An Improved Bernstein-type Inequality for C-Mixing-type Processes and Its Application to Kernel Smoothing

There are many processes, particularly dynamic systems, that cannot be described as strong mixing processes. \citet{maume2006exponential} introduced a new mixing coefficient called C-mixing, which includes a large class of dynamic systems. Based on this, \citet{hang2017bernstein} obtained a Bernstein-type inequality for a geometric C-mixing process, which, modulo a logarithmic factor and some constants, coincides with the standard result for the iid case. In order to honor this pioneering work, we conduct follow-up research in this paper and obtain an improved result under more general preconditions. We allow for a weaker requirement for the semi-norm condition, fully non-stationarity, non-isotropic sampling behavior. Our result covers the case in which the index set of processes lies in $\mathbf{Z}^{d+}$ for any given positive integer $d$. Here $\mathbf{Z}^{d+}$ denotes the collection of all nonnegative integer-valued $d$-dimensional vector. This setting of index set takes both time and spatial data into consideration. For our application, we investigate the theoretical guarantee of multiple kernel-based nonparametric curve estimators for C-Mixing-type processes. More specifically we firstly obtain the $L^{\infty}$-convergence rate of the kernel density estimator and then discuss the attainability of optimality, which can also be regarded as an upate of the result of \citet{hang2018kernel}. Furthermore, we investigate the uniform convergence of the kernel-based estimators of the conditional mean and variance function in a heteroscedastic nonparametric regression model. Under a mild smoothing condition, these estimators are optimal. At last, we obtain the uniform convergence rate of conditional mode function.