TITLE: Gaussian and the convex hull of moment curves
QUESTION [7 upvotes]: Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a sufficiently large $b$? 
Here is another more involved question: Is there a way we can tell how $b$ scales with $d$ (maybe we can even get a closed form solution)?
A modified version of this problem (related to this question: Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian) is, for what kind of $\delta$, the point $(c_1,\dots, c_d+\delta)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a sufficiently large $b$?

REPLY [2 votes]: For any finite sequence $c_1, \ldots , c_d$ which can be a moment sequence at all (which is characterized by a positive definiteness condition), there is a description of all probability measures $\mu$ that satisfy $\int t^j\, d\mu(t)=c_j$ for $j=1, \ldots , d$.
This is sometimes called the Nevanlinna parametrization, and it goes as follows, in outline: Form the orthogonal polynomials $p_j(t)$ with respect to your moment sequence. These satisfy a three term recurrence relation
$$
a_{n-1}p_{n-1} + a_n p_{n+1} + b_np_n = tp_n .\quad\quad\quad\quad (1)
$$
Now the solutions to the (truncated) moment problem are exactly the "spectral measures" of the associated finite Jacobi matrix (the difference operator that acts like the LHS of (1)). I put quotes because this term must be taken in a very general sense: basically, you consider arbitrary extensions of (1) from $\{ 1,2, \ldots , d\}$ to a half line (and the "extension" need not be a difference operator beyond $d$), and then take the spectral measure of this problem.
In particular, you can just impose a boundary condition at $d$, and then your measures will be spectral measures of the problem on a finite-dimensional space, so these will be finitely supported measures, as requested.
The maximum of the support of such a measure is the operator norm of the operator defined by (1), so this suggests an approach to your additional question.