Empirical Measures, Empirical Processes
We analyze properties of empirical measures and empirical processes. In particular, point-wise consistency and asymptotic normality of empirical measures are proven, where a point refers to a given Borel set. These results are extended to functionals. Furthermore, the Glivenko-Cantelli Theorem yields uniform consistency of empirical cumulative distribution functions. The principle of quantile transformation is discussed in detail, allowing for reducing the theory of empirical processes to the case of uniform parent distributions, at least in the case that continuous cumulative distribution functions are under consideration. Convergence in distribution of reduced empirical processes is established.
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