Moderate Deviations and Large Deviations for Kernel Density Estimators

Let fn 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 ℝd. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{ \sup _{x \in \mathbb{R}^d } |f_n (x) - E(f_n (x))|,n \geqslant 1\} $$ \end{document} hold.