Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 40 / 2 / a / iii / Solution

Let $Y_i=\theta (X_i)f(X_i)/g(X_i)$. The independent summands have expectation $\mu$ and second momentWhen this integral is finite, independence givesIf the second-moment integral diverges, the estimator can still be unbiased and consistent, but its variance is infinite; the finite-variance assumption cannot be omitted.