i_1}), \dots, Z_{\tilde{L}}(h_{i_k}))$ converges weakly to $(G(h_{i_1}), \dots, G(h_{i_k}))$, for every fixed finite collection of functions

$$ \{h_{i_1}, \dots, h_{i_k}\} \subset \mathcal{F}. $$

In order to fix this, it is enough to show that for every fixed $a_1, \dots, a_k \in \mathbb{R}$,

$$ \sum_{j=1}^{k} a_j Z_L(h_{i_j}) \to N(0, \sigma^2), \text{ in distribution,} $$
where

$$ \sigma^2 = \sum_{j=1}^{k} a_j^2 \mathrm{Var}(Z_{\tilde{L}}(h_{i_j})) + \sum_{s \neq r} a_j a_i \mathrm{Cov}(Z_{\tilde{L}}(h_{i_s}), Z_{\tilde{L}}(h_{i_r})). $$

By linearity, and in the same footsteps of the arguments of ([57], Chapter 17), we can prove that

$$ \frac{1}{\sqrt{n}} \sum_{j=1}^{\tilde{L}} \tilde{h}_1(\mathbf{B}_j) \to N(0, \gamma_{\tilde{h}_1}^2), $$

where, under Condition (C5),

$$ \gamma_{\tilde{h}_1}^2 = \alpha \mathbb{E}_A (\tilde{h}_1^2(\mathbf{B}_1)). $$

We readily infer that we have

$$ \sqrt{LS_L}(h) \to N(0, m^2 \mathbb{E}_A (\tilde{h}_1^2(\mathbf{B}_1))). $$

Now, to verify the equicontinuity, we need to check that for every $\epsilon > 0$,

$$ \lim_{\delta \to 0} \lim_{n \to \infty} \mathbb{P} \left( \sup_{d(f,g) \le \delta} |Z_L(f) - Z_L(g)| > \epsilon \right) = 0, $$

where $d(\cdot, \cdot)$ is a pseudo distance for which the class $\mathcal{F}$ is totally bounded, and $f, g$ belong to $\mathcal{F}$. According to [72], we have

$$ \begin{aligned} |Z_L(f - g)| &= \left| \frac{1}{\sqrt{L}} \left[ \sum_{k=1}^{L} (f - g)(\mathbf{B}_k) - \mathbb{P}^m(f - g) \right] \right| \\ &\le \left| \frac{1}{\sqrt{L}} \sum_{a \le k \le b} ((f - g)(\mathbf{B}_k) - \mathbb{P}^m(f - g)) \right| \\ &\quad + \left| \frac{1}{\sqrt{L}} \sum_{1 \le k \le \lfloor n/\mathbb{E}(\tau) \rfloor} ((f - g)(\mathbf{B}_k) - \mathbb{P}^m(f - g)) \right|, \end{aligned} $$

where $a = \min(L, \lfloor n/\mathbb{E}(\tau) \rfloor)$ and $b = \max(L, \lfloor n/\mathbb{E}(\tau) \rfloor)$. For the left-hand part in the last inequality, we have

$$ \left| \sum_{a \le k \le b} ((f - g)(\mathbf{B}_k) - \mathbb{P}^m(f - g)) \right| \le 2 \sup_{f \in \mathcal{F}} \max_{1 \le s \le n} \left\{ 2 \left| \sum_{1 \le k \le s} (f(\mathbf{B}_k) - \mathbb{P}^m(f)) \right| \right\}. $$

Dividing the last inequality by $L^{1/2}$ and using the convergence result in ([72] Lemma 2.11) with Condition (C1), we obtain the desired result. The right-hand part in the inequality is treated using ([72] Lemma 4.2) providing that $\mathbb{E}_A(\tau)^{2+\alpha} < \infty$, for $\alpha > 0$ and the hypothesis of a finite uniform entropy integral. To complete the weak convergence of the regenerative U-statistic, we must treat the remaining terms of its Hoeffding decomposition. For $\zeta \in \mathcal{F}$, let us introduce

$$ \zeta := \omega_{\tilde{h}}(\mathbf{B}_1, \dots, \mathbf{B}_m) - \mathbb{P}^m(h) - \sum_{i=1}^{m} \tilde{h}^{(1)}(\mathbf{B}_i), \quad \mathbf{B}_i \in \mathbb{T}. \quad (A2) $$

Once can see that $\zeta$ is centered, that is

$$ \int \zeta(\mathbf{B}_1, \dots, \mathbf{B}_m) dP(\mathbf{B}_1) \dots dP(\mathbf{B}_i) \dots dP(\mathbf{B}_m) = 0. \quad (A3) $$
where

$$ \sigma^2 = \sum_{j=1}^{k} a_j^2 \mathrm{Var}(Z_{\tilde{L}}(h_{i_j})) + \sum_{s \neq r} a_j a_i \mathrm{Cov}(Z_{\tilde{L}}(h_{i_s}), Z_{\tilde{L}}(h_{i_r})). $$

By linearity, and in the same footsteps of the arguments of ([57], Chapter 17), we can prove that

$$ \frac{1}{\sqrt{n}} \sum_{j=1}^{\tilde{L}} \tilde{h}_1(\mathbf{B}_j) \to N(0, \gamma_{\tilde{h}_1}^2), $$

where, under Condition (C5),

$$ \gamma_{\tilde{h}_1}^2 = \alpha \mathbb{E}_A(\tilde{h}_1^2(\mathbf{B}_1)). $$

We readily infer that we have

$$ \sqrt{LS_L}(h) \to N(0, m^2 \mathbb{E}_A(\tilde{h}_1^2(\mathbf{B}_1))). $$

Now, to verify the equicontinuity, we need to check that for every $\epsilon > 0$,

$$ \lim_{\delta \to 0} \lim_{n \to \infty} \mathbb{P} \left( \sup_{d(f,g) \le \delta} |Z_L(f) - Z_L(g)| > \epsilon \right) = 0, $$

where $d(\cdot, \cdot)$ is a pseudo distance for which the class $\mathcal{F}$ is totally bounded, and $f, g$ belong to $\mathcal{F}$. According to [72], we have

$$ \begin{aligned} |Z_L(f - g)| &= \left| \frac{1}{\sqrt{L}} \left[ \sum_{k=1}^{L} (f - g)(\mathbf{B}_k) - \mathbb{P}^m(f - g) \right] \right| \\ &\le \left| \frac{1}{\sqrt{L}} \sum_{a \le k \le b} ((f - g)(\mathbf{B}_k) - \mathbb{P}^m(f - g)) \right| \\ &\quad + \left| \frac{1}{\sqrt{L}} \sum_{1 \le k \le \lfloor n/\mathbb{E}(\tau) \rfloor} ((f - g)(\mathbf{B}_k) - \mathbb{P}^m(f - g)) \right|, \end{aligned} $$

where $a = \min(L, \lfloor n/\mathbb{E}(\tau) \rfloor)$ and $b = \max(L, \lfloor n/\mathbb{E}(\tau) \rfloor)$. For the left-hand part in the last inequality, we have

$$ \left| \sum_{a \le k \le b} ((f - g)(\mathbf{B}_k) - \mathbb{P}^m(f - g)) \right| \le 2 \sup_{f \in \mathcal{F}} \max_{1 \le s \le n} \left\{ 2 \left| \sum_{1 \le k \le s} (f(\mathbf{B}_k) - \mathbb{P}^m(f)) \right| \right\}. $$

