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There are many processes, particularly dynamic systems, that cannot bedescribed as strong mixing processes. \citet{maume2006exponential} introduced anew mixing coefficient called C-mixing, which includes a large class of dynamicsystems. Based on this, \citet{hang2017bernstein} obtained a Bernstein-typeinequality for a geometric C-mixing process, which, modulo a logarithmic factorand some constants, coincides with the standard result for the iid case. Inorder to honor this pioneering work, we conduct follow-up research in thispaper and obtain an improved result under more general preconditions. We allowfor a weaker requirement for the semi-norm condition, fully non-stationarity,non-isotropic sampling behavior. Our result covers the case in which the indexset 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 spatialdata into consideration. For our application, we investigate the theoreticalguarantee of multiple kernel-based nonparametric curve estimators forC-Mixing-type processes. More specifically we firstly obtain the$L^{\infty}$-convergence rate of the kernel density estimator and then discussthe attainability of optimality, which can also be regarded as an upate of theresult of \citet{hang2018kernel}. Furthermore, we investigate the uniformconvergence of the kernel-based estimators of the conditional mean and variancefunction in a heteroscedastic nonparametric regression model. Under a mildsmoothing condition, these estimators are optimal. At last, we obtain theuniform convergence rate of conditional mode function.

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