Abstract
In Chapter 9 we investigate how to control the supremum of the empirical process over a class of functions. The fundamental theoretical question in this direction is whether there exists a “best possible” method to control this supremum at a given size of the random sample. We offer a natural candidate for such a “best possible” method, in the spirit of the Bednorz-Latała result of Chapter 5. Whether this natural method is actually optimal is a major open problem. To illustrate that meditating on these theoretical questions might help to solve practical problems, we present a somewhat streamlined proofs of two deep recent results.
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Talagrand, M. (2014). Theory and Practice of Empirical Processes. In: Upper and Lower Bounds for Stochastic Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54075-2_9
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DOI: https://doi.org/10.1007/978-3-642-54075-2_9
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