Dividing the last inequality by $L^{1/2}$ and using the convergence result in ([72] Lemma 2.11) with Condition (C1), we obtain the desired result. The right-hand part in the inequality is treated using ([72] Lemma 4.2) providing that $\mathbb{E}_A(\tau)^{2+\alpha} < \infty$, for $\alpha > 0$ and the hypothesis of a finite uniform entropy integral. To complete the weak convergence of the regenerative U-statistic, we must treat the remaining terms of its Hoeffding decomposition. For $\zeta \in \mathcal{F}$, let us introduce

$$ \zeta := \omega_{\tilde{h}}(\mathbf{B}_1, \dots, \mathbf{B}_m) - \mathbb{P}^m(h) - \sum_{i=1}^{m} \tilde{h}^{(1)}(\mathbf{B}_i), \quad \mathbf{B}_i \in \mathbb{T}. \quad (A2) $$

Once can see that $\zeta$ is centered, that is

$$ \int \zeta(\mathbf{B}_1, \dots, \mathbf{B}_m) dP(\mathbf{B}_1) \dots dP(\mathbf{B}_i) \dots dP(\mathbf{B}_m) = 0. \quad (A3) $$
By the randomization theorem, according to [7] (for $r = 2$):

$$ \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le L} \zeta(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) \right\| = \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le L} \varepsilon_{i_1}^{(1)} \varepsilon_{i_2}^{(2)} \zeta(\mathbf{B}_{i_1}^{(1)}, \dots, \mathbf{B}_{i_m}^{(1)}) \right\|. $$

Hence, for $\mathcal{C}$ a constant:

$$ \mathbb{E} \left\| \mathcal{L}^{-1/2} \sum_{1 \le i_1 < \dots < i_m \le L} \zeta(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) \right\|_{\mathcal{F}} \le \mathcal{C} \mathbb{E} \int_0^{\infty} \mathcal{L}^{-1/2} \log N_{n,2}(\varepsilon, \mathcal{F}) d\varepsilon. $$

It is sufficient now to use the theorem hypothesis of a uniform entropy integral to complete the proof of the theorem. $\square$

**Proof of Theorem 2.** We have

$$ \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \zeta_{i_1} \dots \zeta_{i_m} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}. $$

By decoupling of the $U$-process, due to [6],

$$ \le C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \zeta_{i_1} \dots \zeta_{i_m} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}. $$

By symmetrization, due to [6], we have

$$ \le 2^m C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \zeta_{i_1} \dots \zeta_{i_m} \varepsilon_{i_1}^{(1)} \dots \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}. $$

for $(\operatorname{sgn}(\zeta_1)\varepsilon_1^*, \dots, \operatorname{sgn}(\zeta_n)\varepsilon_n^*)$ a sequence independent and with the same distribution as $(\zeta_1, \dots, \zeta_n)$. By the invariance of $(\mathbb{P}_\varepsilon \otimes \mathbb{P})^{mn}$ and the fact that $\zeta$ is independent of $X, \varepsilon$, we have that

$$ \begin{aligned} & = 2^m C_m \mathbb{E}_{x,\varepsilon} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} |\zeta_{i_1}| \dots |\zeta_{i_m}| \operatorname{sgn}(\zeta_{i_1}) \varepsilon_{i_1}^{(1)} \dots \operatorname{sgn}(\zeta_{i_m}) \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}} \\ & = 2^m C_m \mathbb{E}_{x,\varepsilon} \left\| \sum_{1 \le i_1 < \dots < i_m \le l_n-1} |\zeta_{i_1}| \dots |\zeta_{i_m}| \varepsilon_{i_1}^{(1)} \dots \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}, \end{aligned} $$

using the reversed order statistics of $\{|\zeta_i|\}_{i=1}^n, |\zeta_{(1)}| \ge \dots \ge |\zeta_{(n)}|$, and the permutations between the different sequences of random variables, and in the same footsteps as [46],

$$ = 2^m C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} |\zeta_{(i_1)}| \dots |\zeta_{(i_m)}| \varepsilon_{i_1}^{(1)} \dots \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}. $$

substituting $\zeta_{(i)}$ by $\sum_{k=i}^n \zeta_{(k)} - \zeta_{(k+1)}$, with $|\zeta_{(n+1)}| = 0$, we have

$$ \begin{aligned} & \le 2^m C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \sum_{k_j \ge i_j, 1 \le j \le r} (|\zeta_{(l_1)}| - |\zeta_{(l_1+1)}|) \dots (|\zeta_{(l_m)}| - |\zeta_{(l_m+1)}|) \right. \\ & \qquad \left. \times \varepsilon_{i_1}^{(1)} \dots \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}} \\ & \le 2^m C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \sum_{\ell_k \ge i_k, 1 \le k \le m} (|\zeta_{(\ell_1)}| - |\zeta_{(\ell_1+1)}|) \dots (|\zeta_{(\ell_m)}| - |\zeta_{(\ell_m+1)}|) \right\|. \end{aligned} $$
By the randomization theorem, according to [7] (for $r = 2$):

$$ \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le L} \zeta(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) \right\| = \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le L} \varepsilon_{i_1}^{(1)} \varepsilon_{i_2}^{(2)} \zeta(\mathbf{B}_{i_1}^{(1)}, \dots, \mathbf{B}_{i_m}^{(1)}) \right\|. $$

Hence, for $\mathcal{C}$ a constant:

$$ \mathbb{E} \left\| \mathcal{L}^{-1/2} \sum_{1 \le i_1 < \dots < i_m \le L} \zeta(\mathbf{B}_{i_1}, \dots, \mathbf{B}_{i_m}) \right\|_{\mathcal{F}} \le \mathcal{C} \mathbb{E} \int_0^{\infty} \mathcal{L}^{-1/2} \log N_{n,2}(\varepsilon, \mathcal{F}) d\varepsilon. $$

It is sufficient now to use the theorem hypothesis of a uniform entropy integral to complete the proof of the theorem. $\square$

**Proof of Theorem 2.** We have

$$ \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \zeta_{i_1} \dots \zeta_{i_m} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}. $$

By decoupling of the $U$-process, due to [6],

$$ \le C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \zeta_{i_1} \dots \zeta_{i_m} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}. $$

By symmetrization, due to [6], we have

$$ \le 2^m C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \zeta_{i_1} \dots \zeta_{i_m} \varepsilon_{i_1}^{(1)} \dots \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}. $$

for $(\operatorname{sgn}(\zeta_1)\varepsilon_1^*, \dots, \operatorname{sgn}(\zeta_n)\varepsilon_n^*)$ a sequence independent and with the same distribution as $(\zeta_1, \dots, \zeta_n)$. By the invariance of $(\mathbb{P}_\varepsilon \otimes \mathbb{P})^{mn}$ and the fact that $\zeta$ is independent of $X, \varepsilon$, we have that

$$ \begin{aligned} & = 2^m C_m \mathbb{E}_{x,\varepsilon} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} |\zeta_{i_1}| \dots |\zeta_{i_m}| \operatorname{sgn}(\zeta_{i_1}) \varepsilon_{i_1}^{(1)} \dots \operatorname{sgn}(\zeta_{i_m}) \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}} \\ & = 2^m C_m \mathbb{E}_{x,\varepsilon} \left\| \sum_{1 \le i_1 < \dots < i_m \le l_n-1} |\zeta_{i_1}| \dots |\zeta_{i_m}| \varepsilon_{i_1}^{(1)} \dots \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}, \end{aligned} $$

using the reversed order statistics of $\{|\zeta_i|\}_{i=1}^n, |\zeta_{(1)}| \ge \dots \ge |\zeta_{(n)}|$, and the permutations between the different sequences of random variables, and in the same footsteps as [46],

