 Generic chaining

Generic chaining bounds the supremum of a stochastic process by choosing an optimized hierarchy of approximations to its index metric space. A standard complexity iswhere $|T_0|=1$, $|T_n|\le 2^{2^n}$ and $d(t,T_n)={\inf }_{s\in T_n}d(t,s)$. Sparse early nets are cheap; progressively larger nets reduce the approximation error. The infimum allows the hierarchy to adapt to the geometry instead of using a single uniform covering scale.

For a centered Gaussian process, its canonical distance is $d(s,t)=(\mathbb{E}|X_s-X_t{|}^2{)}^{1/2}$. Under standard separability and boundedness assumptions, the expected supremum is comparable, up to universal constants, to ${\gamma }_2(T,d)$. For $N$ independent unit-variance Gaussian variables, this scale is $\sqrt{\log N}$ for $N\ge 2$, explaining why a maximum grows much more slowly than the number of variables. Generic chaining refines chaining bounds that depend only on global covering numbers.^[ref]