A famous generic method for approximating distributions is Stein’s method of exchangeable pairs (Stein 1986, 1972). Wikipedia is good on this.

Heuristically, the univariate method of exchangeable pairs goes as follows. Let \(W\) be a random variable conjectured to be approximately Gaussian; assume that \(\mathbb{E} W=0\) and \(\mathbb{E} W^{2}=1 .\) From \(W,\) construct a new random variable \(W^{\prime}\) such that the pair \(\left(W, W^{\prime}\right)\) has the same distribution as \(\left(W^{\prime}, W\right) .\) This is usually done by making a "small random change" in \(W\), so that \(W\) and \(W^{\prime}\) are close. Let \(\Delta=W^{\prime}-W\). If it can be verified that there is a \(\lambda>0\) such that \[ \begin{aligned} \mathbb{E}[\Delta \mid W]=-\lambda W+E_{1} \\ \mathbb{E}\left[\Delta^{2} \mid W\right]=2 \lambda+E_{2} \\ \mathbb{E}|\Delta|^{3}=E_{3} \end{aligned} \] with the random quantities \(E_{1}, E_{2}\) and the deterministic quantity \(E_{3}\) being small compared to \(\lambda,\) then \(W\) is indeed approximately Gaussian, and its distance to Gaussian (in some metric) can be bounded in terms of the \(E_{i}\) and \(\lambda\).

This comes out very nicely where there are natural symmetries to exploit, e.g. in low-d projections.

## Non-Gaussian Stein

## Multivariate Stein

The work of Elizabeth Meckes 1980—2020 whose papers serve as the canonical textbook in the area for now.
Two foundational ones are (**ChatterjeeMultivariate2008?**) and Meckes (2009) and there is a kind of user guide in (**MeckesApproximation2012?**); These bounds are mostly about random projections although that is not required.
The exchangeable pairs are very natural in that case though, you can just switch off you brain and turn the handle to produce results, or easier yet, a computer algebra system that can handle noncommutative algebra.

Two useful presentations:

## References

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*Complex Stochastic Systems*, 235–75. Boca Raton: Chapman & Hall/CRC. http://www.stats.ox.ac.uk/ reinert/papers/episemrevnew.pdf.

*An Introduction to Stein’s Method*, Volume 4:183–221. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Volume 4. CO-PUBLISHED WITH SINGAPORE UNIVERSITY PRESS. https://doi.org/10.1142/9789812567680_0004.

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*Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory*, January, 583–602. https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-Sixth-Berkeley-Symposium-on-Mathematical-Statistics-and/chapter/A-bound-for-the-error-in-the-normal-approximation-to/bsmsp/1200514239.

*Approximate Computation of Expectations*. Vol. 7. IMS. http://www.jstor.org/stable/4355512.

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