# The asymptotic properties of $V$-statistic for mixing multivariate process

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!!