generic chaining

Generic chaining is a method for proving maximal inequalities. It was invented by Michel Talagrand, whose beard I hope to be able to mimic some day.

We are in the setting that was described in chaining. We are trying to obtain a maximal inequality for the sub-Gaussian process $(X_{\theta })$, $\theta \in \Theta$. Chaining gave a bound of the form

$$\mathbb{E}\underset{\theta }{\sup }{X}_{\theta }\le \underset{\theta \in \Theta }{\sup }\sum _{0\le n\le N}{2}^{n/2}\rho (\theta ,{\Theta }_n),$$

where ${\Theta }_0\subset {\Theta }_1\subset \cdots \subset {\Theta }_N=\Theta$ is a chain of sets approximating $\Theta$, where $|{\Theta }_n|\le 2^{2^n}$.
Dudley chaining moves the supremum inside the sum and then states the bound in terms of entropy numbers. In generic chaining we leave the supremum outside the sum.