Law of the iterated logarithm for perturbed empirical distribution functions evaluated at a random point for nonstationary random variables

Abstract

We consider perturbed empirical distribution functions $\hat{F}_n (x) = 1/n\sum^n_{i=1} G_n (x − X_i)$ , where {Gi$nn$, n≥1} is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at 0, and ${X_i, i≥1}$ is a non-stationary sequence of absolutely regular random variables. We derive the almost sure representation and the law of the iterated logarithm for the statistic $\hat{F}_n (U_n)$ where $U_n$ is a $U$-statistic based on $X_1, ... , X_n$. The results obtained extend or generalize the results of Nadaraya,$^{(7)}$ Winter,$^{(16)}$ Puri and Ralescu,$^{(9,10)}$ Oodaira and Yoshihara,$^{(8)}$ and Yoshihara,$^{(19)}$ among others.