The central limit theorem for degenerate variable U-statistics under dependence

Abstract

The central limit theorem (CLT) for degenerate U-statistics with a variable symmetric kernel function has been studied under dependence by many authors, since it has many important applications in nonparametric estimation and testing problems [see, e.g. Takahata, H., and Yoshihara, K. (1987), ‘Central Limit Theorems for Integrated Square Error of Nonparametric Density Estimators Based on a Absolutely Regular Random Sequences’, Yokohama Mathematical Journal, 35, 95–111; Yoshihara, K. (1989), ‘Limiting Behavior of Generalized Quadratic Forms Generated by Absolutely Regular Sequences II’, Yokohama Mathematical Journal, 37, 109–123. Yoshihara, K. (1992), ‘Limiting Behavior of Generalized Quadratic Forms Generated by Absolutely Regular Sequences III’, Yokohama Mathematical Journal, 40, 1–9; Fan, J., and Li, Q. (1999), ‘Central Limit Theorem for Degenerate U-Statistics of Absolutely Regular Processes with Applications to Model Specification Testing’, Journal of Nonparametric Statistics, 10, 245–271; Gao, J., and King, M.L. (2004), ‘Adaptive Testing in Continuous-time Diffusion Models’, Econometric Theory, 20, 844–882; Gao, J. (2007), Nonlinear Time Series: Semiparametric and Nonparametric Methods, Chapman & Hall/CRC; Gao, J., and Hong, Y. (2008), ‘Central Limit Theorem for Generalized U-statistics with Applications in Nonparametric Specification’, Journal of Nonparametric Statistics, 20, 61–76]. In this paper, we provide an improved version with the asymmetric kernel method which is quite useful for application to nonparametric methods in various situations. As an illustration of the usefulness of our result, CLTs for quadratic errors of a nonparametric density estimator are developed under dependency, which is meaningful in its own right.