### 3.4.2. Uniform Law of Vapnik-Chervonenkis Subgraph Classes

empirical process asymptotics Empirical Processes in M-estimation

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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### 3.4.1. Examples of Sufficiencies of the Uniform Law

empirical process asymptotics Empirical Processes in M-estimation

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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### 3.3. Uniform Law Under Random Entropy

empirical process asymptotics Empirical Processes in M-estimation

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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