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Let $w(x)=f(x)/g(x)$ where $g(x)>0$, and set it to zero on the g-null set where $g=0$. The domination assumption makes $f=0$ there and gives $0\le w\le M$. Integrating it also gives $M\ge 1$. For iid proposals $X_i\sim g$, the importance sampling estimator isCauchy-Schwarz inequality under $f$ gives $|\theta |\le (\int {\varphi }^{2}f{)}^{1/2}<\infty$. Direct integration proves unbiasedness, ${\mathbb{E}}_g[\varphi (X)w(X)]=\theta$. MoreoverThusThe iid central limit theorem applies to these finite-variance weighted observations:If the variance is zero, this denotes the point mass at zero. This is the bounded-weight importance-sampling moment bound.

Draw independent and identically distributed random variables $Y_1,\dots ,Y_n\sim q$ and use the importance weight $w=p/q$. The importance sampling estimator isThe envelope supplies the support condition for importance sampling. Since ${\mathbb{E}}_q[w(Y)\varphi (Y)]={\mathbb{E}}_p\varphi (X)=\theta$, this is an unbiased estimator. Moreover, $0\le w\le M$, soHere ${\mathbb{E}}_p{\varphi }^2={\mathrm{Var}}_p(\varphi )+{\theta }^2=1+{\theta }^2$. Subtracting the squared expected value and dividing by $n$ proves the bounded-weight importance-sampling moment bound in the requested form: