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...

umbral polynomials---" maybe that's what makes the difference here? – Suvrit Sep 21 '12 at 8:11