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 ^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) 0 as |x| , then the moderate deviation principle and large deviation principle for hold.