A General Projection Property for Distribution Families

Part of Advances in Neural Information Processing Systems 22 (NIPS 2009)

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Yao-liang Yu, Yuxi Li, Dale Schuurmans, Csaba Szepesvári


We prove that linear projections between distribution families with fixed first and second moments are surjective, regardless of dimension. We further extend this result to families that respect additional constraints, such as symmetry, unimodality and log-concavity. By combining our results with classic univariate inequalities, we provide new worst-case analyses for natural risk criteria arising in different fields. One discovery is that portfolio selection under the worst-case value-at-risk and conditional value-at-risk criteria yields identical portfolios.