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The last example of the ULLN that van de Geer (2000) present regards the convex hull of a function class. Take this subsection as a cherry on top.
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In the previous article, results from lemma 3.1 were covered. Here, the uniform law for Vapnik-Chervonenkis subgraph class, a useful class of functions that has a close relationship with classification problem, will be derived from theorem 3.7.
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The ULLN under finite bracketing entropy condition (lemma 3.1) and vanishing random entropy condition (theorem 3.7) paves the way to the uniform law of frequently used function classes. Here, three examples from lemma 3.1 will be presented. An example from theorem 3.7 will be covered in the next article.
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Using all the lemmas from 3.2 to 3.5, we prove another sufficiency of the uniform law: the vanishing random entropy condition. This condition is fairly weaker than the finite bracketing entropy condition in two senses. First, instead of the (sup-normed) envelope condition, it only requires the envelope to be integrable. In $L^p(Q)$ where $Q$ is a finite measure, this is clearly implied by the envelop condition. In addition, as we already saw before, the vanishing random entropy is implied by the finite bracketing entropy condition.
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The part of the second necessity of the ULLN we would like to prove is the vanishing random entropy condition:
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