Let $A$ be a parametric family of probability distributions that include all distributions in the form of $\phi(X)$ where $X\sim\mathcal{N}(0,\mathbf{I})$ is jointly Gaussian and $\phi:\mathbb{R}^d\to \mathbb{R}^d$ belongs to a parametric set of functions $G$.

To uniquely determine a distribution in $A$, how many moments do we need to know? For example, if $G$ is linear, A includes all Gaussian distributions. Thus, we need first and second moments to uniquely determine a distribution in the set $A$. I wonder if such a result can be extended for other $G$ such as polynomials with degree $k$.