Moment Problems & Measure Recovery

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Abstractions about recovering measures from their moment sequences, including the Hausdorff, Hamburger and Stieltjes moment problems, related transforms, and probabilistic inequalities governing measure and event probabilities.

6 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Hamburger moment problem — The problem of deciding whether a given sequence is the sequence of moments of a positive Borel measure on the whole real line, and whether that representing measure is unique.
• Hausdorff moment problem — The problem of characterizing sequences that are moments of a positive measure on the unit interval, with a unique representing measure whenever one exists.
• Moment problem — The inverse problem of deciding whether a sequence is represented by moments of a measure, and whether that representing measure is unique.
• Stieltjes moment problem — The problem of deciding whether a sequence is the moment sequence of a positive measure on the nonnegative half-line and whether that measure is unique.
• Stieltjes transformation — Map a measure to an analytic function off its support by integrating the resolvent kernel 1/(t−z), with boundary limits recovering density and encoding moments and spectral information.
• Van den Berg–Kesten inequality — A product-measure inequality bounding the probability of disjoint occurrence of two events by the product of their individual probabilities.