For any symmetric function $f\left(g,g^{\prime }\right)$ defined on $\mathcal{G}\times \mathcal{G}$, define $D$ as a $\mathcal{U}$-centered operator, and

$Df\left(g,g^{\prime }\right)=f\left(g,g^{\prime }\right)-Ef\left(g,G^{\prime }\right)-Ef\left(G,g^{\prime }\right)+Ef\left(G,G^{\prime }\right),$

where $G$ is a random element valued in Hilbert space $\mathcal{G}$, $G^{\prime }$ is an independent copy of $G$. The functional class $ℱ=\left\{Df:f\text{is}\text{a}\text{symmetric}\text{function}\right\}$ is a linear space. Define the inner product on $ℱ$ as

$D{f}_1\circ D{f}_2=ED{f}_1\left(G,G^{\prime }\right)D{f}_2\left(G,G^{\prime }\right).$

It can be verified that the map satisfies the conditions of an inner product. In addition, define the norm on $ℱ$ as $\Vert Df\Vert =\sqrt{Df\circ Df}$. Take $\mathcal{G}=\mathcal{Z}\times \mathcal{Y}$. Let $K_Z\left(\left(z,y\right),\left(z^{\prime },y^{\prime }\right)\right)=K_Z\left(z,z^{\prime }\right)$ and ${\varphi }_Z\left(\left(z,y\right),\left(z^{\prime },y^{\prime }\right)\right)={\varphi }_Z\left(z,z^{\prime }\right)$ for any $\left(z,y\right),\left(z^{\prime },y^{\prime }\right)\in \mathcal{Z}\times \mathcal{Y}$. Then ${\varphi }_Z\in ℱ$. Similarly, ${\psi }_Y\in ℱ$. Thus $\text{KCMD}\left(Y|Z\right)$ in equation (3) can also be written as

$\text{KCMD}\left(Y|Z\right)={\varphi }_Z\circ {\psi }_Y.$ (7)

This implies that the KCMD measures the conditional mean dependence of $Y$ on $Z$ through the inner product of ${\varphi }_Z$ and ${\psi }_Y$ in a linear space. The cosine value

$\text{cos}\left({\varphi }_Z,{\psi }_Y\right)=\frac{{\varphi }_Z\circ {\psi }_Y}{\Vert {\varphi }_Z\Vert \cdot \Vert {\psi }_Y\Vert }$

measures the strength of conditional mean dependence.

Lemma 2. Suppose $\Vert {\varphi }_Z\Vert \ne 0$ and $\Vert {\psi }_Y\Vert \ne 0$, then ${\psi }_Y$ can be decomposed into two orthogonal parts,

${\psi }_Y=\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z+\left({\psi }_Y-\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\right).$ (8)

The first term in (8) represents the part of $Y$’s conditional mean that is affected by $Z$, the second term represents the part that can not be interpreted by $Z$, and in a sense it corresponds to $U=Y-E\left(Y|Z\right)$ since $E\left(U|Z\right)=0$. Next we define $W=\left(X,Z\right)$ and $\mathcal{W}=\mathcal{X}\times \mathcal{Z}$. One way to measure the additional contribution of $X$ to the conditional mean of $Y$ controlling for $Z$, is to measure $E\left(U|W\right)$.

Inspired by [10] , we provide definitions for the Kernel-based Partial Conditional Mean Dependence and the Kernel-based Partial Conditional Mean Correlation. Define ${\varphi }_W$ similar to ${\varphi }_Z$, replacing $K_Z\left(z,z^{\prime }\right)$ with $K_W\left(w,w^{\prime }\right)$ in ${\varphi }_Z$ is ${\varphi }_W$.

