3 Gaussian limit process for the kernel den- sity estimator in the space of generalized functions We now describe the limit process for the kernel estimator (3) as ¯ h = max 1 ≤ j ≤ $_{k}$h$_{j}$ → 0 with n → ∞ , as a generalized random process. Such a de- scription was in Zinde-Walsh (2008), but there was an error in the variance computation that is corrected here. The main result here is that in the gen- eralized functions space convergence of the kernel density estimator can be at a parametric rate for a suitable selection of the kernel and bandwidth; unlike the usual case in the literature this selection alone provides the result independently of any properties (smoothness) of the distribution. Recall that convergence of generalized random functions is defined (see, e.g. Gel’fand and Vilenkin, 1964 or summary in Zinde-Walsh, 2008) as weak convergence of random linear continuous functionals on the space D$_{·}$ (for any of the D$_{k}$, D$_{∞}$, etc. spaces here) that are indexed by the functions in D$_{·}$ : stochastic convergence of random functionals, ˆ f, follows from stochastic con- vergence of random vectors of values of the functional (( ˆ f, ψ$_{1}$ ) , ..., ( ˆ f, ψ$_{m}$ )) ′ for any finite set ( ψ$_{1}$, ..., ψ$_{m}$ ) with ψ$_{l}$ ∈ D$_{·}$. Thus we need to consider the be- havior of such random vectors. Theorem 2 in Zinde-Walsh (2008) gives the convergence rate O ( ¯ h $^{l}$) for the generalized bias function of the kernel estimator based on a random sample 13