STRONG L_P-CONSISTENCY OF ONEDIMENSIONAL NEAREST NEIGHBOR DENSITY ESTIMATES

Suppose that X_1, \cdots, X_n are i.i.d. onedimensional random variables with a density f(x). If 2 a_n(x) is the length of the smallest interval centered at x and containing at least k_n points of X_1, \cdots, X_n, then f_n(x)=\frack_n2 n a_n(x) is a nearest neighbor density estimate of f(x). For p>1, we conclude that if \int f^p(x) d x< \infty, \lim _n \rightarrow \infty k_n / r=0 and \lim _n \rightarrow \infty k_n / \log n=\infty, then \lim _n \rightarrow \infty \int\left|f_n(x)-f(x)\right|^p d x=0, a.s.; conversely, if \lim _n \rightarrow \infty \int\left|f_n(x)-f(x)\right|^p d x=0, a.s., then \int f^p\left(x . d x< \infty, \lim _n \rightarrow \infty k_n^\prime n=0\right. and \lim _n \rightarrow \infty k_n=\infty.