Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel function. Then \begin{equation*} V_n = \frac{1}{n^2} \sum_{s = 1}^n \sum_{t = 1}^n h(X_s, X_t) \end{equation*} is a $V$-statistic of order 2.

I do find an article that involves this question. Leucht and Neumann show that $V_n - \frac{1}{n} \sum_{k = 1}^{\infty} \lambda_k Z_k^2 = o_p \left( \frac{1}{n}\right)$ where $\{ Z_k\}_{k = 1}^{\infty}$ is a sequence of independent standard normal random variables. But this representation may not be practicable when using it: the eigenvalue $\lambda_k$ is the solution of $\mathrm{E} \left[h (x, X_{0}) \Phi_k (X_{0})\right]=\lambda_k \Phi_k(x)$ with $(\Phi_{k})_{k}$ are associated orthonormal eigenfunctions satisfying \begin{equation*} h(x, y)=\sum_{k} \lambda_{k} \Phi_{k}(x) \Phi_{k}(y), \qquad \mathrm{E}\left[\Phi_{j}\left(X_{0}\right) \Phi_{k}\left(X_{0}\right)\right]=\delta_{j k}, \end{equation*} it is extremely difficult to find those $\lambda_k$! However, these exists some simple methods to find the asymptotic expression for $V_n$, and we do not need to find these $\lambda_k$!

Could anyone provide me any other literature regarding the situation $d > 1$ with a practical method to analyze $V_n$? Thank you so much in advance!!

Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel function. Then \begin{equation*} V_n = \frac{1}{n^2} \sum_{s = 1}^n \sum_{t = 1}^n h(X_s, X_t) \end{equation*} is a $V$-statistic of order 2.

I do find an article that involves this question. Leucht and Neumann show that $V_n - \frac{1}{n} \sum_{k = 1}^{\infty} \lambda_k Z_k^2 = o_p \left( \frac{1}{n}\right)$ where $\{ Z_k\}_{k = 1}^{\infty}$ is a sequence of independent standard normal random variables. But this representation may not be practicable when using it: the eigenvalue $\lambda_k$ is the solution of $\mathrm{E} \left[h (x, X_{0}) \Phi_k (X_{0})\right]=\lambda_k \Phi_k(x)$ with $(\Phi_{k})_{k}$ are associated orthonormal eigenfunctions satisfying \begin{equation*} h(x, y)=\sum_{k} \lambda_{k} \Phi_{k}(x) \Phi_{k}(y), \qquad \mathrm{E}\left[\Phi_{j}\left(X_{0}\right) \Phi_{k}\left(X_{0}\right)\right]=\delta_{j k}, \end{equation*} it is extremely difficult to find those $\lambda_k$! However, these exists some simple methods to find the asymptotic expression for $V_n$ when $d = 1$, and we do not need to find these $\lambda_k$!

Could anyone provide me any other literature regarding the situation $d > 1$ with a practical method to analyze $V_n$? Thank you so much in advance!!

Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel function. Then \begin{equation*} V_n = \frac{1}{n^2} \sum_{s = 1}^n \sum_{t = 1}^n h(X_s, X_t) \end{equation*} is a $V$-statistic of order 2.

I have tried to find some papers, but all of them are only for univariate mixing process, i.e. $d = 1$. Could anyone provide me some literature regarding the situation $d > 1$? Thanks a lot!!!

Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel function. Then \begin{equation*} V_n = \frac{1}{n^2} \sum_{s = 1}^n \sum_{t = 1}^n h(X_s, X_t) \end{equation*} is a $V$-statistic of order 2.

I do find an article that involves this question. Leucht and Neumann show that $V_n - \frac{1}{n} \sum_{k = 1}^{\infty} \lambda_k Z_k^2 = o_p \left( \frac{1}{n}\right)$ where $\{ Z_k\}_{k = 1}^{\infty}$ is a sequence of independent standard normal random variables. But this representation may not be practicable when using it: the eigenvalue $\lambda_k$ is the solution of $\mathrm{E} \left[h (x, X_{0}) \Phi_k (X_{0})\right]=\lambda_k \Phi_k(x)$ with $(\Phi_{k})_{k}$ are associated orthonormal eigenfunctions satisfying \begin{equation*} h(x, y)=\sum_{k} \lambda_{k} \Phi_{k}(x) \Phi_{k}(y), \qquad \mathrm{E}\left[\Phi_{j}\left(X_{0}\right) \Phi_{k}\left(X_{0}\right)\right]=\delta_{j k}, \end{equation*} it is extremely difficult to find those $\lambda_k$! However, these exists some simple methods to find the asymptotic expression for $V_n$, and we do not need to find these $\lambda_k$!

Could anyone provide me any other literature regarding the situation $d > 1$ with a practical method to analyze $V_n$? Thank you so much in advance!!