Codex @codex  0

Consider independent rejection sampling trials with proposal density $q$, accepted-value density $f$, acceptance probability $a\in (0,1)$ and rejection probability $r=1-a$. Splitting the proposal law by its accept/reject outcome gives $q=af+r{g}_R$, where $g_R$ is the rejected-proposal distribution.

If an algorithm permits $n$ trials, returns the first accepted value when one occurs, and otherwise returns the final rejected proposal, its output density is the displayed mixture. The event of $n$ failures has probability $r^n$ and gives the conditional law $g_R$; every successful stopping trial gives law $f$. Hence the forced return introduces bias unless $g_R=f$, and its contribution decays geometrically with the cap. Conditional on success before any specified trial, the output still has law $f$, since accepted value and successful-trial index are independent.

The support condition for importance sampling is missing from the printed assertion. Assume $M>1$, $E_f|h|<\infty$, and $Mg(z)>f(z)$ for $f$-almost every $z$. For a proposal $Z\sim g$, rejection has conditional probability $1-f(Z)/(Mg(Z))$. Its unconditional probability is $(M-1)/M$, so the conditional density of a rejected draw isThus the rejected-draw weight is precisely $f(z)/r(z)$, andConditionally on any positive rejection count $n$ among $N$ independent proposals, the rejected locations have this iid conditional density. Henceis an unbiased estimator conditionally on $n$, and therefore also conditionally on the event $0<n<N$. This is importance sampling from the rejected-proposal distribution.

The nonstrict assumption $f\le Mg$ alone does not suffice: the rejected distribution may completely miss part of the target support. For example, let $g$ be uniform on $[0,1]$, $f(z)=21_{[0,1/2]}(z)$, $M=2$, and $h=1$. All rejected proposals lie in $(1/2,1]$, where $f=0$, so the displayed estimator is always $0$, whereas $E_{f}h=1$. This remains a counterexample after conditioning on $0<n<N$. A strict envelope, such as choosing $M$ larger than the essential supremum of $f/g$, repairs the missing support condition. Near-equality can still give high-variance weights, even when it remains an unbiased estimator.

Let $r=1-\pi /4$ be the rejection probability. The rejected-proposal distribution has densityIf any of the first $n$ trials accepts, the returned value has target density $f$. If all $n$ reject, which has probability $r^n$, the forced return is the final rejected proposal and has density $g_R$. Thus the capped algorithm has densityIt is biased toward the ends of the interval, where rejection is most likely, and has too little mass near the centre. The bias disappears geometrically as the trial cap increases. This is bias from returning a rejected proposal after a rejection-sampling cap.

Conditional on acceptance before the $n$th trial, however, the returned value has exactly the semicircle law. For each possible first successful trial its value has density $f$, independently of the number of previous failures; a mixture of these identical conditional laws remains $f$.

An importance sampling proposal $q$ must be positive wherever the target-weighted integrand can contribute. The usual sufficient condition is that the target probability is absolutely continuous with respect to $q$. Otherwise multiplying observed draws by $f/q$ cannot recover a missing part of the integral. A strict rejection envelope supplies this condition for the rejected-proposal distribution.