The Equivalence of Weak, Strong and Complete Convergence in $L_1$ for Kernel Density Estimates

Abstract

Let $f$ be a density on $R^d$, and let $f_n$ be the kernel estimate of $f$, $f_n(x) = (nh^d)^{-1} \sum^n_{i=1} K((x - X_i)/h)$ where $h = h_n$ is a sequence of positive numbers, and $K$ is an absolutely integrable function with $\int K(x) dx = 1$. Let $J_n = \int |f_n(x) - f(x)| dx$. We show that when $\lim_nh = 0$ and $\lim_nnh^d = \infty$, then for every $\varepsilon > 0$ there exist constants $r, n_0 > 0$ such that $P(J_n \geq \varepsilon) \leq \exp(-rn), n \geq n_0$. Also, when $J_n \rightarrow 0$ in probability as $n \rightarrow \infty$ and $K$ is a density, then $\lim_nh = 0$ and $\lim_nnh^d = \infty$.