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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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