# Generically approximating probability distributions

March 12, 2021 — March 22, 2021

approximation

functional analysis

metrics

model selection

optimization

probability

statistics

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 that 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 these 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.

## 1 Stein’s method

See Stein’s method.

## 2 References

Chatterjee, and Meckes. 2008. “Multivariate Normal Approximation Using Exchangeable Pairs.”

*arXiv:math/0701464*.
Meckes. 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*.
Reinert, and Röllin. 2007. “Multivariate Normal Approximation with Stein’s Method of Exchangeable Pairs Under a General Linearity Condition.”

Stein. 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*.
———. 1986.

*Approximate Computation of Expectations*.