I have a certain $n$-dimensional multivariate random variable $X$ I wish to study. I can compute its characteristic function, and can show that
$$ \lVert \vec t\rVert_\infty \leq 1\implies \exp(-\frac{1}{2}\frac{\lVert \vec t\rVert_2^2}{n}) \leq \varphi_X(\vec t) \leq \exp(-\frac{1}{4}\frac{\lVert \vec t\rVert_2^2}{n}) $$
In particular, for such bounded $\vec t$ the characteristic function of $X$ looks like that of a Gaussian.
Does this give me any information about $X$ itself? If I had such an inequality for all $\vec t\in\mathbb{R}^n$, I could use Fourier inversion to argue that the density of $X$ is (approximately) the density of a Gaussian, which would be interesting for me. I don't know if the weaker bound that I currently have can be used to make some weaker statement relating $X$ to a Gaussian.