NOTE: If you're here for my notes on the Cambridge course, read parts 1 and 2. If you are reading this website in order from start to finish, you should be familiar with part 1 - part 1 and the central limit theorem corollary are part of level 6. The proof of the change of variables theorem from my probability notes and the differentiation under the integral sign theorem from my differential equations notes are also technically in this course.
See the IA probability course (level 8.4) for more detail on properties of the multivariate normal and the proof that almost sure convergence implies convergence in probability and for the first variant of the strong law of large numbers.