$$ = 2^m C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} |\zeta_{(i_1)}| \dots |\zeta_{(i_m)}| \varepsilon_{i_1}^{(1)} \dots \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}}. $$

substituting $\zeta_{(i)}$ by $\sum_{k=i}^n \zeta_{(k)} - \zeta_{(k+1)}$, with $|\zeta_{(n+1)}| = 0$, we have

$$ \begin{aligned} & \le 2^m C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \sum_{k_j \ge i_j, 1 \le j \le r} (|\zeta_{(l_1)}| - |\zeta_{(l_1+1)}|) \dots (|\zeta_{(l_m)}| - |\zeta_{(l_m+1)}|) \right. \\ & \qquad \left. \times \varepsilon_{i_1}^{(1)} \dots \varepsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}} \\ & \le 2^m C_m \mathbb{E} \left\| \sum_{1 \le i_1 < \dots < i_m \le n} \sum_{\ell_k \ge i_k, 1 \le k \le m} (|\zeta_{(\ell_1)}| - |\zeta_{(\ell_1+1)}|) \dots (|\zeta_{(\ell_m)}| - |\zeta_{(\ell_m+1)}|) \right\|. \end{aligned} $$
$$
\begin{aligned}
& \times \epsilon_{i_1}^{(1)} \cdots \epsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \Big\|_{\mathcal{F}} \\
\le & 2^m \mathbf{C}_m \mathbb{E} \left[ \sum_{1 \le \ell_1, \dots, \ell_m \le n} (|\xi_{(\ell_1)}| - |\xi_{(\ell_1+1)}|) \cdots (|\xi_{(\ell_m)}| - |\xi_{(\ell_m+1)}|) \right. \\
& \left. \times \mathbb{E} \left\| \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \epsilon_{i_1}^{(1)} \cdots \epsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}} \right] \\
\le & 2^m \mathbf{C}_m \mathbb{E} \left[ \sum_{1 \le \ell_1, \dots, \ell_m \le n} \int_{|\xi_{(\ell_1+1)}|}^{|\xi_{(\ell_1)}|} \cdots \int_{|\xi_{(\ell_m+1)}|}^{|\xi_{(\ell_m)}|} \psi_n(\ell_1, \dots, \ell_m) dt_m \cdots dt_1 \right] \\
\le & 2^m \mathbf{C}_m \mathbb{E} \left[ \sum_{1 \le \ell_1, \dots, \ell_m \le n} \int_{|\xi_{(\ell_1+1)}|}^{|\xi_{(\ell_1)}|} \cdots \int_{|\xi_{(\ell_m+1)}|}^{|\xi_{(\ell_m)}|} \psi_n(|i: |\xi_i| > t_1|, \dots, |i: |\xi_i| > t_m|) dt_m \cdots dt_1 \right] \\
\le & 2^m \mathbf{C}_m \mathbb{E} \left[ \int_{\mathbb{R}_{\ge 0}^m} \psi_n(|i: |\xi_i| > t_1|, \dots, |i: |\xi_i| > t_m|) dt_1 \cdots dt_m \right] \\
\le & 2^m \mathbf{C}_m \int_{\mathbb{R}_{\ge 0}^m} \mathbb{E} \psi_n \left( \sum_{i=1}^n 1_{|\xi_i| > t_1}, \dots, \sum_{i=1}^n 1_{|\xi_i| > t_m} \right) dt_1 \cdots dt_m. \quad (\text{By Fubini's theorem.})
\end{aligned}
$$

Now, suppose that $\psi_n(\ell_1, \dots, \ell_m) = \bar{\psi}_n(\prod_{k=1}^m \ell_k)$. Then, we may further bound the above equation by

$$
\begin{aligned}
& \int_{\mathbb{R}_{\ge 0}^m} \mathbb{E} \bar{\psi}_n \left( \prod_{k=1}^m \sum_{i=1}^n 1_{|\bar{\xi}_i| > t_k} \right) dt_1 \dots dt_m \\
&= \int_{\mathbb{R}_{\ge 0}^m} \mathbb{E} \bar{\psi}_n \left( \sum_{1 \le i_1, \dots, i_m \le n} \prod_{k=1}^m 1_{|\bar{\xi}_{i_k}| > t_k} \right) dt_1 \dots dt_m \\
&\le \int_{\mathbb{R}_{\ge 0}^m} \bar{\psi}_n \left( \sum_{1 \le i_1, \dots, i_m \le n} \mathbb{E} \prod_{k=1}^m 1_{|\bar{\xi}_{i_k}| > t_k} \right) dt_1 \dots dt_m \quad (\text{by Jensen's inequality}) \\
&\le \int_{\mathbb{R}_{\ge 0}^m} \bar{\psi}_n \left( \sum_{1 \le i_1, \dots, i_m \le n} \prod_{k=1}^m P(|\bar{\xi}_{i_k}| > t_k)^{1/m} \right) dt_1 \dots dt_m,
\end{aligned}
$$

where the last inequality follows from the generalized Hölder inequality and the assumption that $\bar{\psi}_n$ is nondecreasing. $\square$

**Proof of Lemma 2.** For

$$
\psi_n(\ell_1, \dots, \ell_m) \equiv a(\ell_1, \dots, \ell_m) \left( \prod_{k=1}^{m} \ell_k \right)^{1/2},
$$

Theorem 2 implies that:

$$
\begin{aligned}
& \left\| \sum_{1 \le i_1, \dots, i_m \le n} \bar{\xi}_{i_1} \cdots \bar{\xi}_{i_m} f(Y_{i_1}, \dots, Y_{i_m}) \right\|_{\mathcal{F}_{(n, \dots, n), n}} \\
\le & K_m \int_{\mathbb{R}_{\ge 0}^m} \mathbb{E} \left[ a \left( \sum_{i=1}^n 1_{|\bar{\xi}_i| > t_1}, \dots, \sum_{i=1}^n 1_{|\bar{\xi}_i| > t_m} \right) \prod_{k=1}^m \left( \sum_{i=1}^n 1_{|\bar{\xi}_i| > t_k} \right)^{1/2} \right] dt_1 \cdots dt_m \\
\le & K_m \int_{\mathbb{R}_{\ge 0}^m} A_{2,n}(t_1, \dots, t_m) \left\{ \mathbb{E} \prod_{k=1}^m \sum_{i=1}^n 1_{|\bar{\xi}_i| > t_k} \right\}^{1/2} dt_1 \cdots dt_m
\end{aligned}
$$
$$
\begin{aligned}
& \times \epsilon_{i_1}^{(1)} \cdots \epsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \Big\|_{\mathcal{F}} \\
\le & 2^m \mathbf{C}_m \mathbb{E} \left[ \sum_{1 \le \ell_1, \dots, \ell_m \le n} (|\xi_{(\ell_1)}| - |\xi_{(\ell_1+1)}|) \cdots (|\xi_{(\ell_m)}| - |\xi_{(\ell_m+1)}|) \right. \\
& \left. \times \mathbb{E} \left\| \sum_{1 \le i_k \le \ell_k, 1 \le k \le m} \epsilon_{i_1}^{(1)} \cdots \epsilon_{i_m}^{(m(n))} f(Y_{i_1}^{(1)}, \dots, Y_{i_m}^{(m)}) \right\|_{\mathcal{F}} \right] \\
\le & 2^m \mathbf{C}_m \mathbb{E} \left[ \sum_{1 \le \ell_1, \dots, \ell_m \le n} \int_{|\xi_{(\ell_1+1)}|}^{|\xi_{(\ell_1)}|} \cdots \int_{|\xi_{(\ell_m+1)}|}^{|\xi_{(\ell_m)}|} \psi_n(\ell_1, \dots, \ell_m) dt_m \cdots dt_1 \right] \\
\le & 2^m \mathbf{C}_m \mathbb{E} \left[ \sum_{1 \le \ell_1, \dots, \ell_m \le n} \int_{|\xi_{(\ell_1+1)}|}^{|\xi_{(\ell_1)}|} \cdots \int_{|\xi_{(\ell_m+1)}|}^{|\xi_{(\ell_m)}|} \psi_n(|\{i : |\xi_i| > t_1\}|, \dots, |\{i : |\xi_i| > t_m\}|) dt_m \cdots dt_1 \right] \\
\le & 2^m \mathbf{C}_m \mathbb{E} \left[ \int_{\mathbb{R}_{\ge 0}^m} \psi_n(|\{i : |\xi_i| > t_1\}|, \dots, |\{i : |\xi_i| > t_m\}|) dt_1 \cdots dt_m \right] \\
\le & 2^m \mathbf{C}_m \int_{\mathbb{R}_{\ge 0}^m} \mathbb{E} \psi_n \left( \sum_{i=1}^n 1_{|\xi_i| > t_1}, \dots, \sum_{i=1}^n 1_{|\xi_i| > t_m} \right) dt_1 \cdots dt_m. \quad (\text{By Fubini's theorem.})
\end{aligned}
$$

