Random-count central limit theorem for accepted rejection samples ($\sqrt{n}({\hat{\theta }}_{\mathrm{rej}}-\theta )\Rightarrow N(0,M{\mathrm{Var}}_f\varphi )$)

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.