## 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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### References:

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