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Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows:

Let $p \in \mathbb{R}[x_1, x_2, ...]$ be a polynomial such that, for any sequence $X_1, X_2, ...$ of i.i.d. random variables on the real line all of whose moments are finite, we have $\mathbb{E}(p(X_1, X_2, ...)) \ge 0$. Then there exists $q \in \mathbb{R}[x_1, x_2, ...]$ such that $\mathbb{E}(p(X_1, X_2, ...)) = \mathbb{E}(q(X_1, X_2, ...))$ for all possible $X_i$ and such that $q$ is a sum of squares.

The idea is that $\mathbb{E}(p)$ is a polynomial function of the moments of $X_1$. The motivating example is when $\mathbb{E}(p)$ is a Hankel determinant $h_n$, which is a positive scalar multiple of

$$\mathbb{E} \left[ \prod_{1 \le i < j \le n} (X_i - X_j)^2 \right].$$

What work has been done on this conjecture? Looking at the papers citing this one didn't turn up anything promising, and neither did various Google searches.

Edit: There is a familial resemblance to Hilbert's 17th problem, but it's probably worth noting that a counterexample to the strong version of Hilbert's 17th problem is not automatically a counterexample to this problem. For example, it's known that

$$p(x, y, z) = x^6 + y^4 z^2 + y^2 z^4 - 3x^2 y^2 z^2$$

is everywhere non-negative (by AM-GM) so in particular satisfies the hypotheses of the problem, but is not a sum of squares of polynomials. However, $\mathbb{E}(p) = \mathbb{E}(q)$ where

$$q(x, y, z) = (x^3 - xy^2)^2 + \frac{3}{2} (x(y^2 - z^2))^2.$$

Edit #2: I'm now a little concerned that I'm misstating the conjecture because there appears to be a very small counterexample. Taking $p = x_1 x_2$ we have $\mathbb{E}(p) = m_1^2$ (where $m_1 = \mathbb{E}(X_1)$), which is non-negative. However, no sum of squares $q$ exists such that $\mathbb{E}(q) = m_1^2$: any such $q$ must be a sum of squares of linear polynomials, and in that case $\mathbb{E}(q)$ must contain a term $m_2 = \mathbb{E}(X_1^2)$ with positive coefficient. Perhaps one should allow in addition sums of squares of polynomials in the moments...

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Ah, good that you made the edit; I was wondering about the Motzkin polynomial; I quickly skimmed the Rota paper, and while I don't really know the details, I thought the conjecture was asking for "a sum of squares of umbral polynomials ---" maybe that's what makes the difference here? – Suvrit Sep 21 '12 at 8:11
@Suvrit: that is equivalent to the conjecture as I have formulated it (as far as I can tell). An umbral polynomial is a polynomial before taking expectations. – Qiaochu Yuan Sep 21 '12 at 17:48
up vote 4 down vote accepted

I think your reformulation of the conjecture in the Rota-Shen paper is correct. Also your counterexample is correct. So I guess the conjecture was not stated properly in that article. Perhaps one should restrict to translation invariant polynomials only. By that I mean polynomials in the moments which remain invariant if we change the basic random variable $X$ to $X+c$ where $c$ is some constant, the first example being the variance $m_2-m_1^2$. I think one can make the conjecture more precise by asking the polynomial to be a positive linear combination of evaluations of squares of products of differences of umbral letters.

It is hard to know what Rota's motivation was. It is not clear to me if for him this conjecture was a matter of probability theory or of (real) classical invariant theory. The Hankel determinants which determine moment sequences and motivate this conjecture are essentially the catalecticants of binary forms and their umbral expression with squared Vandermondes is the standard representation of these catalecticants using the classical symbolic method. My feeling is Rota's motivation might be to understand a big problem in classical invariant theory which is how does one know that the evaluation of an umbral polynomial is nonzero (see this very interesting article by Alexandersson and Shapiro).

Also for the proper formulation of the conjecture, I suggest contacting J. Shen. Her latest article, "Least-squares halftoning via human vision system and Markov gradient descent (LS-MGD): algorithm and analysis", SIAM Rev. 51 (2009), no. 3, 567–589, has a contact email address for her. Another person who works in this area and might know what exactly Rota's conjecture was is Elvira Di Nardo who gave lectures at the SLC 67 on cumulants and umbral calculus.

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So does this mean that my comment about requiring umbral polynomials is correct? or it does not matter? – Suvrit Sep 21 '12 at 16:20
@Suvrit: Sorry, I don't yet understand your comment. When you say sums of squares of umbral polynomials, versus which other interpretation might there be a confusion? – Abdelmalek Abdesselam Sep 21 '12 at 17:40

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