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 f̂n denotes a Wegman-Davies recursive density estimator f̂n (x) =1nhn1/2∑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.