Binomial count and iid values in fixed-budget rejection sampling ($N\sim \mathrm{Bin}(n,1/M)$)

For n independent proposals and an envelope constant $M$, each rejection decision succeeds with probability $1/M$. The accepted count is Bin(n,1/M). Conditional on any acceptance pattern, the accepted values have independent target density f; summing over patterns preserves that product law conditional on the count. The empty-output probability is $(1-1/M{)}^n$.

Conditional iid accepted values have the ordinary target-sample CLT at scale $\sqrt{N}$. Since $N/n\to 1/M$, the same estimator has asymptotic variance $M{\mathrm{Var}}_f\varphi$ at scale $\sqrt{n}$. The centered count has limiting variance $(1/M)(1-1/M)$. Any fixed convention on an empty output is asymptotically negligible.