In probability theory, central limit theorems (CLTs), broadly speaking, state that the distribution of the sum of a sequence of random variables (r.v.'s), suitably normalized, converges to a normal distribution as their number <italic>n</italic> increases indefinitely. However, the preceding convergence in...

Let {(Xi, Yi); I = 1,2,...} be a sequence of i.i.d. r.v.'s and denote by m(y x0), -[infinity] y [infinity], the conditional distribution function of Y given X = x0, -[infinity] x0 [infinity]. In this paper we propose and discuss certain smooth variants (based both on single as well as double...

Some known kernel type estimates of a density and its derivatives f(p) are considered. Strong uniform consistency properties over the whole real line are studied. Improved rate of convergence results are established under substantially weaker smoothness assumptions of f(p), p [greater-or-equal,...