Chaining ($X_t-X_{t_0}={\sum }_k(X_{{\pi }_k(t)}-X_{{\pi }_{k-1}(t)})$)

A multiscale method for bounding the supremum of a stochastic process. Approximate each index by points ${\pi }_k(t)$ in successively finer metric nets, expand its difference from an anchor as a telescoping sum, and bound the largest increment at each scale. For a finite index set the finest net can be the entire set, making this decomposition exact. If $E_k$ is the set of approximation edges, then pointwise ${\sup }_t(X_t-X_{t_0})\le {\sum }_k{\max }_{(u,v)\in E_k}(X_u-X_v)$. Finite maxima controlled by moment-generating functions turn the sizes of the nets into metric entropy sums and integrals. The Dudley entropy integral and dyadic increment chaining are related instances.

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]