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Limiting behavior of the perturbed empirical distribution functions evaluated at \(U\)-statistics for strongly mixing sequences of random variables. (English) Zbl 0873.62050
Summary: We prove the almost sure representation, a law of the iterated logarithm and an invariance principle for the statistic \(\widehat{F}_n(U_n)\) for a class of strongly mixing sequences of random variables \(\{X_i, i\geq 1\}\). Stationarity is not assumed. Here \(\widehat{F}_n\) is the perturbed empirical distribution function and \(U_n\) is a \(U\)-statistic based on \(X_1,\dots,X_n\).

MSC:
62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
62E20 Asymptotic distribution theory in statistics
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