There are various approximations we might use for a a probability distribution. Empirical CDFs, Kernel density estimates, variational approximation, Edgeworth expansions, Laplace approximations…

From each of these we might get close in some metric to the desired target.

This is a broad topic which I cannot hope to cover in full generality. Special cases of interest include

- Statements about where the probability mass is with high probability (concentration theorems)
- statements about the asymptotic distributions of variables eventually approaching some distribution as some parameter goes to infinity (limit theorems. Most famously a lot of things approach normal distributions, but there are many limit theorems

There are other types of results besides, in this domain. I am quite interested in collecting here, some results that tell me about how various combinations of variables approach a limiting distribution in some probability metric.

## Stein’s method

See Stein’s method.

## References

Chatterjee, Sourav, and Elizabeth Meckes. 2008. “Multivariate Normal Approximation Using Exchangeable Pairs.” January 15, 2008. http://arxiv.org/abs/math/0701464.

Meckes, Elizabeth. 2006. “An Infinitesimal Version of Stein’s Method of Exchangeable Pairs.”

———. 2009. “On Stein’s Method for Multivariate Normal Approximation.” In

*High Dimensional Probability V: The Luminy Volume*, 153–78. Beachwood, Ohio, USA: Institute of Mathematical Statistics. https://doi.org/10.1214/09-IMSCOLL511.
Reinert, Gesine, and Adrian Röllin. 2007. “Multivariate Normal Approximation with Stein’s Method of Exchangeable Pairs Under a General Linearity Condition,” November. https://doi.org/10.1214/09-AOP467.

Stein, Charles. 1972. “A Bound for the Error in the Normal Approximation to the Distribution of a Sum of Dependent Random Variables.”

*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.
———. 1986.

*Approximate Computation of Expectations*. Vol. 7. IMS. https://www.jstor.org/stable/4355512.
Vershynin, Roman. 2018.

*High-Dimensional Probability: An Introduction with Applications in Data Science*. 1st ed. Cambridge University Press. https://doi.org/10.1017/9781108231596.
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