A Note on the k-NN Density Estimate

k-NN (k Nearest Neighbour) density estimate as a nonparametric estimation method is widely used in machine learning or data analysis. The convergence problem of k-NN approach has been intensively investigated. In particular, the equivalence of convergence in weak or strong sense (i.e. in probability sense or in almost surely sense) has been respectively developed. In this note, we will show that the k-NN estimator converges in probability is equivalent to converge in the L-2 sense. Moreover, some relevant asymptotic results about the expectations of k-NN estimator will be established.