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