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MATH 5050/6815: Homework 3 Due on Tuesday, Mar. 27, by the end of lecture. Problem 1 Let X1 , X2 , X3 , ... be i.i.d. with a a distribution which is exponential with parameter λ. (This means that they have a density function λe−λx on [0, ∞)). Show that almost surely Xn lim sup = λ−1 . n→∞ log n Problem 2 Let X1 , X2 , ... be i.i.d. with E|X1| < ∞. Show that almost surely max{X1 , X2 , ..., Xn } = 0. n Problem 3 (1) Prove that the transition densities of Brownian Motion satisfy the Chapman-Kolmogorov equations. (2) Do Problem 8.1 p.195 in your book. (3) Do Problem 8.10 (a) p.197 in your book. Problem 4 Problem 8.4 , page 195 from your book. Problem 5 10 points, Extra Credit: Prove that if P{En } → 0 as n → ∞ and ∞ X n=1 1 P{Enc ∩ En } < ∞ then P{En i.o.} = 0.