= Normal variance mismatch in importance sampling
{c}
{title2=$w_\sigma(x)=\sigma^{-1}e^{(1-\sigma^{-2})x^2/2}$}
For target $p=N(0,\sigma^2)$ and proposal $q=N(0,1)$, this is the <importance weight>. Assuming only $E_pf^2<\infty$, the weighted estimator has guaranteed finite variance for $0<\sigma\leq1$, since $E_q(w_\sigma^2f^2)=E_p(w_\sigma f^2)\leq\sigma^{-1}E_pf^2$. For every $\sigma>1$, choosing $f^2=e^{ax^2}$ 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 $\sigma<\sqrt2$. The integrand hypothesis therefore matters.
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