Timeline for Coupling Marginals of Distributions on the Sphere
Current License: CC BY-SA 3.0
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| Aug 26, 2015 at 23:39 | comment | added | AustinC | These are good points. Some interesting examples are found by taking a dirac delta $\delta_{x^n}$, and averaging it over all permutation. Convex combinations of such averages preserve $\|X^n\|^2 = n$. And indeed we can't recover a distribution of this form by Dominik's methods. I'm interested in marginals since I'm interested in which distributions $P_Y$ can be induced through a transform $P_{Y|X}$ of the marginal. Specifically, I'm maximizing some function of $P_Y$ over such distributions. But I thought the question of which distributions could be coupled is interesting alone. | |
| Aug 26, 2015 at 22:54 | comment | added | Anthony Quas | You shouldn't expect to be able to do this for atomic distributions. If you had a distribution supported on values whose squares never sum to $n$, you would have a counterexample. The same would be true if the distribution is `almost atomic' (e.g. 99.99% of the mass is within $10^{-100}$ of a collection of values with the above property. | |
| Aug 26, 2015 at 22:34 | comment | added | Dominik | The class of such distributions will probably be fairly large. For example, for each i.i.d. sequence $X_i$ with $\mathbb{P}(X_1 = 0) = 0$ the distribution of $\sqrt{n}X_i\left(X_1^2 + \ldots + X_n^2\right)^{-\tfrac{1}{2}}$ satisfies the condition. Why are you only interested in those distributions that have the same marginals for all coordinates? This property is certainly less restrictive than the permutation invariance. | |
| Aug 26, 2015 at 21:55 | history | edited | AustinC | CC BY-SA 3.0 |
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| Aug 26, 2015 at 21:52 | review | First posts | |||
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| Aug 26, 2015 at 21:50 | history | asked | AustinC | CC BY-SA 3.0 |