Normal variance mismatch in importance sampling ($w_{\sigma }(x)={\sigma }^{-1}{e}^{(1-{\sigma }^{-2})x^2/2}$)

For target $p=N(0,{\sigma }^2)$ and proposal $q=N(0,1)$, this is the importance weight. Assuming only $E_p{f}^2<\infty$, the weighted estimator has guaranteed finite variance for $0<\sigma \le 1$, since $E_q(w_{\sigma }^2{f}^2)=E_p(w_{\sigma }{f}^2)\le {\sigma }^{-1}{E}_p{f}^2$. For every $\sigma >1$, choosing $f^2=e^{a{x}^2}$ with $\max (0,{\sigma }^{-2}-1/2)<a<(2{\sigma }^2{)}^{-1}$ gives finite target second moment but infinite weighted second moment. With the additional assumption that $f$ is bounded, the sufficient range is instead $\sigma <\sqrt{2}$. The integrand hypothesis therefore matters.