Past exam of the simulation and statistical programming course of the University of Oxford / 2023 / 2 / b / iii / Solution

For a measurable set $C$, (i) gives the joint acceptance probabilityDivide by $Z_p/(M{Z}_q)$ from (ii). The resulting conditional probability isThus every accepted value has target probability density function $p$. Repeating rejected trials with fresh independent random variables does not change this law: the contribution from acceptance on trial $j$ is multiplied by $(1-Z_p/(M{Z}_q){)}^{j-1}$, and summing that geometric series gives the same conditional law. Independent repetitions produce independent target draws.