  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.