Definition 1. The population partial KCMD of $Y$ given $X$, after controlling for the effect of $Z$, i.e., $\text{pKCMD}\left(Y|X;Z\right)$ is defined as

$\text{pKCMD}\left(Y|X;Z\right)={\varphi }_W\circ \left({\psi }_Y-\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\right).$

If $\Vert {\varphi }_Z\Vert =0$, then we define $\text{pKCMD}\left(Y|X;Z\right)={\varphi }_W\circ {\psi }_Y$.

The population Kernel-based Partial Conditional Mean Correlation(pKCMC) is defined as

$\text{pKCMC}\left(Y|X;Z\right)=\frac{\text{pKCMD}\left(Y|X;Z\right)}{\Vert {\varphi }_W\Vert \cdot \Vert {\psi }_Y-\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\Vert }.$

If $\Vert {\varphi }_W\Vert \cdot \Vert {\psi }_Y-\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\Vert =0$, then we define $\text{pKCMC}\left(Y|X;Z\right)=0$.

After performing some straightforward calculations, we obtain an equivalent expression for pKCMD, which is given by

$\begin{array}{c}\text{pKCMD}\left(Y|X;Z\right)={\varphi }_W\circ {\psi }_Y-\frac{\left({\varphi }_Z\circ {\psi }_Y\right)\left({\varphi }_W\circ {\varphi }_Z\right)}{{\Vert {\varphi }_Z\Vert }^2}\\ =\text{KCMD}\left(Y|W\right)-\frac{\text{KCMD}\left(Y|Z\right)\text{HSIC}\left(Z,W\right)}{\text{HSIC}\left(Z,Z\right)}.\end{array}$ (9)

The $\text{HSIC}\left(Z,W\right)$ is the Hilbert-Schmidt Independence Criterion(HSIC) between $Z$ and $W$, it measures the dependence between these two random elements. The kernel functions used in $\text{HSIC}\left(Z,W\right)$ are $K_Z\left(\cdot ,\cdot \right):\mathcal{Z}\times \mathcal{Z}\to R$ and $K_W\left(\cdot ,\cdot \right):\mathcal{W}\times \mathcal{W}\to R$, which, like in KCMD, depend on the variables in their subscripts. The content about HSIC can be found in [15] [17] - [19] and so on. We reviewed the specific form of HSIC in the Appendix and derive the last equation of (9). When the conditional mean of $Y$ does not depend on $Z$ or $Z$ is a constant, we have $\text{KCMD}\left(Y|Z\right)=0$ or $\text{HSIC}\left(Z,Z\right)=0$. As a result, we have $\text{pKCMD}\left(Y|X;Z\right)=\text{KCMD}\left(Y|W\right)=\text{KCMD}\left(Y|X\right)$.

Given the sample ${\left(X_i,Y_i,Z_i\right)}_{i=1}^n$, we want to define sample partial KCMD, denoted as ${\text{pKCMD}}_n\left(Y|X;Z\right)$ as the sample analog of population partial KCMD. Let $W_i=\left(X_i,Z_i\right)$, and define $\tilde{C}$ be $n\times n$ matrix with entries ${\tilde{C}}_{ij}$,

${\tilde{C}}_{ij}=\{\begin{array}{ll}{c}_{ij}-c_{i\cdot }-c_{\cdot j}+c_{\cdot \cdot },\hfill & i\ne j\hfill \\ 0,\hfill & i=j\hfill \end{array},$

where $c_{ij}=K_W\left(W_i,W_j\right)$, and $c_{i\cdot }$, $c_{\cdot j}$ and $c_{\cdot \cdot }$ are defined similarly to $a_{i\cdot }$, $a_{\cdot j}$ and $a_{\cdot \cdot }$.

