# Reference request: discretisation of probability measures on $mathbb R^d$

MathOverflow Asked by MB2009 on December 9, 2020

Given a probability measures $mu$ on $mathbb R^d$ with finite first movement, i.e.

$$int_{mathbb R^d}|x|mu(dx)~~<~~+infty.$$

My concern is to approximate $mu$ some $mu_n$ that is countably or finitely supported. Of course, a generic way is to take such a $mu_n$ concentrated on the grid ${vec{k}/n}_{vec{k}in mathbb Z^d}$. I wonder whether there exists more literature dealing with this issue, especially from the viewpoint of implementation. Many thanks for answers and comments.

PS: Thanks for the reply. To summarise, I’m interested in the $mu_n$ such that:

(1) the computation of $mu_n[{vec{k}/n}]$ is tractable;

(2) the Wasserstein distance $W_1(mu,mu_n)$ is easy to estimate.

Of course, the quantisation approach provides a good upper bound for $W_1(mu,mu_n)$, but the computation of $mu_n[{vec{k}/n}]$ is not obvious. So my question is whether there exists some explicit “discretisation” of $mu$ such that the “discretised weights” are easy to obtain?

The keyword to look for might be "quantization", see e.g. G. Pagès' review :

https://doi.org/10.1051/proc/201448002

Answered by paz on December 9, 2020

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