Berry-Esseen bounds of asymptotic normality of kernel density estimator for long-span high-frequency data with α-mixing

High frequency data has a wide range of applications, and its statistical problems have attracted widespread attention. In this paper, we study the convergence rates of the asymptotic normality of kernel density estimation for long-span high-frequency data under the condition of alpha-mixing dependence, which is the widest class of mixing dependent random variables, and provide the Berry-Esseen upper bounds for kernel density estimation. To derive these results, some moment inequalities for alpha-mixing high-frequency data are established. Our numerical simulations demonstrate that the repeated estimates of kernel density estimation at each fixed point exhibit asymptotic normality. In empirical analysis, the kernel density estimates of the returns of China's Shanghai and Shenzhen stock indices are provided. We use these two kernel density estimates to compare the differences in returns and risks between the Shanghai and Shenzhen stock markets, and fit these return distributions using a skew generalized T-distribution.