Now, suppose that $\psi_n(\ell_1, \dots, \ell_m) = \bar{\psi}_n(\prod_{k=1}^m \ell_k)$. Then, we may further bound the above equation by

$$
\begin{aligned}
& \int_{\mathbb{R}_{\ge 0}^m} \mathbb{E} \bar{\psi}_n \left( \prod_{k=1}^m \sum_{i=1}^n 1_{|\xi_i| > t_k} \right) dt_1 \dots dt_m \\
&= \int_{\mathbb{R}_{\ge 0}^m} \mathbb{E} \bar{\psi}_n \left( \sum_{1 \le i_1, \dots, i_m \le n} \prod_{k=1}^m 1_{|\xi_{i_k}| > t_k} \right) dt_1 \dots dt_m \\
&\le \int_{\mathbb{R}_{\ge 0}^m} \bar{\psi}_n \left( \sum_{1 \le i_1, \dots, i_m \le n} \mathbb{E} \prod_{k=1}^m 1_{|\xi_{i_k}| > t_k} \right) dt_1 \dots dt_m \quad (\text{by Jensen's inequality}) \\
&\le \int_{\mathbb{R}_{\ge 0}^m} \bar{\psi}_n \left( \sum_{1 \le i_1, \dots, i_m \le n} \prod_{k=1}^m P(|\xi_{i_k}| > t_k)^{1/m} \right) dt_1 \dots dt_m,
\end{aligned}
$$

where the last inequality follows from the generalized Hölder inequality and the assumption that $\bar{\psi}_n$ is nondecreasing. $\square$

**Proof of Lemma 2.** For

$$ \psi_n(\ell_1, \dots, \ell_m) \equiv a(\ell_1, \dots, \ell_m) \left( \prod_{k=1}^{m} \ell_k \right)^{1/2}, $$

Theorem 2 implies that:

$$
\begin{aligned}
& \left\| \sum_{1 \le i_1, \dots, i_m \le n} \xi_{i_1} \cdots \xi_{i_m} f(Y_{i_1}, \dots, Y_{i_m}) \right\|_{\mathcal{F}_{(n, \dots, n), n}} \\
\le & K_m \int_{\mathbb{R}_{\ge 0}^m} \mathbb{E} \left[ a \left( \sum_{i=1}^n 1_{|\xi_i| > t_1}, \dots, \sum_{i=1}^n 1_{|\xi_i| > t_m} \right) \prod_{k=1}^m \left( \sum_{i=1}^n 1_{|\xi_i| > t_k} \right)^{1/2} \right] dt_1 \cdots dt_m \\
\le & K_m \int_{\mathbb{R}_{\ge 0}^m} A_{2,n}(t_1, \dots, t_m) \left\{ \mathbb{E} \prod_{k=1}^m \sum_{i=1}^n 1_{|\xi_i| > t_k} \right\}^{1/2} dt_1 \cdots dt_m
\end{aligned}
$$
$$ 
\begin{aligned}
&\leq K_m \int_{\mathbb{R}_{\geq 0}^m} A_{2,n}(t_1, \dots, t_m) \left( \sum_{1 \leq i_1, \dots, i_m \leq n} \prod_{k=1}^m \mathbb{P}(|\xi_{i_k}| > t_k)^{1/m} \right)^{1/2} dt_1 \cdots dt_m \\
&= n^{m/2} K_m \int_{\mathbb{R}_{\geq 0}^m} A_{2,n}(t_1, \dots, t_m) \prod_{k=1}^m \mathbb{P}(|\xi_1| > t_k)^{1/2m} dt_1 \cdots dt_m.
\end{aligned}
$$

Here

$$ A_{2,n}(t_1, \dots, t_m) \equiv \left\{ \mathbb{E} \left[ a^2 \left( \sum_{i=1}^n \mathbf{1}_{|\xi_i| > t_1}, \dots, \sum_{i=1}^n \mathbf{1}_{|\xi_i| > t_m} \right) \right] \right\}^{1/2} \to 0 $$

as long as none of $\{\mathbb{P}(|\xi_1| > t_k) : 1 \leq k \leq m\}$ vanishes. The claim now follows from the dominated convergence theorem. □

**Corollary A1 ([6]).** Let $X(t), t \in T$, be a (weak) Gaussian or Rademacher chaos process of degree $m$ and let

$$ d_X(s,t) := [\mathbb{E}(X(t) - X(s))^2]^{1/2}, s, t \in T. $$

If

$$ \int_0^D (\log N(T, d_X, \varepsilon))^{m/2} d\varepsilon < \infty, $$

then there is a version of $X$, which we keep denoting $X$, with almost all of its sample paths in $C_u(T, d_X)$ and such that

$$ \left\| \sup_{t \in T} |X(t)| \right\|_{\psi_{2/m}} \leq \|X(t_0)\|_{\psi_2} + K \int_0^D (\log(N(T, d_X, \varepsilon)))^{m/2} d\varepsilon, $$

and

$$ \left\| \sup_{d_X(s,t) \leq \delta \atop s,t \in T} |X(t) - X(s)| \right\|_{\psi_{2/m}} \leq K \int_0^\delta (\log(N(T, d_X, \varepsilon)))^{m/2} d\varepsilon, $$

for all $0 < \delta \leq D$, where $K$ is a universal constant and $D$ is the diameter of $T$ for the pseudodistance $d_X$. In fact, every separable version of $X$ satisfies these properties.

**Theorem A1 ([73]).** For any random elements $Y_n$ with values in a metric space $(S, d)$, where $Y$ is measurable and has a separable range, the following are equivalent:

1. $Y_n$ converge in law to $Y$;

2. $d_{BL}(Y_n, Y) \to 0$ as $n \to \infty$;

## References

1. Halmos, P.R. The theory of unbiased estimation. *Ann. Math. Stat.* **1946**, *17*, 34–43. [CrossRef]

2. Hoeffding, W. A class of statistics with asymptotically normal distribution. *Ann. Math. Stat.* **1948**, *19*, 293–325. [CrossRef]

3. van der Vaart, A.W. *Asymptotic Statistics*; Cambridge Series in Statistical and Probabilistic Mathematics; Cambridge University Press: Cambridge, MA, USA, 1998; Volume 3, p. xvi+443.

