A multiscale method for bounding the supremum of a stochastic process. Approximate each index by points 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 is the set of approximation edges, then pointwise . 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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