MATH 5050/6815: Homework 3
Due on Tuesday, Mar. 6, by the end of lecture.
For this homework set, you need the Jensen’s inqualities. The have to do with convex
functions:
A function f (x) is convex if for all 0 ≤ p ≤ 1,
f (px + (1 − p)y) ≤ pf (x) + (1 − p)f (y).
For more practice, please do Problem 5.3 page 125 (not for credit :-) ).
Proposition 0.1 (Jensen’s Inequalities). Let f be a convex function so that the following
expectations exist. For any random variable X and σ− algebra G, the following hold:
(1)
E(f (X)) ≥ f (E(X)),
(2)
E(f (X)|G) ≥ f (E(X|G)).
Problem 1
a) Let Mn a mean µ positive martingale. Show that if the variance of the martingale
V ar(Mn ) < C for al n then the martingale is uniformly integrable.
b) Problem 5.4 page 126 from the book.
Problem 2: Branching processes Let Xn be the population size of the n-th generation
of a branching process with X0 = 1, with offspring distribution Z, with E(Z) = µ > 1 and
V ar(Z) = σ 2 < ∞. For notation purposes, assume that the i-th individual of generation n
PXn−1 (i)
(i)
gave birth to Zn many children and so Xn = i=1
Zn−1 .
2
2
(1) Find the E(Xn ) in terms of E(Xn−1 ). (Hint: Condition on Xn−1 )
(2) Find the E(Xn2 ) by solving the recursion you found in the previous question.
(3) Find the V ar(Xn ).
(4) Show that the martingale Mn = Xn /µn is uniformly integrable.
(5) Show that the probability of the branching process surviving is positive.
Problem 3 Let X a F-measurable random variable and {Fn }n≥0 a filtration of σ− algebras,
so that Fn ⊆ F for all n ∈ N. Also assume E|X| < ∞ and EX = µ, V ar(X) = σ 2 < ∞.
(1) Let Mn = E(X|Fn ). Show that Mn is a mean µ, Fn − martingale.
(2) Decide whether Mn is uniformly integrable.
S
(3) Explain why you can apply martingale convergence theorem. Suppose that n≥0 Fn =
F. Can you guess what M∞ should be? Why?
Problem 4
Do problem 5.17, page 129 in your book.
Problem 5 10 points, Extra Credit:
Problem 5.6. , page 126 from your book.
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