Random weighting estimation of kernel density

Let X1,X2,…,Xn be independent and identically distributed random variables with common probability density function f(x). The kernel density estimation of f(x) can be defined as fn(x)=(1/nhn)∑i=1nK((x−Xi)/hn), where K(u) is a kernel function and hn>0 is a series of positive constants that satisfy limn→∞hn=0. A theory is established to approximate kernel density estimation fn(x) by using random weighting estimation H^n(x) of f(x). Under certain conditions, it rigorously proves that nhn(H^n(x)−fn(x)) and nhn(fn(x)−f(x)) have the same limiting distribution for any random series X1,X2,…,Xn.