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We consider the minimization problem of φ-divergences between a given probability measure P and subsets Ω of the vector space M F of all signed measures which integrate a given class F of bounded or unbounded measurable functions. The vector space M F is endowed with the weak topology induced by the class F ∪ B b where B b is the class of all bounded measurable functions. We treat the problems of existence and characterization of the φ-projections of P on Ω. We also consider the dual equality and the dual attainment problems when Ω is defined by linear constraints.

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decomposition of lattice distributions into convolutions of Poisson signed measures , Theory of Probability & Its Applications , 49 ( 3 ) ( 2005 ), 545 – 552 . [20

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Abstract  

Let (X, d) be a compact metric space and let

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(X) denote the space of all finite signed Borel measures on X. Define I:
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(X) → ℝ by I(μ) = ∫XX d(x, y)dμ(x)dμ(y), and set M(X) = sup I(μ), where μ ranges over the collection of measures in
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(X) of total mass 1. The space (X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in
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(X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure. This paper explores the constant M(X) and other geometric aspects of X in the case when the space X is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are L 1-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [11] [13].

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. [5] Andrievskii , V. V. Blatt , H.-P. 2002 Discrepancy of Signed Measures and Polynomial Approximation Springer

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] Hochberg , K.J. , A signed measure on path space related to Wiener measure , Annals of Probability , 6 ( 1978 ) 433 – 458 . [17

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.” A technical complication is the sign of Q or μ *. Yeung ( 2008 , p. 59) noted that one has to be cautious in referring to this information measure as a signed measure instead of a measure (because the latter can assume only nonnegative values

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