Definition 2. Given a random sample from the joint distribution $\left(X,Y,Z\right)$, the sample partial kernel-based conditional mean dependence of $Y$ given $X$, after controlling for the effect of $Z$, is given by

${\text{pKCMD}}_n\left(Y|X;Z\right)=\left(\tilde{C}\cdot \tilde{B}\right)-\frac{\left(\tilde{A}\cdot \tilde{B}\right)\left(\tilde{A}\cdot \tilde{C}\right)}{\left(\tilde{A}\cdot \tilde{A}\right)},$

assuming $\left(\tilde{A}\cdot \tilde{A}\right)\ne 0$ and $\left(\tilde{C}\cdot \tilde{B}\right)$ otherwise. The sample partial kernel-based conditional mean correlation is given by

${\text{pKCMC}}_n\left(Y|X;Z\right)=\frac{{\left|\tilde{A}\right|}^2{\text{pKCMD}}_n\left(Y|X;Z\right)}{\left|\tilde{C}\right|\cdot \left|{\left(\left|\tilde{A}\right|\right)}^2\tilde{B}-\left(\tilde{A}\circ \tilde{B}\right)\tilde{A}\right|}$

If $\left|\tilde{C}\right|\cdot \left|{\left(\left|\tilde{A}\right|\right)}^2\tilde{B}-\left(\tilde{A}\circ \tilde{B}\right)\tilde{A}\right|=0$, then we define ${\text{pKCMC}}_n\left(Y|X;Z\right)=0$.

We next outline theoretical properties of the sample pKCMD. Analogous results hold for the sample pKCMC, which we omit discussing further here.

Theorem 1. If one of the following two conditions holds,

a) $K_Z\left(\cdot ,\cdot \right)$ and $K_W\left(\cdot ,\cdot \right)$ are bounded kernels, $E\Vert Y\Vert <\infty$;

b) $E{K}_Z^2\left(Z,Z^{\prime }\right)<\infty$, $E{K}_W^2\left(W,W^{\prime }\right)<\infty$, and $E{\Vert Y\Vert }^2<\infty$.

Then, as $n\to \infty$, we have ${\text{pKCMD}}_n\left(Y|X;Z\right)\to \text{pKCMD}\left(Y|X;Z\right)$ a.s..

We also show that ${\text{pKCMD}}_n\left(Y|X;Z\right)$ is concentrated. To obtain the bounds of the deviation ${\text{pKCMD}}_n\left(Y|X;Z\right)-\text{pKCMD}\left(Y|X;Z\right)$, we impose the following condition.

(C1) There exists a constant $s_0>0$ such that for all $0<s\le 2s_0$, $E\exp \left(s{\Vert Y\Vert }^2\right)<\infty$.

Condition (C1) follows immediately when $Y$ is bounded uniformly, or when it has a Gaussian distribution. Condition (C1) is widely used in statistical research, for example, in [11] [20] [21] , to analyze the theoretical properties of feature screening.

Theorem 2. If $K_Z\left(\cdot ,\cdot \right)$ and $K_W\left(\cdot ,\cdot \right)$ are bounded kernels, $E\Vert Y\Vert <\infty$, and Condition (C1) holds, then for any $ϵ>0$, there exist constants $0<\beta <1/2$, $r_1>0$ and $r_2>0$ such that

$\begin{array}{l}P\left(\left|{\text{pKCMD}}_n\left(Y|X;Z\right)-\text{pKCMD}\left(Y|X;Z\right)\right|\ge ϵ\right)\\ \le O\left(\text{exp}\left(-r_1{n}^{1-2\beta }{ϵ}^2\right)+n\exp \left\{-r_2{n}^{\beta }\right\}\right).\end{array}$

Take $ϵ=\eta n^{-\gamma }$ with $0<\gamma <1/2$ and a constant $\eta >0$. According to Theorem 2, there exists $0<\beta +\gamma <1/2$, such that $P\left(\left|{\text{pKCMD}}_n\left(Y|X;Z\right)-\text{pKCMD}\left(Y|X;Z\right)\right|\ge \eta n^{-\gamma }\right)=o\left(1\right)$. This implies that ${\text{pKCMD}}_n\left(Y|X;Z\right)$ is concentrated, and the deviation between ${\text{pKCMD}}_n\left(Y|X;Z\right)$ and $\text{pKCMD}\left(Y|X;Z\right)$ is less than $\eta n^{-\gamma }$ with probability at least $1-O\left(\text{exp}\left(-r_1{n}^{1-2\beta }{ϵ}^2\right)+n\exp \left\{-r_2{n}^{\beta }\right\}\right)$.

