Suppose $Z_1, \ldots, Z_k$ are random variables with mean $0$ and variance $1$ that are "approximately jointly Gaussian" in the sense that for any scalars $c_1, \ldots, c_k$, we have that $\mathbb P(c_1Z_1 + \cdots + c_kZ_k\leq t)$ is equal to the probability of a mean $0$ and certain variance (depending on the scalars $c_i$ and the covariances of the $Z_i$, which we may assume are smaller than some small absolute constant and actually going to $0$ as $k$ increases) Gaussian being at most $t$, up to an error which is bounded by $C_Ak^{-A}$ for any $A>0$, where $c_A$ is a constant depending on $A$. (You should think that $k$ is going to $\infty$ and for each $k$, I have a different collection of $k$ random variables.)
Do you know if this information is enough to deduce a quantitative multivariate central limit theorem for $Z = (Z_1, \ldots, Z_k)$? It feels like there should be a way, especially since the distribution of the linear combinations of the $Z_i$ fully determines the distribution of the random vector $Z$.
I am aware of the existence of Stein's method and that it would probably work here, but I am trying to avoid it for now and see if there's an easier and more straightforward way to get the distribution of $Z$ quantitatively close to that of a multivariate normal, from the information of each of the linear combinations being quantitatively close to a Gaussian.
Thank you in advance!
