Bounded-weight importance-sampling moment bound
= Bounded-weight importance-sampling moment bound
{title2=$\mathbb E_g[\phi^2w^2]\leq M\mathbb E_f\phi^2$}
If $f\leq Mg$, the importance weight $w=f/g$ is bounded by $M$, and $\mathbb E_g[\phi^2w^2]\leq M\mathbb E_f\phi^2$. A finite target second moment therefore guarantees finite variance and the iid central limit theorem for the unnormalized importance-sampling average. Its proposal-budget variance is $\mathbb E_g[\phi^2w^2]-(\mathbb E_f\phi)^2$.