Definition A1. (HSIC) The Hilbert-Schmidt Independent Criterion of random elements $Z\in \mathcal{Z}$ and $W\in \mathcal{W}$ is defined as

$\begin{array}{c}\text{HSIC}\left(Z,W\right)=E\left[K_Z\left(Z,Z^{\prime }\right)K_W\left(W,W^{\prime }\right)\right]+E\left[K_Z\left(Z,Z^{\prime }\right)\right]E\left[K_W\left(W,W^{\prime }\right)\right]\\ -2E\left\{E\left[K_Z\left(Z,Z^{\prime }\right)|Z\right]E\left[K_W\left(W,W^{\prime }\right)|W\right]\right\},\end{array}$

where $\left(Z^{\prime },W^{\prime }\right)$ is an independent copy of $\left(Z,W\right)$, $K_Z\left(\cdot ,\cdot \right)$ and $K_W\left(\cdot ,\cdot \right)$ are two kernel functions.

Note that in Definition A1, the kernel functions $K_Z\left(\cdot ,\cdot \right)$ and $K_W\left(\cdot ,\cdot \right)$ are mutable and depend on $Z$ and $W$, respectively. HSIC was developed to test the independence of $Z$ and $W$, and has many good properties. a) It is non negative; b) If $K_Z\left(\cdot ,\cdot \right)$ and $K_W\left(\cdot ,\cdot \right)$ are characteristic, then $\text{HSIC}\left(Z,W\right)=0$ if and only if $Z$ and $W$ are independent. Using the symbol ${\varphi }_Z$ defined in (2), symbol ${\varphi }_W$ is similar to ${\varphi }_Z$ , but the kernel function used in $K_W\left(\cdot ,\cdot \right)$ is ${\varphi }_W$, we have

$\begin{array}{l}{\varphi }_Z\circ {\varphi }_W\\ =E{\varphi }_Z\left(Z,Z^{\prime }\right){\varphi }_W\left(W,W^{\prime }\right)\\ =E(\left\{K_Z\left(Z,Z^{\prime }\right)-E\left[K_Z\left(Z,Z^{\prime }\right)|Z\right]-E\left[K_Z\left(Z,Z^{\prime }\right)|Z^{\prime }\right]+E{K}_Z\left(Z,Z^{\prime }\right)\right\}\\ \times \left\{K_W\left(W,W^{\prime }\right)-E\left[K_W\left(W,W^{\prime }\right)|W\right]-E\left[K_W\left(W,W^{\prime }\right)|W^{\prime }\right]+E{K}_W\left(W,W^{\prime }\right)\right\})\\ =E\left[K_Z\left(Z,Z^{\prime }\right)K_W\left(W,W^{\prime }\right)\right]+E\left[K_Z\left(Z,Z^{\prime }\right)\right]E\left[K_W\left(W,W^{\prime }\right)\right]\\ -2E\left\{E\left[K_Z\left(Z,Z^{\prime }\right)|Z\right]E\left[K_W\left(W,W^{\prime }\right)|W\right]\right\}\\ =\text{HSIC}\left(Z,W\right),\end{array}$

thus (9) holds.

