Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 3 / c / Solution

For $K>0$, the Cauchy-Schwarz inequality and part a giveThus $(Y_n^{-})$ is uniformly integrable. Combining this with the weak convergence of random variables from part b yields convergence of the first moments:By symmetry of the standard normal density,