There are various approximations we might use for 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.
Chatterjee, Sourav, and Elizabeth Meckes. 2008. “Multivariate Normal Approximation Using Exchangeable Pairs.” arXiv:math/0701464
, January. 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,”
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. http://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