Proof of Lemma 1. The Lemma 1 is the Proposition 1 in [13] . □

Proof of Lemma 2. Because

$\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\left(z,z^{\prime }\right)\in ℱ,$

and

$\begin{array}{l}\left({\psi }_Y-\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\right)\left(\left(y,z\right),\left(y^{\prime },z^{\prime }\right)\right)\\ ={\psi }_Y\left(y,y^{\prime }\right)-\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\left(z,z^{\prime }\right)\in ℱ,\end{array}$

their inner product is

$\begin{array}{l}\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\circ \left({\psi }_Y-\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }{\varphi }_Z\right)\\ =\frac{\Vert {\psi }_Y\Vert \text{cos}\left({\varphi }_Z,{\psi }_Y\right)}{\Vert {\varphi }_Z\Vert }\left({\varphi }_Z\circ {\psi }_Y\right)-\frac{{\Vert {\psi }_Y\Vert }^2{\left[\text{cos}\left({\varphi }_Z,{\psi }_Y\right)\right]}^2}{{\Vert {\varphi }_Z\Vert }^2}\left({\varphi }_Z\circ {\varphi }_Z\right)\\ ={\Vert {\psi }_Y\Vert }^2\text{cos}\left({\varphi }_Z,{\psi }_Y\right)\times \frac{\left({\varphi }_Z\circ {\psi }_Y\right)}{\Vert {\varphi }_Z\Vert \cdot \Vert {\psi }_Y\Vert }-{\Vert {\psi }_Y\Vert }^2{\left[\text{cos}\left({\varphi }_Z,{\psi }_Y\right)\right]}^2\\ =0,\end{array}$

thus these two parts in (8) are orthogonal. □

Proof of Theorem 1. a) When $K_Z\left(\cdot ,\cdot \right)$ and $K_W\left(\cdot ,\cdot \right)$ are bounded kernels. We suppose that $K_Z\left(\cdot ,\cdot \right)<M<\infty$ and $K_W\left(\cdot ,\cdot \right)<M<\infty$. First, we consider $\left(\tilde{A}\cdot \tilde{B}\right)$ in ${\text{pKCMD}}_n\left(Y|X;Z\right)$. Because $\left(\tilde{A}\cdot \tilde{B}\right)$ is an unbiased estimate of ${\varphi }_Z\circ {\psi }_Y$ and can be written as an U-statistic by equation (6), and

$\begin{array}{c}E\left|h\left(V_1,V_2,V_3,V_4\right)\right|\le E\left|a_{uv}\left(b_{uv}-b_{uq}-b_{vr}+b_{qr}\right)\right|\\ \le ME\left|b_{uv}-b_{uq}-b_{vr}+b_{qr}\right|\\ \le 4ME\left|b_{uv}\right|\\ \le 4ME\Vert Y\Vert E\Vert Y^{\prime }\Vert \\ <\infty .\end{array}$

By the strong law of large number for U-statistics,

$\left(\tilde{A}\cdot \tilde{B}\right)\to {\varphi }_Z\circ {\psi }_Y,a.s\mathrm{..}$

Similarly, we have

$\left(\tilde{C}\cdot \tilde{B}\right)\to {\varphi }_W\circ {\psi }_Y,a.s.,$

$\left(\tilde{A}\cdot \tilde{C}\right)\to {\varphi }_Z\circ {\varphi }_W,a.s.,$

$\left(\tilde{A}\cdot \tilde{A}\right)\to {\varphi }_Z\circ {\psi }_Z,a.s\mathrm{..}$

According to the continuous mapping theorem, we have $\text{pKCMD}\left(Y|X;Z\right)\to \text{pKCMD}\left(Y|X;Z\right)$ a.s..

b) For general kernel.

$\begin{array}{c}E\left|h\left(V_1,V_2,V_3,V_4\right)\right|\le E\left|a_{uv}\left(b_{uv}-b_{uq}-b_{vr}+b_{qr}\right)\right|\\ \le E{a}_{uv}{b}_{uv}+E{a}_{uv}{b}_{uq}+E{a}_{uv}{b}_{vr}+E{a}_{uv}{b}_{qr}\\ \le 2E{a}_{uv}^2+2E{b}_{uv}^2\\ \le 2E{K}_Z^2\left(Z_u,Z_v\right)+2{\left(E{\Vert Y\Vert }^2\right)}^2\\ \left(\text{because}E{K}_Z^2\left(Z,Z^{\prime }\right)<\infty \text{and}E{\Vert Y\Vert }^2<\infty \right)\\ <\infty .\end{array}$

