The multidimensional truncated Moment Problem: Gaussian and Log-Normal Mixtures, their Carath\'eodory Numbers, and Set of Atoms

We study truncated moment sequences of distribution mixtures, especially from Gaussian and log-normal distributions and their Carath\'eodory numbers. For $\mathsf{A} = \{a_1,\dots,a_m\}$ continuous (sufficiently differentiable) functions on $\mathbb{R}^n$ we give a general upper bound of $m-1$ and a general lower bound of $\left\lceil \frac{2m}{(n+1)(n+2)}\right\rceil$. For polynomials of degree at most $d$ in $n$ variables we find that the number of Gaussian and log-normal mixtures is bounded by the Carath\'eodory numbers in \cite{didio17Cara}. Therefore, for univariate polynomials $\{1,x,\dots,x^d\}$ at most $\left\lceil\frac{d+1}{2}\right\rceil$ distributions are needed. For bivariate polynomials of degree at most $2d-1$ we find that $\frac{3d(d-1)}{2}+1$ Gaussian distributions are sufficient. We also treat polynomial systems with gaps and find, e.g., that for $\{1,x^2,x^3,x^5,x^6\}$ 3 Gaussian distributions are enough for almost all truncated moment sequences. For log-normal distributions the number is bounded by half of the moment number. We give an example of continuous functions where more Gaussian distributions are needed than Dirac delta measures. We show that any inner truncated moment sequence has a mixture which contains any given distribution.