ASYMPTOTIC NORMALITY OF RECURSIVE KERNEL DENSITY ESTIMATES UNDER DEPENDENT ASSUMPTIONS

Let X₁, …, X_n be random samples from an unknown density funotion f（x）. The recursive kernel density function estimator can be obtained by putting f_n(x)=\frac1n \sum_j=1^n h_j^-1 K\left(\fracx-X_jh_j\right), where K is a univariate kernel funotion, and h_n is a sequence of positive numbers converging to zero. In the paper, the asymptotio multi-normality of g_n（x） is given in the case of dependent sample, where g_n（x）=（f（x）-Ef_n（x））/（var（f_n（x）））^1/2.