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Given a distribution $P_X$ on $\mathbb{R}$, when does there exist a coupling (i.e. joint distribution) $P_{X^n}$ of $X_1,...,X_n$, each distributed according to $P_X$, such that $\sum X_i^2 = n$ almost surely?

The motivation comes from the following: let $P_{X^n}$ be an arbitrary distribution on $\sqrt{n}S^{n-1}$ that is permutation invariant, i.e. we can assume all marginal distributions $X_i$ have the same probability law. Then how can we characterize the marginal distribution of $P_{X^n}$? E.g. Clearly $\mathbb{E}[X^2] = 1$, and $X^2 \leq n$ a.s., but given a random variable satisfying these properties, can I couple $n$ copies of them onto the sphere?

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  • $\begingroup$ 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. $\endgroup$
    – Dominik
    Commented Aug 26, 2015 at 22:34
  • $\begingroup$ 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. $\endgroup$
    – Anthony Quas
    Commented Aug 26, 2015 at 22:54
  • $\begingroup$ 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. $\endgroup$
    – AustinC
    Commented Aug 26, 2015 at 23:39

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