Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / S / 1 / b / Solution

For unknown normalization constants, use self-normalized importance sampling with $\tilde{w}=\tilde{p}/\tilde{q}$:Under the same support condition, $E_q\tilde{w}=Z_p/Z_q>0$ and $E_q[\tilde{w}f]=(Z_p/Z_q)\mu$. Moreover $E_q|\tilde{w}f|=(Z_p/Z_q)E_p|f|<\infty$, by Cauchy-Schwarz inequality and the supplied $E_p{f}^2<\infty$. The strong law of large numbers applies to numerator and denominator sample averages. Their ratio tends almost surely, hence in probability, to $\mu$. If a finite sample happens to give zero denominator, it can be assigned an arbitrary value; this does not affect the eventual almost sure limit. The ratio estimator is generally biased at finite $N$.