Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / S / 1 / c / Solution

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 $f$.