exercises for 10 Feb to 16 Feb

This week we have finished up the proof of the Portmanteau theorem. Then we proved a lemma showing that for weak convergence to take place, it is enough that the probability measure involved converge for all sets in a pi-system that are such that all open sets can be written as finite unions of sets in this pi-system. That weak convergence is equivalent to c.d.f. convergence is a corollary of this lemma. Next we proved various lemmata combining convergence in distribution and convergence in probability, ending with the Cram{\'e}r--Slutsky rules.Then we got to tightness. We have proved Helly's theorem in the real line, and Prokhorov's theorem. Lastly, we introduced characteristic functions (c.f.), and found the c.f. of a standard normal random variable. 

Exercises: 2.22--2.26, as well as A.36, A.39, and A.40