Probability Prelim, January 2013
1. Let (Fn:n∈N) be an increasing sequence of σ-algebras, and let F∞=σ(∪∞
n=1Fn).
Show that for any F∞-measurable integrable random variable X,E[X|Fn]→X, a.s.
and in L1.
2. For each n∈N, let Xn,j ,j= 1, . . . , n be a sequence of independent random vari-
ables with EXn,j = 0. Let Sn=Pn
j=1 Xn,j, and assume that E(S2
n) = 1 and
Pn
j=1 E(|Xn,j|3)→0. Prove that Snconverges weakly to a standard normal.
3. (Glivenko-Cantelli Theorem). Suppose that (Xn:n∈N) are IID uniform on [0,1].
For n∈N, let ˆ
Fn:R→[0,1] be defined as ˆ
Fn(t) = Pn
j=1 1(−∞,t](Xj)
n. Prove that
supt∈[0,1]
ˆ
Fn(t)−t
→0, a.s.
(Hint: Consider first tof the form k
M, k = 0, . . . , M , for some large M. Estimate the
differences for other choices of tfrom these).
4. Let p∈(0,1). Let (Dn:n∈N) be IID with P(Dn= 1) = p, and P(Dn=−1) = 1−p.
Suppose that X0= 0, and for n∈N, let Xn=Pn
j=1 Dj.
(a) Find a function Λ : R→Rsuch that for each θ, (eθXn−Λ(θ)n:n∈Z+) is a
martingale.
(b) Suppose that p≥1
2. Let τ= inf{n∈Z+:Xn= 1}. Find the Laplace transform
Ee−ρτ , ρ > 0.
5. Let (Bt:t≥0) be standard BM. For > 0, consider the event A={|B1|< }. For
0≤s≤t < 1, find lim→0E(BsBt1A)
P(A).
(Hint: Show that (Bs−sB1, Bt−tB1) is independent of B1and that E(BsB11A) =
O(3
2), as →0).