The $L_1$ Convergence of Kernel Density Estimates

Abstract

Let $X_1, \cdots, X_n$ be a sequence of independent random vectors taking values in $\mathbf{\mathbb{R}}^d$ with a common probability density $f$. If $f_n(x) = (1/h) h^{-d}_n\sum^n_{i = 1}K((x - X_i)/h_n)$ is the kernel estimate of $f$ from $X_1, \cdots, X_n$ then conditions on $K$ and $\{h_n\}$ are given which insure that $\int|f_n(x) - f(x)|dx \rightarrow_n 0$ in probability or with probability one. No continuity conditions are imposed on $f$.