= Generic chaining
{title2=$\gamma_2(T,d)$}
<Generic chaining> bounds the <supremum> of a <stochastic process> by choosing an optimized hierarchy of approximations to its index <metric space>. A standard complexity is
$$
\gamma_2(T,d)=\inf_{(T_n)}\sup_{t\in T}\sum_{n\ge0}2^{n/2}d(t,T_n),
$$
where $|T_0|=1$, $|T_n|\le2^{2^n}$ 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 $N\ge2$, 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.<High-Dimensional Probability by Roman Vershynin>[First edition, Section 8.5, Talagrand's functional and generic chaining]{ref}
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