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We consider perturbed empirical distribution functions$$\hat F_n (x) = 1/n\sum\nolimits_{i = 1}^n {G_n (x - X_i )} $$, where {Ginn, n≥1} is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at 0, and {Xi, 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$$\hat F_n (U_n )$$ whereUn is aU-statistic based onX1,...,Xn. 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.

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