4. Mises, R.V. On the asymptotic distribution of differentiable statistical functions. *Ann. Math. Stat.* **1947**, *18*, 309–348. [CrossRef]

5. Serfling, R.J. *Approximation Theorems of Mathematical Statistics*; John Wiley & Sons: Hoboken, NJ, USA, 2009; Volume 162.

6. de la Peña, V.H.; Giné, E. *Decoupling*; From dependence to independence, Randomly stopped processes. *U-statistics and processes*. Martingales and beyond; Probability and its Applications (New York); Springer: New York, NY, USA, 1999; p. xvi+392.

7. Arcones, M.A.; Giné, E. Limit theorems for *U*-processes. *Ann. Probab.* **1993**, *21*, 1494–1542. [CrossRef]

8. Lee, A.J. *U-statistics*; Volume 110, *Statistics: Textbooks and Monographs*; Theory and practice; Marcel Dekker, Inc.: New York, NY, USA, 1990; p. xii+302.

9. Stute, W. Almost sure representations of the product-limit estimator for truncated data. *Ann. Statist.* **1993**, *21*, 146–156. [CrossRef]

10. Arcones, M.A.; Wang, Y. Some new tests for normality based on *U*-processes. *Statist. Probab. Lett.* **2006**, *76*, 69–82. [CrossRef]
$$ 
\begin{aligned}
&\leq K_m \int_{\mathbb{R}_{\geq 0}^m} A_{2,n}(t_1, \dots, t_m) \left( \sum_{1 \leq i_1, \dots, i_m \leq n} \prod_{k=1}^m \mathbb{P}(|\xi_{i_k}| > t_k)^{1/m} \right)^{1/2} dt_1 \cdots dt_m \\
&= n^{m/2} K_m \int_{\mathbb{R}_{\geq 0}^m} A_{2,n}(t_1, \dots, t_m) \prod_{k=1}^m \mathbb{P}(|\xi_1| > t_k)^{1/2m} dt_1 \cdots dt_m.
\end{aligned}
$$

Here

$$ A_{2,n}(t_1, \dots, t_m) \equiv \left\{ \mathbb{E} \left[ a^2 \left( \sum_{i=1}^n \mathbf{1}_{|\xi_i| > t_1}, \dots, \sum_{i=1}^n \mathbf{1}_{|\xi_i| > t_m} \right) \right] \right\}^{1/2} \to 0 $$

as long as none of {$\mathbb{P}(|\xi_1| > t_k) : 1 \leq k \leq m$} vanishes. The claim now follows from the dominated convergence theorem. $\square$

**Corollary A1 ([6]).** Let $X(t), t \in T$, be a (weak) Gaussian or Rademacher chaos process of degree $m$ and let

$$ d_X(s,t) := [\mathbb{E}(X(t) - X(s))^2]^{1/2}, s, t \in T. $$

If

$$ \int_{0}^{D} (\log N(T, d_X, \varepsilon))^{m/2} d\varepsilon < \infty, $$

then there is a version of $X$, which we keep denoting $X$, with almost all of its sample paths in $C_u(T, d_X)$ and such that

$$ \left\| \sup_{t \in T} |X(t)| \right\|_{\psi_{2/m}} \leq \|X(t_0)\|_{\psi_2} + K \int_0^D (\log(N(T, d_X, \varepsilon)))^{m/2} d\varepsilon, $$

and

$$ \left\| \sup_{d_X(s,t) \leq \delta \atop s,t \in T} |X(t) - X(s)| \right\|_{\psi_{2/m}} \leq K \int_0^\delta (\log(N(T, d_X, \varepsilon)))^{m/2} d\varepsilon, $$

for all $0 < \delta \leq D$, where $K$ is a universal constant and $D$ is the diameter of $T$ for the pseudodistance $d_X$. In fact, every separable version of $X$ satisfies these properties.

**Theorem A1 ([73]).** For any random elements $Y_n$ with values in a metric space $(S, d)$, where $Y$ is measurable and has a separable range, the following are equivalent:

1. $Y_n$ converge in law to $Y$;

2. $d_{BL}(Y_n, Y) \to 0$ as $n \to \infty$;

## References

1. Halmos, P.R. The theory of unbiased estimation. *Ann. Math. Stat.* **1946**, *17*, 34–43. [CrossRef]

2. Hoeffding, W. A class of statistics with asymptotically normal distribution. *Ann. Math. Stat.* **1948**, *19*, 293–325. [CrossRef]

3. van der Vaart, A.W. *Asymptotic Statistics*; Cambridge Series in Statistical and Probabilistic Mathematics; Cambridge University Press: Cambridge, MA, USA, 1998; Volume 3, p. xvi+443.

4. Mises, R.V. On the asymptotic distribution of differentiable statistical functions. *Ann. Math. Stat.* **1947**, *18*, 309–348. [CrossRef]

5. Serfling, R.J. *Approximation Theorems of Mathematical Statistics*; John Wiley & Sons: Hoboken, NJ, USA, 2009; Volume 162.

6. de la Peña, V.H.; Giné, E. *Decoupling*; From dependence to independence, Randomly stopped processes. *U-statistics and processes*. Martingales and beyond; Probability and its Applications (New York); Springer: New York, NY, USA, 1999; p. xvi+392.

7. Arcones, M.A.; Giné, E. Limit theorems for *U*-processes. *Ann. Probab.* **1993**, *21*, 1494–1542. [CrossRef]

8. Lee, A.J. *U-statistics*; Volume 110, *Statistics: Textbooks and Monographs*; Theory and practice; Marcel Dekker, Inc.: New York, NY, USA, 1990; p. xii+302.

9. Stute, W. Almost sure representations of the product-limit estimator for truncated data. *Ann. Statist.* **1993**, *21*, 146–156. [CrossRef]

10. Arcones, M.A.; Wang, Y. Some new tests for normality based on *U*-processes. *Statist. Probab. Lett.* **2006**, *76*, 69–82. [CrossRef]
11. Giné, E.; Mason, D.M. Laws of the iterated logarithm for the local U-statistic process. *J. Theoret. Probab.* **2007**, *20*, 457–485.
[CrossRef]

12. Giné, E.; Mason, D.M. On local *U*-statistic processes and the estimation of densities of functions of several sample variables. *Ann. Statist.* **2007**, *35*, 1105–1145.
[CrossRef]

13. Schick, A.; Wang, Y.; Wefelmeyer, W. Tests for normality based on density estimators of convolutions. *Statist. Probab. Lett.* **2011**, *81*, 337–343.
[CrossRef]

14. Joly, E.; Lugosi, G. Robust estimation of *U*-statistics. *Stochastic Process. Appl.* **2016**, *126*, 3760–3773.
[CrossRef]

15. Lee, S.; Linton, O.; Whang, Y.J. Testing for stochastic monotonicity. *Econometrica* **2009**, *77*, 585–602.

16. Ghosal, S.; Sen, A.; van der Vaart, A.W. Testing monotonicity of regression. *Ann. Statist.* **2000**, *28*, 1054–1082.
[CrossRef]

17. Abrevaya, J.; Jiang, W. A nonparametric approach to measuring and testing curvature. *J. Bus. Econom. Statist.* **2005**, *23*, 1–19.
[CrossRef]

18. Nolan, D.; Pollard, D. *U*-processes: Rates of convergence. *Ann. Statist.* **1987**, *15*, 780–799.
[CrossRef]

19. Sherman, R.P. Maximal inequalities for degenerate *U*-processes with applications to optimization estimators. *Ann. Statist.* **1994**, *22*, 439–459.
[CrossRef]

