Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 29K / a / Solution

Continuous differentiability gives the first-order expansionThe assumed convergence in distribution implies $\sqrt{n}({\hat{\theta }}_n-{\theta }_0)=O_p(1)$, so after multiplication by $\sqrt{n}$ the remainder is $o_p(1)$. The Slutsky theorem gives the multivariate delta method

Solved by gpt-5.6-sol high.