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 interested in collecting results that tell me about how various combinations of variables approach a limiting distribution in some probability metric.
See Stein’s method.
Chatterjee, Sourav, and Elizabeth Meckes. 2008. “Multivariate Normal Approximation Using Exchangeable Pairs.” arXiv:math/0701464, January.
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.
Reinert, Gesine, and Adrian Röllin. 2007. “Multivariate Normal Approximation with Stein’s Method of Exchangeable Pairs Under a General Linearity Condition,” November.
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.
———. 1986. Approximate Computation of Expectations. Vol. 7. IMS.
Vershynin, Roman. 2018. High-Dimensional Probability: An Introduction with Applications in Data Science. 1st ed. Cambridge University Press.
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