20. Yoshihara, K.i. Limiting behavior of *U*-statistics for stationary, absolutely regular processes. *Z. Wahrscheinlichkeitstheorie und Verw. Gebiete* **1976**, *35*, 237–252.
[CrossRef]

21. Borovkova, S.; Burton, R.; Dehling, H. Limit theorems for functionals of mixing processes with applications to *U*-statistics and dimension estimation. *Trans. Amer. Math. Soc.* **2001**, *353*, 4261–4318.
[CrossRef]

22. Denker, M.; Keller, G. On *U*-statistics and *v*. Mises' statistics for weakly dependent processes. *Z. Wahrsch. Verw. Gebiete* **1983**, *64*, 505–522.
[CrossRef]

23. Leucht, A. Degenerate *U*- and *V*-statistics under weak dependence: Asymptotic theory and bootstrap consistency. *Bernoulli* **2012**, *18*, 552–585.
[CrossRef]

24. Leucht, A.; Neumann, M.H. Degenerate *U*- and *V*-statistics under ergodicity: Asymptotics, bootstrap and applications in statistics. *Ann. Inst. Statist. Math.* **2013**, *65*, 349–386.
[CrossRef]

25. Bouzebda, S.; Nemouchi, B. Weak-convergence of empirical conditional processes and conditional *U*-processes involving functional mixing data. In *Statistical Inference for Stochastic Processes*; Springer: New York, NY, USA, 2022; pp. 1–56.

26. Efron, B. Bootstrap methods: Another look at the jackknife. *Ann. Statist.* **1979**, *7*, 1–26.
[CrossRef]

27. Hall, P. *The Bootstrap and Edgeworth Expansion*; Springer Series in Statistics; Springer: New York, NY, USA, 1992; p. xiv+352.

28. Bickel, P.J.; Freedman, D.A. Some asymptotic theory for the bootstrap. *Ann. Statist.* **1981**, *9*, 1196–1217.
[CrossRef]

29. Arcones, M.A.; Giné, E. On the bootstrap of *U* and *V* statistics. *Ann. Statist.* **1992**, *20*, 655–674.
[CrossRef]

30. Dehling, H.; Mikosch, T. Random quadratic forms and the bootstrap for *U*-statistics. *J. Multivariate Anal.* **1994**, *51*, 392–413.
[CrossRef]

31. Leucht, A.; Neumann, M.H. Consistency of general bootstrap methods for degenerate *U*-type and *V*-type statistics. *J. Multivariate Anal.* **2009**, *100*, 1622–1633.
[CrossRef]

32. Politis, D.N.; Romano, J.P. A circular block-resampling procedure for stationary data. In *Exploring the Limits of Bootstrap* (East Lansing, MI, 1990); Wiley Ser. Probab. Math. Statist. Probab. Math. Statist.; Wiley: New York, NY, USA, 1992; pp. 263–270.

33. Carlstein, E. The use of subseries values for estimating the variance of a general statistic from a stationary sequence. *Ann. Statist.* **1986**, *14*, 1171–1179.
[CrossRef]

34. Politis, D.N.; Romano, J.P. The stationary bootstrap. *J. Amer. Statist. Assoc.* **1994**, *89*, 1303–1313.
[CrossRef]

35. Rubin, D.B. The Bayesian bootstrap. *Ann. Statist.* **1981**, *9*, 130–134.
[CrossRef]

36. Lo, A.Y. A Bayesian method for weighted sampling. *Ann. Statist.* **1993**, *21*, 2138–2148.
[CrossRef]

37. Mason, D.M.; Newton, M.A. A rank statistics approach to the consistency of a general bootstrap. *Ann. Statist.* **1992**, *20*, 1611–1624.
[CrossRef]

38. Præstgaard, J.; Wellner, J.A. Exchangeably weighted bootstraps of the general empirical process. *Ann. Probab.* **1993**, *21*, 2053–2086.
[CrossRef]

39. van der Vaart, A. New Donsker classes. *Ann. Probab.* **1996**, *24*, 2128–2140.
[CrossRef]

40. Alvarez-Andrade, S.; Bouzebda, S. Strong approximations for weighted bootstrap of empirical and quantile processes with applications. *Stat. Methodol.* **2013**, *11*, 36–52.
[CrossRef]

41. Bouzebda, S. On the strong approximation of bootstrapped empirical copula processes with applications. *Math. Methods Statist.* **2012**, *21*, 153–188.
[CrossRef]

42. Bouzebda, S.; Elhattab, I.; Ferfache, A.A. General *M*-Estimator Processes and their *m* out of *n* Bootstrap with Functional Nuisance Parameters. In *Methodology and Computing in Applied Probability*; Springer: New York, NY, USA, 2022; pp. 1–45.

43. Huskova, M.; Janssen, P. Consistency of the generalized bootstrap for degenerate U-statistics. *Ann. Stat.* **1993**, *21*, 1811–1823.
[CrossRef]

44. Janssen, P. Weighted bootstrapping of U-statistics. *J. Stat. Plan. Inference* **1994**, *38*, 31–41.
[CrossRef]

45. Alvarez-Andrade, S.; Bouzebda, S. Cramér's type results for some bootstrapped *U*-statistics. *Statist. Papers* **2020**, *61*, 1685–1699.
[CrossRef]

46. Han, Q. Multiplier *U*-processes: Sharp bounds and applications. *Bernoulli* **2022**, *28*, 87–124.
[CrossRef]

47. Radulović, D. Renewal type bootstrap for Markov chains. *Test* **2004**, *13*, 147–192.
[CrossRef]
11. Giné, E.; Mason, D.M. Laws of the iterated logarithm for the local U-statistic process. *J. Theoret. Probab.* **2007**, *20*, 457–485.
[CrossRef]

12. Giné, E.; Mason, D.M. On local *U*-statistic processes and the estimation of densities of functions of several sample variables. *Ann. Statist.* **2007**, *35*, 1105–1145.
[CrossRef]

13. Schick, A.; Wang, Y.; Wefelmeyer, W. Tests for normality based on density estimators of convolutions. *Statist. Probab. Lett.* **2011**, *81*, 337–343.
[CrossRef]

14. Joly, E.; Lugosi, G. Robust estimation of *U*-statistics. *Stochastic Process. Appl.* **2016**, *126*, 3760–3773.
[CrossRef]

15. Lee, S.; Linton, O.; Whang, Y.J. Testing for stochastic monotonicity. *Econometrica* **2009**, *77*, 585–602.

16. Ghosal, S.; Sen, A.; van der Vaart, A.W. Testing monotonicity of regression. *Ann. Statist.* **2000**, *28*, 1054–1082.
[CrossRef]

17. Abrevaya, J.; Jiang, W. A nonparametric approach to measuring and testing curvature. *J. Bus. Econom. Statist.* **2005**, *23*, 1–19.
[CrossRef]

18. Nolan, D.; Pollard, D. *U*-processes: Rates of convergence. *Ann. Statist.* **1987**, *15*, 780–799.
[CrossRef]

19. Sherman, R.P. Maximal inequalities for degenerate *U*-processes with applications to optimization estimators. *Ann. Statist.* **1994**, *22*, 439–459.
[CrossRef]

20. Yoshihara, K.i. Limiting behavior of *U*-statistics for stationary, absolutely regular processes. *Z. Wahrscheinlichkeitstheorie und Verw. Gebiete* **1976**, *35*, 237–252.
[CrossRef]

