Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 47 / 3 / a / Solution

For rejection sampling, independently generate a proposal $Y$ with density $g$ and a uniform $U$ on $(0,1)$. Accept $Y$ if $U\le f(Y)/(Mg(Y))$; otherwise generate a fresh independent pair and repeat. Set the ratio to zero on $g=0$ where necessarily $f=0$. The envelope condition makes the acceptance threshold at most one.

For any measurable set $A$, a single attempt satisfiesTaking the whole space shows that acceptance has probability $1/M$. Conditioning on acceptance therefore gives density $f$. Repeating rejected independent attempts preserves that conditional law and terminates almost surely; the number of attempts is geometric with mean $M$.

For fixed proposal density, admissibility requires $M\ge {\sup }_{g(x)>0}f(x)/g(x)$ under the pointwise convention in the question. Since $1/M$ decreases with $M$, the smallest valid envelope gives the greatest acceptance probability:For densities specified only almost everywhere, replace the supremum by the essential supremum. Integration of the envelope also gives $M\ge 1$.