Uniform Glivenko–Cantelli Classes
نویسنده
چکیده
A class of sets, or functions, is said to be P–Glivenko–Cantelli if the empirical measure Pn converges in some sense to the true measure, P , as n → ∞, uniformly over the class of sets or functions. Thus, the notions of Glivenko–Cantelli, and likewise uniform Glivenko–Cantelli are for the most part qualitative assessments of how “well–behaved” a collection of sets or functions is, in the sense of convergence of empirical measures. Since the ability to positively or negatively identify this property in a given family of sets or functions is quite useful, it is natural to seek a more specific, or rather, more tractably calculable criterion, or set of criteria, that is equivalent with some form of Glivenko–Cantelli.
منابع مشابه
A Counterexample Concerning the Extension of Uniform Strong Laws to Ergodic Processes
We present a construction showing that a class of sets C that is Glivenko-Cantelli for an i.i.d. process need not be Glivenko-Cantelli for every stationary ergodic process with the same one dimensional marginal distribution. This result provides a counterpoint to recent work extending uniform strong laws to ergodic processes, and a recent characterization of universal Glivenko Cantelli classes.
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