1. Let X, Y , and Z be independent real random variables with EeX ∞, E |X | ∞, EY = 0, and EZ 2 ∞. Show that {X, eX + Y + Z 2} is a submartingale where B1 is the smallest σ-algebra making X measurable, and find its Doob decomposition. Hint: See Problem 10.1.5.
2. Let fn := an 1[0,1/n] on [0,1] with Lebesgue measure. For what sequences {an } is { fn } uniformly integrable?
3. Let {Xn } be a submartingale. Show that EXn is a nondecreasing sequence of numbers, and give an example to show that Xn - EXn is not necessarily a martingale. Hint: Consider the uniqueness in the Doob decomposition.