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I read that rejection sampling might fail in high dimensional settings, as the rejection rate becomes too low. Intuitively - i can understand this - but i would like to understand the formal proof as well. From the excellent answer on How does the proof of Rejection Sampling make sense? I understand that the rejection rate is $1/M$. I understand as well that it is required that $\sigma_q > \sigma_p$ and that in D dimensions hence M should be $(\sigma_p/\sigma_q)^D$, but i am struggling to combines these facts to e.g. proof the example. Any tips welcome!

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    $\begingroup$ Have you computed that $1/20000\approx (1 / (1+0.01))^{1000} = 1/(\sigma_p/\sigma_q)^D$? $\endgroup$ – whuber Jul 26 '15 at 19:26
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    $\begingroup$ Thanks! i was multiplying the individual acceptance rates on the marginal distributions (which probably should give the same answer) - but made calculation error. thanks. $\endgroup$ – Wouter Jul 26 '15 at 19:51

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