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 $(\Omega,\mathcal{F})$.

  • $\begingroup$ 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 $X$. $\endgroup$ – Xi'an Feb 22 at 17:54

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