On some limit laws for perturbed empirical distribution functions (Q1341363)

The convergence of random variables of the form \(S_ n = \int F_ n (T_ n - s) v_ n (ds)\) is established, where \(T_ n\) is some random variable, \(F_ n\) is an empirical distribution function based on an independent sample of size \(n\) and \(v_ n\) is some measure. As an example we state the following result. If \(a \in R\) is a real constant such that \(T_ n \to a\) a.s. (in probability, respectively), \(v_ n \to v\) weakly, then \(S_ n \to b = \int F(a - s) v(ds)\) a.s. (in probability, respectively). Here \(F\) is a continuous d.f. of a sample.