Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 36 / 4 / a

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.