21. Borovkova, S.; Burton, R.; Dehling, H. Limit theorems for functionals of mixing processes with applications to *U*-statistics and dimension estimation. *Trans. Amer. Math. Soc.* **2001**, *353*, 4261–4318.
[CrossRef]

22. Denker, M.; Keller, G. On *U*-statistics and *v*. Mises' statistics for weakly dependent processes. *Z. Wahrsch. Verw. Gebiete* **1983**, *64*, 505–522.
[CrossRef]

23. Leucht, A. Degenerate *U*- and *V*-statistics under weak dependence: Asymptotic theory and bootstrap consistency. *Bernoulli* **2012**, *18*, 552–585.
[CrossRef]

24. Leucht, A.; Neumann, M.H. Degenerate *U*- and *V*-statistics under ergodicity: Asymptotics, bootstrap and applications in statistics. *Ann. Inst. Statist. Math.* **2013**, *65*, 349–386.
[CrossRef]

25. Bouzebda, S.; Nemouchi, B. Weak-convergence of empirical conditional processes and conditional *U*-processes involving functional mixing data. In *Statistical Inference for Stochastic Processes*; Springer: New York, NY, USA, 2022; pp. 1–56.

26. Efron, B. Bootstrap methods: Another look at the jackknife. *Ann. Statist.* **1979**, *7*, 1–26.
[CrossRef]

27. Hall, P. *The Bootstrap and Edgeworth Expansion*; Springer Series in Statistics; Springer: New York, NY, USA, 1992; p. xiv+352.

28. Bickel, P.J.; Freedman, D.A. Some asymptotic theory for the bootstrap. *Ann. Statist.* **1981**, *9*, 1196–1217.
[CrossRef]

29. Arcones, M.A.; Giné, E. On the bootstrap of *U* and *V* statistics. *Ann. Statist.* **1992**, *20*, 655–674.
[CrossRef]

30. Dehling, H.; Mikosch, T. Random quadratic forms and the bootstrap for *U*-statistics. *J. Multivariate Anal.* **1994**, *51*, 392–413.
[CrossRef]

31. Leucht, A.; Neumann, M.H. Consistency of general bootstrap methods for degenerate *U*-type and *V*-type statistics. *J. Multivariate Anal.* **2009**, *100*, 1622–1633.
[CrossRef]

32. Politis, D.N.; Romano, J.P. A circular block-resampling procedure for stationary data. In *Exploring the Limits of Bootstrap* (East Lansing, MI, 1990); Wiley Ser. Probab. Math. Statist. Probab. Math. Statist.; Wiley: New York, NY, USA, 1992; pp. 263–270.

33. Carlstein, E. The use of subseries values for estimating the variance of a general statistic from a stationary sequence. *Ann. Statist.* **1986**, *14*, 1171–1179.
[CrossRef]

34. Politis, D.N.; Romano, J.P. The stationary bootstrap. *J. Amer. Statist. Assoc.* **1994**, *89*, 1303–1313.
[CrossRef]

35. Rubin, D.B. The Bayesian bootstrap. *Ann. Statist.* **1981**, *9*, 130–134.
[CrossRef]

36. Lo, A.Y. A Bayesian method for weighted sampling. *Ann. Statist.* **1993**, *21*, 2138–2148.
[CrossRef]

37. Mason, D.M.; Newton, M.A. A rank statistics approach to the consistency of a general bootstrap. *Ann. Statist.* **1992**, *20*, 1611–1624.
[CrossRef]

38. Præstgaard, J.; Wellner, J.A. Exchangeably weighted bootstraps of the general empirical process. *Ann. Probab.* **1993**, *21*, 2053–2086.
[CrossRef]

39. van der Vaart, A. New Donsker classes. *Ann. Probab.* **1996**, *24*, 2128–2140.
[CrossRef]

40. Alvarez-Andrade, S.; Bouzebda, S. Strong approximations for weighted bootstrap of empirical and quantile processes with applications. *Stat. Methodol.* **2013**, *11*, 36–52.
[CrossRef]

41. Bouzebda, S. On the strong approximation of bootstrapped empirical copula processes with applications. *Math. Methods Statist.* **2012**, *21*, 153–188.
[CrossRef]

42. Bouzebda, S.; Elhattab, I.; Ferfache, A.A. General *M*-Estimator Processes and their *m* out of *n* Bootstrap with Functional Nuisance Parameters. In *Methodology and Computing in Applied Probability*; Springer: New York, NY, USA, 2022; pp. 1–45.

43. Huskova, M.; Janssen, P. Consistency of the generalized bootstrap for degenerate U-statistics. *Ann. Stat.* **1993**, *21*, 1811–1823.
[CrossRef]

44. Janssen, P. Weighted bootstrapping of U-statistics. *J. Stat. Plan. Inference* **1994**, *38*, 31–41.
[CrossRef]

45. Alvarez-Andrade, S.; Bouzebda, S. Cramér's type results for some bootstrapped *U*-statistics. *Statist. Papers* **2020**, *61*, 1685–1699.
[CrossRef]

46. Han, Q. Multiplier *U*-processes: Sharp bounds and applications. *Bernoulli* **2022**, *28*, 87–124.
[CrossRef]

47. Radulović, D. Renewal type bootstrap for Markov chains. *Test* **2004**, *13*, 147–192.
[CrossRef]
48. Giné, E.; Zinn, J. Bootstrapping general empirical measures. *Ann. Probab.* **1990**, *18*, 851-869. [CrossRef]

49. Nummelin, E. *General Irreducible Markov Chains and Nonnegative Operators*; Cambridge Tracts in Mathematics; Cambridge University Press: Cambridge, UK, 1984; Volume 83, p. xi+156.

50. Athreya, K.B.; Ney, P. A new approach to the limit theory of recurrent Markov chains. *Trans. Amer. Math. Soc.* **1978**, *245*, 493-501.
[CrossRef]

51. Nummelin, E. A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete **1978**, *43*, 309-318. [CrossRef]

52. Chung, K.L. *Markov Chains with Stationary Transition Probabilities*, 2nd ed.; Die Grundlehren der mathematischen Wissenschaften, Band 104; Springer: New York, NY, USA, 1967; p. xi+301.

53. van der Vaart, A.W.; Wellner, J.A. *Weak Convergence and Empirical Processes*; Springer Series in Statistics; With applications to statistics; Springer: New York, NY, USA, 1996; p. xvi+508.

54. Pollard, D. *Convergence of Stochastic Processes*; Springer Series in Statistics; Springer: New York, NY, USA, 1984; p. xiv+215.

55. Douc, R.; Guillin, A.; Moulines, E. Bounds on regeneration times and limit theorems for subgeometric Markov chains. *Ann. Inst. Henri Poincaré Probab. Stat.* **2008**, *44*, 239-257. [CrossRef]

56. Meyn, S.; Tweedie, R.L. *Markov Chains and Stochastic Stability*, 2nd ed.; Cambridge University Press: Cambridge, MA, USA, 2009.

57. Meyn, S.P.; Tweedie, R.L. *Markov Chains and Stochastic Stability*; Communications and Control Engineering Series; Springer: London, UK, 1993; p. xvi+548.

58. Bertail, P.; Clémençon, S. Regeneration-based statistics for Harris recurrent Markov chains. In *Dependence in Probability and Statistics*; Springer: New York, NY, USA, 2006; Volume 187, pp. 3-54.

