Sets that Support a Joint Distribution

Christopher Coscia; Martin Tassy; Peter Winkler

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arXiv · 2608.04135

Sets that Support a Joint Distribution

Christopher Coscia·Martin Tassy·Peter Winkler

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Abstract

Given probability distributions $μ$ and $ν$ on measure spaces $X$ and $Y$, and a closed set $S \subseteq X \times Y$, when is there a probability distribution on $X \times Y$ whose marginals are $μ$ and $ν$, and whose support is precisely $S$? We answer the question when the marginals are discrete, and when the marginals are continuous distributions on the real line. Of special interest is the case where $S \subseteq [0,1]^2$ and $μ$ and $ν$ are Lebesgue measure; then the above question is tantamount to ``when is $S$ the support of a doubly stochastic measure?". The discrete case is generalized to determine when a (possibly infinite) edge-capacitated, node-weighted graph supports a full, nowhere-zero flow; for the continuous case we provide a particularly straightforward characterization when the set in question is regular (i.e., is the closure of its interior).

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Christopher Coscia, Martin Tassy, Peter Winkler. 2026-08-04. Sets that Support a Joint Distribution. https://arxiv.org/abs/2608.04135

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