A light-tailed proposal can give ordinary importance sampling infinite variance

On $\mathbb{R}$ with Lebesgue measure, take target density $p(x)=\tilde{p}(x)=\frac{1}{2}{e}^{-\mathrm{\mid }x\mathrm{\mid }}$, proposal density $q(x)=e^{-2\mathrm{\mid }x\mathrm{\mid }}$, and $f(x)=1$. Direct integration gives $\int p=\frac{1}{2}({\int }_{-\mathrm{\infty }}^0{e}^{x}dx+{\int }_0^{\mathrm{\infty }}{e}^{-x}dx)=1$ and $\int q=2{\int }_0^{\mathrm{\infty }}{e}^{-2x}dx=1$. Also $q(x)>0$ for every $x$, while $f=1$ has target mean $1$ and variance $0$.

Use [F7] with density $r=q$ for integration against $Q$. For every $x$, the importance weight is $W(x)=p(x)\mathrm{/}q(x)=\frac{1}{2}{e}^{\mathrm{\mid }x\mathrm{\mid }}$. By [F2], ${\mathbb{E}}_{Q}W={\int }_{\mathbb{R}}p(x)dx=1$ and ${\mathbb{E}}_Q[W^2]={\int }_{\mathbb{R}}\frac{1}{4}{e}^{2\mathrm{\mid }x\mathrm{\mid }}{e}^{-2\mathrm{\mid }x\mathrm{\mid }}dx={\int }_{\mathbb{R}}\frac{1}{4}dx=\mathrm{\infty }$. The proposal has full target support, but the finite-second-moment hypothesis in [F3] fails.

By [F5], the ordinary estimate ${\bar{W}}_n=n^{-1}{\sum }_i{W}_i$ has mean $1$. For each finite $n$, nonnegativity gives ${\bar{W}}_n\ge W_1\mathrm{/}n$, hence $W_1^2\mathrm{/}{n}^2\le {\bar{W}}_n^2$. Monotonicity [F6] and step 2.1 imply ${\mathbb{E}}_Q[{\bar{W}}_n^2]\ge n^{-2}{\mathbb{E}}_Q[W_1^2]=\mathrm{\infty }$. If its centered second moment were finite, the inequality $x^2\le 2(x-1{)}^2+2$ would imply ${\mathbb{E}}_Q[{\bar{W}}_n^2]<\mathrm{\infty }$, a contradiction. Thus the centered second moment is infinite, so the estimate does not have finite variance under [F4]. This refutes the sufficiency claim.