# Empirical Process in M-Estimation (Chapter 3)

## March 21, 2018

In Chapter 3, one of the most important theorem in empirical process theory, namely, the Uniform Law of Large Numbers (ULLN), is stated and discussed.
Summary If a function class $\mathcal{G}$ is said to satisfy the ULLN, then $$ \sup_{g \in \mathcal{G}} \left| \int gd(P_n - P) \right| \rightarrow 0, \quad \text{a.s.} $$ The ULLN means that the expected and empirical average become closer as the number of samples gets larger for any $g \in \mathcal{G}$, which is explained in the following figure.
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