Timeline for Existence of a map connecting two marginals of a product measure
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2014 at 2:25 | comment | added | SBF | @MichaelGreinecker: thanks, but Nate's example (appied to $X$ being an arbitrary finite set and $\bar X = [0,1]$) is something I'm also dealing with, so surely in general in the situation I'm interested in $\bar p$ may not be a pushforward of $p$. Btw, you shall know that one I guess. | |
| Jan 14, 2014 at 2:23 | comment | added | Michael Greinecker | In optimal transport theory one minimzes an integral over probabilities on a product space with given marginals. If the solution is supported on the graph of a function, one has a Monge-solution. There is a very, very large and active literature on the topic that might provide useful sufficient conditions. | |
| Jan 14, 2014 at 2:21 | vote | accept | SBF | ||
| Jan 14, 2014 at 2:19 | answer | added | Nate Eldredge | timeline score: 2 | |
| Jan 14, 2014 at 2:15 | comment | added | SBF | @NateEldredge: Thanks. Nope, your example applies - you can post it as an answer (unless the question will be closed before that as a trivial one) | |
| Jan 14, 2014 at 2:09 | comment | added | Nate Eldredge | Are some conditions missing? Take $X = \{0\}$, $\bar{X} = [0,1]$, $A = X \times \bar{X}$, $p = \delta_0$ is a point mass, $\bar{p}$ is Lebesgue measure, and $P = p \times \bar{p}$. Then the marginals are as required but clearly $\bar{p}$ cannot be the pushforward of $p$. | |
| Jan 14, 2014 at 1:40 | history | asked | SBF | CC BY-SA 3.0 |