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

Authors:	Michel Harel Madan L Puri

Institution:	(1) U.R.A. C.N.R.S. 743, Orsay, France;(2) Indiana University, 47405 Bloomington, Indiana

Abstract:	We consider perturbed empirical distribution functions $\hat F_n (x) = 1/n\sum\nolimits_{i = 1}^n {G_n (x - X_i )}$ , where {Ginn, n1} 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 $\hat F_n (U_n )$ whereU _n is aU-statistic based onX 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.Research supported by the Office of Naval Research Contract N00014-91-J-1020.

Keywords:	Perturbed empirical distribution functions absolutely regular processes strong mixing almost sure representation U-statistic law of the iterated logarithm
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