I have a sum $$ S = \sum_{i=1}^{n} d_i X_i^2, $$ where $X_i$ are independent standard normals, and $d_i > 0$ are fixed real numbers, for example $d_i = i$. The asymptotic distribution of this sum is given by the central limit theorem as $\mathrm{N}(\sum d_i, 2\sum d_i^2)$, and according to the Berry-Esseen theorem, this approximation has error on the order $n^{-1/2}$.

When $n$ is not large enough that I can ignore terms $O(n^{-1/2})$, is there a way to derive a more accurate asymptotic distribution in a tractable closed form, such that the error would be $O(n^{-1})$ or better? For fixed $n$, the CLT also gives a poor approximation of the tail of the distribution.

Does there exist a "universal" distribution that I could use for $S$ in the general cases (like the normal distribution in the CLT, but with extra accuracy). For example, what if $X_i^2$ are i.i.d. Beta random variables instead?