Similar to (a), we can also prove that $\text{pKCMD}\left(Y|X;Z\right)\to \text{pKCMD}\left(Y|X;Z\right)$ a.s.. This completes the proof. □

Proof of Theorem 2. Denote $w={\varphi }_Z\circ {\psi }_Y$ and $\hat{w}=\widehat{\text{KCMD}}\left(Y|Z\right)$. By employing the Markov inequality, we obtain that for any $ϵ>0$ and $t>0$,

$\begin{array}{c}P\left(\hat{w}-w>ϵ\right)=P\left(\text{exp}\left(\hat{w}t-wt\right)>\text{exp}\left(ϵt\right)\right)\\ \le \text{exp}\left(-ϵt\right)E\exp \left(\hat{w}t-wt\right)\\ =\text{exp}\left(-ϵt-wt\right)E\exp \left(\hat{w}t\right).\end{array}$

Note that $\hat{w}$ admits a U-statistic expression

$\hat{w}=\frac{1}{\left(\begin{array}{l}n\hfill \\ 4\hfill \end{array}\right)}\underset{i<j<s<t}{{\displaystyle \sum }}h\left(V_i,V_j,V_s,V_t\right),$

with the kernel $h\left(V_i,V_j,V_s,V_t\right)=\frac{1}{4!}{\displaystyle {\sum }_{\left(u,v,q,r\right)}^{\left(i,j,s,t\right)}\left(a_{uv}{b}_{uv}-a_{uv}{b}_{uq}-a_{uv}{b}_{vr}+a_{uv}{b}_{qr}\right)}$, where

$V_i=\left(Z_i,Y_i\right)$ and the sum is over all 4! permutations of $\left(i,j,s,t\right)$. Following [23] (Section 5.1.6), we write

${\hat{w}}_k={\left(n!\right)}^{-1}\underset{n!}{{\displaystyle \sum }}\underset{i=1}{\overset{m}{{\displaystyle \sum }}}h\left(V_{\left(4i-3\right)},V_{\left(4i-2\right)},V_{\left(4i-1\right)},V_{\left(4i\right)}\right)/m,$

where where ${{\displaystyle \sum }}_{n!}$ denotes the summation over all possible permutations of $\left(1,\cdots ,n\right)$, $V_{\left(i\right)}$ is the $i$-th element under the permutation, $m=\lfloor n/4\rfloor$ is the integer part of $n/4$. Denote $h_i=h\left(V_{\left(4i-3\right)},V_{\left(4i-2\right)},V_{\left(4i-1\right)},V_{\left(4i\right)}\right)$, write $\hat{w}={\hat{w}}_1+{\hat{w}}_2$, where

${\hat{w}}_1={\left(n!\right)}^{-1}\underset{n!}{{\displaystyle \sum }}\underset{i=1}{\overset{m}{{\displaystyle \sum }}}{h}_{i}I\left(\left|h_i\right|\le M_0\right)/m,$

${\hat{w}}_2={\left(n!\right)}^{-1}\underset{n!}{{\displaystyle \sum }}\underset{i=1}{\overset{m}{{\displaystyle \sum }}}{h}_{i}I\left(\left|h_i\right|>M_0\right)/m,$

and $M_0>0$. Correspondingly, its population counterpart can also be decomposed as $w=E{h}_{1}I\left(\left|h_1\right|\le M_0\right)+E{h}_{1}I\left(\left|h_1\right|>M_0\right)=w_1+w_2$.

