  In this paper, we construct a moment inequality for mixing dependent random
variables, it is of independent interest. As applications, the consistency of
the kernel density estimation is investigated. Several limit theorems are
established: First, the central limit theorems for the kernel density estimator
$f_{n,K}(x)$ and its distribution function are constructed. Also, the
convergence rates of $\|f_{n,K}(x)-Ef_{n,K}(x)\|_{p}$ in sup-norm loss and
integral $L^{p}$-norm loss are proved. Moreover, the a.s. convergence rates of
the supremum of $|f_{n,K}(x)-Ef_{n,K}(x)|$ over a compact set and the whole
real line are obtained. It is showed, under suitable conditions on the mixing
rates, the kernel function and the bandwidths, that the optimal rates for
i.i.d. random variables are also optimal for dependent ones.
