  Discrete kernel smoothing is now gaining importance in nonparametric
statistics. In this paper, we investigate some asymptotic properties of the
normalized discrete associated-kernel estimator of a probability mass function.
We show, under some regularity and non-restrictive assumptions on the
associated-kernel, that the normalizing random variable converges in mean
square to 1. We then derive the consistency and the asymptotic normality of the
proposed estimator. Various families of discrete kernels already exhibited
satisfy the conditions, including the refined CoM-Poisson which is
underdispersed and of second-order. Finally, the first-order binomial kernel is
discussed and, surprisingly, its normalized estimator has a suitable asymptotic
behaviour through simulations.
