Asymptotic properties of perturbed empirical distribution functions evaluated at a random point (Q1100824)

Given a sequence \(X_ i\), \(i\geq 1\), of independent and identically distributed random variables, the usual perturbed empirical distribution function is \[ F^*_ n(x)=n^{=1}\sum^{n}_{i=1}G_ n(x-X_ i) \] where \(G_ n\), \(n\geq 1\), is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at zero. Let \(U_ n\), \(n\geq 1\), be a sequence of U-statistics. We derive the almost sure representation, the law of iterated logarithm, and an invariance principle for the statistic \(F^*_ n(U_ n)\).