# Acceptance-Rejection using Functional

Setup Let $$X\in L^1(\Omega,\mathcal{F},\mathbb{P})$$.
As far as I've seen, Monte-Carlo methods generate $$x_1,\dots,x_n$$ from the distribution of $$X$$ and uses the Glivenko-Cantelli theorem to conclude that $$\frac1{n}\sum_{i=1}^N \delta_{x_i} \overset{D}{\rightarrow} Law(X).$$

Acceptance\rejection sampling follows the same procedure, but given a function $$f:\mathbb{R}\rightarrow \mathbb{R}$$ and a threshold $$M\in \mathbb{R}$$, it extends the above method to obtain $$\frac1{\sum_{i=1}^N I_{f(x_i)\leq M}}\sum_{i=1}^N \delta_{x_i}I_{f(x_i)\leq M} \overset{D}{\rightarrow} Law(X|f(X)\leq M),$$ (here I've assume that $$N$$ was large enought, and $$f$$ was nice enough so that we're not dividing by $$0$$).

Question My question is, if $$f$$ is instead a continuous functional $$f:\mathscr{P}(\Omega,\mathcal{F})\rightarrow \mathbb{R}$$, then can it be used to do the acceptance/rejection?

Here, $$\mathscr{P}(\Omega,\mathcal{F})$$ is the set of probability measures on $$(\Omega,\mathcal{F})$$.

• This is unclear, as $X$ and $f$ need be related to modify the distribution of $X$ into the target. Taking a probability distribution does not impact $X$. – Xi'an Feb 22 at 17:54