Abstract
Strassen [23] established that there exists a two step martingale with marginal distributions μ, ν if and only if μ, ν are in convex order. Recently Choné, Gozlan and Kramarz [6] obtained a transport characterization of the stochastic order defined by convex positively 1-homogeneous functions, in the spirit of Strassen’s theorem under certain technical assumptions. In this note we prove the result of [6] in full generality. We also observe that the restriction of the result to the case where μ, ν are supported on a half space is equivalent to Strassen’s classical theorem.
Funding Statement
This research was funded in whole or in part by by the Austrian Science Fund (FWF) through projects 10.55776/P35197 and 10.55776/P34743. For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission.
Acknowledgments
Both authors thank the anonymous referees for their suggestions and remarks that significantly improved the presentation of this note and Mathias Beiglböck, Leo Brauner, and Gudmund Pammer for many helpful discussions and remarks.
Citation
Stefan Schrott. Daniel Toneian. "On Strassen’s theorem for support functions." Electron. Commun. Probab. 29 1 - 12, 2024. https://doi.org/10.1214/24-ECP618
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