  In this paper we study the ideal variable bandwidth kernel density estimator
introduced by McKay (1993) and Jones, McKay and Hu (1994) and the plug-in
practical version of the variable bandwidth kernel estimator with two sequences
of bandwidths as in Gin\'e and Sang (2013). Based on the bias and variance
analysis of the ideal and true variable bandwidth kernel density estimators, we
study the central limit theorems for each of them.
