Tail behavior of the empirical distribution function of convolutions

Control charts based on convolutions require study of the tail behavior of the empirical distribution function of convolutions. It is well-known that this empirical distribution function at a fixed argument $x$ is asymptotically normal. The asymptotic normality is extended here to sequences $x_n$ tending to infinity at a suitable rate. At still larger $x_n$'s Poisson limiting distributions come in for the classical empirical distribution function. Surprisingly, this property does not generalize to its convolution counterpart, since for those $x_n$'s it is degenerate at 0 with probability tending to 1. Exact inequalities for the tail behavior are presented as well.