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Minimal \(\mathcal{L}^p \)-densities with prescribed marginals. (English) Zbl 1473.46037

Summary: We derive sharp lower bounds for \(\mathcal{L}^p \)-functions on the \(n\)-dimensional unit hypercube in terms of their \(p\)-th marginal moments. Such bounds are the unique solutions of a system of constrained nonlinear integral equations depending on the marginals. For square-integrable functions, the bounds have an explicit expression in terms of the second marginals’ moments.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60E15 Inequalities; stochastic orderings
45G15 Systems of nonlinear integral equations
62P05 Applications of statistics to actuarial sciences and financial mathematics
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