4. [25 points] Consider a sequence of Poisson random variables {X} some An > 0. (a) Assume that An is a positive integer (i.e. An {1,2,3....}). Using the central limit theorem, prove that Xn An D. An Remark. You may use the fact that E[X] = Var[X] variable. Dy N(0, 1) as A, 0. Remark. ~ Poisson(n) for You can also use the fact that if X~ Poisson (A) and Y~ Poisson(), and they are independent, then X+Y~ Poisson ( + A). You don't need to prove the central limit theorem. Xn - An An = An for a Poisson random (b) Now we let An be any positive real sequence. Prove that the previous claim still remains true, i.e. D, N(0, 1) as An . Slutsky's theorem would be helpful to prove this general claim.