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arXiv · 2310.20402

On Strassen's Theorem for support functions

Stefan Schrott·Daniel Toneian
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Abstract

Strassen 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-Kramarz 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 Choné-Gozlan-Kramarz result 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.

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Stefan Schrott, Daniel Toneian. 2025-06-03. On Strassen's Theorem for support functions. https://doi.org/10.1214/24-ecp618

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