The following Theorem gives the limit process for the kernel estimator of density. Theorem 2. For a kernel function K satisfying Assumption A, if h → 0 and ¯ h 2 $^{l}$n = O (1) as n → ∞ the sequence of generalized random processes n 1 2 ( ̂ f − f − ¯ h $^{l}$B ( h, K ) ) converges to a generalized Gaussian process with mean functional zero and covariance functional C which for any ψ$_{1}$, ψ$_{2}$ ∈ D$_{l}$$_{+}$$_{k}$ provides If n ¯ h 2 l → 0 , then ˆ f − f converges at the parametric rate √$_{n}$ to a generalized zero mean Gaussian process with covariance functional C in (9) . Proof. See appendix. The condition on the bandwidth that makes it possible to eliminate the bias asymptotically is less stringent than in the usual topologies and also than that originally stated in Zinde-Walsh (2008). Under this requirement on the bandwidth convergence is actually at a parametric rate and the limit covariance does not involve the kernel function. 4 Distribution function conditional on some 15 ( C, ( ψ$_{1}$, ψ$_{2}$ )) = E ([ ψ$_{1}$ ( x ) − Eψ$_{1}$ ( x )] [ ψ$_{2}$ ( x ) − Eψ$_{2}$x )]) = cov ( ψ$_{1}$, ψ$_{2}$ ) . (9)