Probability Prelim, August 2012
1. Suppose that (Xn:n∈N) is a sequence of centered normal random variables
which converges weakly to some random variable X. Prove that E(X2)<∞and
limn→∞ E(X2
n) = E(X2).
2. Prove Jensen’s inequality for conditional expectation. Specifically, if Gis a sigma-
algebra and ϕ:R→Ris convex and Xis a random variable with X, ϕ(X)∈L1,
then
ϕ(E[X|G]) ≤E[ϕ(X)|G].
3. Let (Xn:n∈N) be a sequence of IID random variables with E|X1|<∞and
E(X1) = 0. Let Sn=Pj≤nXjand let S∗
n= maxk≤nSk.
(a) Conclude from the Law of Large Numbers that S∗
n/n converges to 0 a.s.
(b) Prove that S∗
n/n converges to 0 in L1.
(Hint : Enough to show that (|S∗
n|/n :n∈N) is uniformly integrable)
4. (a) Let (Mn,Fn:n∈N) be a submartingale. Suppose that supn∈NE(Mn∨0) <∞.
Prove that supn∈NE|Mn|<∞.
(b) Let (Mn,Fn:n∈N) be a submartingale, and let Tbe an a.s. finite stopping
time. Prove that (Mn∧T,Fn∧T:n∈N) is a submartingale.
(c) Suppose that (Mn,Fn:n∈N) is a submartingale satisfying Mn+1 −Mn≤1.
Let
C={lim Mnexists and is finite}, D ={lim sup Mn=∞}.
Prove that P(C∪D) = 1.
( Hint : for N∈N, let TN= inf {n:Mn≥N}. Apply the submartingale
convergence theorem to Mn∧TN, and show it implies S∞
N=1{TN=∞} ⊂ C. )
5. Let (Bt:t≥0) be standard Brownian motion. Compute the variance and the third
moment of R1
0Bsds
