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.