Jensen inequality yields

$\begin{array}{c}E\exp \left({\hat{w}}_{1}t\right)=E\exp \left(t{\left(n!\right)}^{-1}\underset{n!}{{\displaystyle \sum }}\underset{i=1}{\overset{m}{{\displaystyle \sum }}}{h}_{i}I\left(\left|h_i\right|\le M_0\right)/m\right)\\ \le E\exp \left(\frac{t}{m}\underset{i=1}{\overset{m}{{\displaystyle \sum }}}{h}_{i}I\left(\left|h_i\right|\le M_0\right)\right)\\ =E^m\exp \left(\frac{t}{m}{h}_{i}I\left(\left|h_i\right|\le M_0\right)\right),\end{array}$

so

$P\left({\hat{w}}_1-w_1>ϵ\right)\le \text{exp}\left(-ϵt\right)E^m\exp \left(\frac{t}{m}{h}_{i}I\left(\left|h_i\right|\le M_0\right)-\frac{t}{m}{w}_1\right).$

$E{h}_{i}I\left(\left|h_i\right|\le M_0\right)=w_1$, and $\left|h_{i}I\left(\left|h_i\right|\le M_0\right)\right|\le M_0$, apply the Lemma 5.6.1.A of [23] ,

$E\exp \left(\frac{t}{m}{h}_{i}I\left(\left|h_i\right|\le M_0\right)-\frac{t}{m}{w}_1\right)\le \text{exp}\left(\frac{t^2{M}_0^2}{2m^2}\right).$

So,

$P\left({\hat{w}}_1-w_1>ϵ\right)\le \text{exp}\left(\frac{t^2{M}_0^2}{2m}-ϵt\right).$

Furthermore, $P\left(\left|{\hat{w}}_1-w_1\right|>ϵ\right)\le 2\exp \left(\frac{t^2{M}_0^2}{2m}-ϵt\right)$ by the symmetry of the U-statistic.

Now we turn to ${\hat{w}}_2$. $w_{k2}^2=E^2{h}_{1}I\left(\left|h_1\right|>M_0\right)\le E{h}_1^{2}P\left(\left|h_1\right|>M_0\right)\le \frac{E{h}_1^{2}E{\left|h_1\right|}^{q_1}}{M_0^{q_1}}$

for any $q_1\in N$. By the assumption that $K_Z\left(\cdot ,\cdot \right)$ is bounded, there exists a positive constant $M$ such that $\left|a_{uv}\right|<M$, and

$\left|h_1\right|=\left|h\left(V_i,V_j,V_s,V_t\right)\right|\le a_{uv}\left|b_{uv}-b_{uq}-b_{vr}+b_{qr}\right|\le M\left({\Vert Y_u\Vert }^2+{\Vert Y_v\Vert }^2+{\Vert Y_q\Vert }^2+{\Vert Y_r\Vert }^2\right)$,

this yields $E{\left|h_1\right|}^{q_1}\le 4^{q_1-1}{A}^{q_1}E{\Vert Y\Vert }^{2q_1}<\infty$ by condition (C1).

Thus, if we choose $M_0=n^{\beta }$ for $0<\beta <1/2$, then $\left|w_2\right|\le ϵ/2$ for sufficiently large $n$. Hence, $P\left(\left|{\hat{w}}_2-w_2\right|\ge ϵ\right)\le P\left(\left|{\hat{w}}_2\right|\ge ϵ/2\right)$.

$\begin{array}{c}P\left(\left|{\hat{w}}_2\right|\ge ϵ/2\right)\le P\left({\displaystyle {\cup }_{i=1}^n\left\{{\Vert Y_i\Vert }^2>\frac{M_0}{4M}\right\}}\right)\\ \le \underset{i=1}{\overset{n}{{\displaystyle \sum }}}P\left({\Vert Y_i\Vert }^2>\frac{M_1}{4M}\right)\\ =nP\left({\Vert Y_i\Vert }^2>\frac{M_0}{4M}\right)\\ \le nC\exp \left\{-r_2{n}^{\beta }\right\},\end{array}$

for $r_2\in \left(0,2s_0\right]$.