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.