Moderate deviations and large deviations for kernel density estimators

Let f(n) be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in R-d. It is proved that if the kernel function is an integrable function with bounded variation, and the common density function f of the random variables is continuous and f(x) --> as \x\ --> infinity , then the moderate deviation principle and large deviation principle for {supx is an element of R-d \f(n)(x)-E( f(n)(x))\, n greater than or equal to 1} hold.