# Reference request: a conjecture of Rota on positive functions of a random variable

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
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.