On the asymptotic normality for integrated square error of Wegman-Davies recursive density estimators

In this article, let {Xn,n >= 1} be a sequence of i.i.d. random variables with common probability density function f, and fn denotes a Wegman-Davies recursive density estimator fn(x)=1nhn1/2 n-ary sumation i=1n1hi1/2K(x-Xihi), where K is a kernel function and {hi,i >= 1} is a sequence of band-width parameters. The asymptotic normality for integrated square error of Wegman-Davies recursive density estimators are established by using martingale method.