59. Bertail, P.; Clémençon, S. A renewal approach to Markovian *U*-statistics. *Math. Methods Statist.* **2011**, *20*, 79-105. [CrossRef]

60. Revuz, D. *Markov Chains*, 2nd ed.; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1984; Volume 11.

61. Cheng, G. Moment consistency of the exchangeably weighted bootstrap for semiparametric M-estimation. *Scand. J. Stat.* **2015**, *42*, 665-684. [CrossRef]

62. Shao, J.; Wu, C. Heteroscedasticity-robustness of jackknife variance estimators in linear models. *Ann. Statist.* **1987**, *15*, 1563-1579.
[CrossRef]

63. Weng, C.S. On a second-order asymptotic property of the Bayesian bootstrap mean. *Ann. Statist.* **1989**, *17*, 705-710. [CrossRef]

64. van Zwet, W.R. The Edgeworth expansion for linear combinations of uniform order statistics. In *Second Prague Symposium on Asymptotic Statistics* (Hradec Králové, 1978); North-Holland: Amsterdam, NY, USA, 1979; pp. 93-101.

65. Pauly, M. Consistency of the subsample bootstrap empirical process. *Statistics* **2012**, *46*, 621-626. [CrossRef]

66. Shao, J.; Tu, D.S. *The Jackknife and Bootstrap*; Springer Series in Statistics; Springer: New York, NY, USA, 1995; p. xviii+516.

67. Bertail, P.; Clémençon, S. Regenerative block bootstrap for Markov chains. *Bernoulli* **2006**, *12*, 689-712. [CrossRef]

68. Bertail, P.; Clémençon, S. Approximate regenerative-block bootstrap for Markov chains. *Comput. Statist. Data Anal.* **2008**, *52*, 2739-2756. [CrossRef]

69. Fan, Y.; Ullah, A. On goodness-of-fit tests for weakly dependent processes using kernel method. *J. Nonparametr. Statist.* **1999**, *11*, 337-360. First NIU Symposium on Statistical Sciences (De Kalb, IL, 1996). [CrossRef]

70. Frees, E.W. Infinite order *U*-statistics. *Scand. J. Statist.* **1989**, *16*, 29-45.

71. Rempala, G.; Gupta, A. Weak limits of *U*-statistics of infinite order. *Random Oper. Stochastic Equations* **1999**, *7*, 39-52. [CrossRef]

72. Levental, S. Uniform limit theorems for Harris recurrent Markov chains. *Probab. Theory Related Fields* **1988**, *80*, 101-118. [CrossRef]

73. Dudley, R.M. Nonlinear functionals of empirical measures and the bootstrap. In *Probability in Banach Spaces*, 7 (Oberwolfach, 1988); Progr. Probab.; Birkhäuser Boston: Boston, MA, USA, 1990; Volume 21, pp. 63-82.
48. Giné, E.; Zinn, J. Bootstrapping general empirical measures. *Ann. Probab.* **1990**, *18*, 851-869. [CrossRef]

49. Nummelin, E. *General Irreducible Markov Chains and Nonnegative Operators*; Cambridge Tracts in Mathematics; Cambridge University Press: Cambridge, UK, 1984; Volume 83, p. xi+156.

50. Athreya, K.B.; Ney, P. A new approach to the limit theory of recurrent Markov chains. *Trans. Amer. Math. Soc.* **1978**, *245*, 493-501.
[CrossRef]

51. Nummelin, E. A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete **1978**, *43*, 309-318. [CrossRef]

52. Chung, K.L. *Markov Chains with Stationary Transition Probabilities*, 2nd ed.; Die Grundlehren der mathematischen Wissenschaften, Band 104; Springer: New York, NY, USA, 1967; p. xi+301.

53. van der Vaart, A.W.; Wellner, J.A. *Weak Convergence and Empirical Processes*; Springer Series in Statistics; With applications to statistics; Springer: New York, NY, USA, 1996; p. xvi+508.

54. Pollard, D. *Convergence of Stochastic Processes*; Springer Series in Statistics; Springer: New York, NY, USA, 1984; p. xiv+215.

55. Douc, R.; Guillin, A.; Moulines, E. Bounds on regeneration times and limit theorems for subgeometric Markov chains. *Ann. Inst. Henri Poincaré Probab. Stat.* **2008**, *44*, 239-257. [CrossRef]

56. Meyn, S.; Tweedie, R.L. *Markov Chains and Stochastic Stability*, 2nd ed.; Cambridge University Press: Cambridge, MA, USA, 2009.

57. Meyn, S.P.; Tweedie, R.L. *Markov Chains and Stochastic Stability*; Communications and Control Engineering Series; Springer: London, UK, 1993; p. xvi+548.

58. Bertail, P.; Clémençon, S. Regeneration-based statistics for Harris recurrent Markov chains. In *Dependence in Probability and Statistics*; Springer: New York, NY, USA, 2006; Volume 187, pp. 3-54.

59. Bertail, P.; Clémençon, S. A renewal approach to Markovian *U*-statistics. *Math. Methods Statist.* **2011**, *20*, 79-105. [CrossRef]

60. Revuz, D. *Markov Chains*, 2nd ed.; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1984; Volume 11.

61. Cheng, G. Moment consistency of the exchangeably weighted bootstrap for semiparametric M-estimation. *Scand. J. Stat.* **2015**, *42*, 665-684. [CrossRef]

62. Shao, J.; Wu, C. Heteroscedasticity-robustness of jackknife variance estimators in linear models. *Ann. Statist.* **1987**, *15*, 1563-1579.
[CrossRef]

63. Weng, C.S. On a second-order asymptotic property of the Bayesian bootstrap mean. *Ann. Statist.* **1989**, *17*, 705-710. [CrossRef]

64. van Zwet, W.R. The Edgeworth expansion for linear combinations of uniform order statistics. In *Second Prague Symposium on Asymptotic Statistics* (Hradec Králové, 1978); North-Holland: Amsterdam, NY, USA, 1979; pp. 93-101.

65. Pauly, M. Consistency of the subsample bootstrap empirical process. *Statistics* **2012**, *46*, 621-626. [CrossRef]

66. Shao, J.; Tu, D.S. *The Jackknife and Bootstrap*; Springer Series in Statistics; Springer: New York, NY, USA, 1995; p. xviii+516.

67. Bertail, P.; Clémençon, S. Regenerative block bootstrap for Markov chains. *Bernoulli* **2006**, *12*, 689-712. [CrossRef]

68. Bertail, P.; Clémençon, S. Approximate regenerative-block bootstrap for Markov chains. *Comput. Statist. Data Anal.* **2008**, *52*, 2739-2756. [CrossRef]

69. Fan, Y.; Ullah, A. On goodness-of-fit tests for weakly dependent processes using kernel method. *J. Nonparametr. Statist.* **1999**, *11*, 337-360. First NIU Symposium on Statistical Sciences (De Kalb, IL, 1996). [CrossRef]

70. Frees, E.W. Infinite order *U*-statistics. *Scand. J. Statist.* **1989**, *16*, 29-45.

71. Rempala, G.; Gupta, A. Weak limits of *U*-statistics of infinite order. *Random Oper. Stochastic Equations* **1999**, *7*, 39-52. [CrossRef]

72. Levental, S. Uniform limit theorems for Harris recurrent Markov chains. *Probab. Theory Related Fields* **1988**, *80*, 101-118. [CrossRef]

73. Dudley, R.M. Nonlinear functionals of empirical measures and the bootstrap. In *Probability in Banach Spaces*, 7 (Oberwolfach, 1988); Progr. Probab.; Birkhäuser Boston: Boston, MA, USA, 1990; Volume 21, pp. 63-82.