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.
Description
Publisher's, offprint version
Keywords
Perturbed empirical distribution functions, absolutely regular processes, strong mixing, almost sure representation, law of the iterated logarithm
Citation
Puri, M. L. “Law of the iterated logarithm for perturbed empirical distribution functions evaluated at a random point for nonstationary random variables.” Journal of Theoretical Probability (1994), Volume 7 Issue 4, 831–855. Co-author: Michel Harel.
DOI
10.1007/BF02214375
Link(s) to data and video for this item
Relation
Rights
Type
Article