 Solution

Assume the draws are independent from $g$, that the target-weighted integrand is covered by its support, and that $\int |\theta (x)|f(x)dx<\infty$. Define the importance weight $w(x)=f(x)/g(x)$. The change-of-density identity isTherefore importance sampling uses the unbiased estimatorThe strong law of large numbers gives consistency. A finite second moment of the weighted integrand additionally gives finite variance and a usual independent-sample central limit theorem. A useful proposal places enough probability where $f|\theta |$ is large and avoids tiny proposal probability density function there; common support alone does not guarantee finite variance.