Gaussian and the convex hull of moment curves

Question

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 $b$?

Answer 1

Let $P_d$ be the set of all probability measures $\mu$ on $\mathbb R$ whose first $d$ moments $c_1,\dots, c_d$ are the same as those of the standard normal distribution $\gamma$. By Theorem 3.1 in [1] (and the sentence following it there), there is an extreme measure $\nu$ of $P_d$. By Theorem 2.1 and Example 2.1 (a) in [1] or Corollary 11 in [2], $\nu$ is a mixture of finitely many Dirac measures. So, the support set of $\nu$ is bounded and hence contained in the interval $[-b,b]$ for some real $b>0$. That is, the point $(c_1,\dots, c_d)$ is in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$.

This provides a positive answer to the question if the $[0,b]$ is replaced by $[-b,b]$. Actually, the answer will remain positive for any probability measure on $\mathbb R$ in place of the standard normal distribution $\gamma$.

The answer to the original question is an obvious no, since for any random variable $X$ with support set contained in $[0,b]$ and $EX^2=\int_{\mathbb R}x^2\nu(dx)=1>0$, one would have $EX>0=\int_{\mathbb R}x\nu(dx)$.

Answer 2

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.