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Superresolution rates in Prokhorov metric. (English) Zbl 0861.60005

Summary: Consider the problem of recovering a probability measure supported by a compact subset \(U\) of \(\mathbb{R}^m\) when the available measurements concern only some of its \(\Phi\)-moments (\(\Phi\) being an \(\mathbb{R}^k\) valued continuous function on \(U\)). When the true \(\Phi\)-moment \(c\) lies on the boundary of the convex hull of \(\Phi(U)\), generalizing the results of the second author and E. Gassiat [SIAM J. Math. Anal. 27, No. 4, 1129-1152 (1996)], we construct a small set \(R_{\alpha,\delta(\varepsilon)}\) such that any probability measure \(\mu\) satisfying \(|\int_U \Phi(x)d\mu(x)-c|\leq\varepsilon\) is almost concentrated on \(R_{\alpha,\delta(\varepsilon)}\). When \(\Phi\) is a pointwise \(T\)-system (extension of \(T\)-systems), the study of the set \(R_{\alpha,\delta(\varepsilon)}\) leads to the evaluation of the Prokhorov radius of the set \(\{\mu: |\int_U \Phi(x)d\mu(x)-c|\leq \varepsilon\}\).

MSC:

60A10 Probabilistic measure theory
43A07 Means on groups, semigroups, etc.; amenable groups
62A01 Foundations and philosophical topics in statistics
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