Let $n\in\mathbb{N}$ and $\sigma>0$ be fixed. I have a certain class $\mathcal{C}$ of random variables I am interested in analyzing. This contains
- $\vec X\sim \mathcal{N}(0, \sigma^2I_n)$
- Sums of (independent) random variables in $\mathcal{C}$, and
- Convolutions of (independent) random variables in $\mathcal{C}$. By this, I mean that if one views the random variables $\vec X$ as a polynomials $\sum_{i = 0}^n \vec X_i y^i$, the operation is polynomial multiplication in $\mathbb{R}[y]/(y^n-1)$.
Practically, authors in my applied area have observed that the resulting random variables look Gaussian. This can be qualitatively justified by noting that the first two points preserve Gaussianity, and the third (by the CLT) should yield something Gaussian-like.
My issue is with trying to quantify this. By standard quantitative versions of the CLT (Berry Esseen theorem), one can get that a random variable in $\mathcal{C}$ is at most $O(1/\sqrt{n})$ (and perhaps improve it to $O(1/n)$) from a Gaussian in the Kolmogorov Smirnov metric. These rates hold in much more generality though, where there is not such a strong assumption that the underlying random variables are related to Gaussians.
Can one obtain better rates in this setting? I would also be interested in rates in other metrics, in particular in the total variation, but if random variables in $\mathcal{C}$ are still quite far from Gaussian this is of second-order concern.
