= Chaining
{title2=$X_t-X_{t_0}=\sum_k(X_{\pi_k(t)}-X_{\pi_{k-1}(t)})$}
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 $\sup_t(X_t-X_{t_0})\leq\sum_k\max_{(u,v)\in E_k}(X_u-X_v)$. 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.
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