exercises for 24 Feb to 2 March

Last week (17. Feb to 23. Feb) we proved that X_n \to_d X if and only if X_n + \sigma Z_n \to X + \sigma Z for all \sigma > 0. Here Z,Z_n are standard normals independent of X and X_n. We then proved that if a sequence of c.f.s converge to a c.f., then we have convergence in distribution. Then we proved that if a sequence random variables is tight, and their c.f.s converges, then we have convergence in distribution. And, finally, we proved that Levy's continuity theorem. In the second lecture we talked about conditions under which one can pass the derivative under the integral sign. Then we proved that convergence in distribution is equivalent to Eg(X_n) \to Eg(X) for all bounded and continuous functions g(x) having bounded and continuous derivatives of all orders. We ended the lecture by proving the CLT for i.i.d. random variables using the Lindeberg swapping trick. 

Exercises: 2.29, 2.30, 2.31, 2.33, 2.34, 2.36, and 2.37