Random weighting estimation of kernel density

Let X 1 , X 2 , … , X n 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 f n ( x ) = ( 1 / n h n ) ∑ i = 1 n K ( ( x − X i ) / h n ) , where K(u) is a kernel function and hn>0 is a series of positive constants that satisfy lim n → ∞ h n = 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 n h n ( H ^ n ( x ) − f n ( x ) ) and n h n ( f n ( x ) − f ( x ) ) have the same limiting distribution for any random series X 1 , X 2 , … , X n .