Impossibility of weak convergence of kernel density estimators to a non-degenerate law in L₂(ℝ^d)

It is well known that the kernel estimator, for the probability density f on ℝ^d has pointwise asymptotic normality and that its weak convergence in a function space, especially with the uniform topology, is a difficult problem. One may conjecture that the weak convergence in L2(ℝ^d) could be possible. In this paper, we deny this conjecture. That is, letting, we prove that for any sequence {r_n} of positive constants such that r_n = o(√n), if the rescaled residual, converges weakly to a Borel limit in L₂(ℝ^d), then the limit is necessarily degenerate.