STA 731 - Homework 8
1. Let {En}be events. Verify
n
X
k=1
1Ek= 1Sn
k=1 Ek
n
X
k=1
1Ek
and then, using the Schwartz inequality, prove that
P
n
[
k=1
Ek≥E(Pn
k=1 1Ek)2
E(Pn
k=1 1Ek)2.
2. Suppose {Xn, n ≥1}is a sequence of random variables satisfying
P(Xn=n) = 1
n,
P(Xn= 0) = 1 −1
n.
(a) Does {Xn}converge in probability? If so, to what? Why?
(b) Does {Xn}converge in distribution? If so, to what? Why?
(c) Suppose in addition that {Xn}is an independent sequence. Does {Xn}converge
almost surely? What is lim supn→∞ Xnand lim inf n→∞ Xnalmost surely? Explain
your answer.
3. Let Xand Ybe independent Bernoulli random variables on a probability space
(Ω,B, P ) with Xd
=Yand
P(X= 0) = 1
2=P(X= 1).
Let Xn=Yfor n≥1. Show that
Xn=⇒X
but that Xndoes NOT converge in probability to X.
4. If Tn>0 satisfies √n(Tn−θ)d
→N(0, τ 2),
find the limiting distribution of
(a) √n(pTn−√θ) and (b) √n(log Tn−log θ) for θ > 0.
5. Suppose Ysis Poisson distributed with parameters sso that
P(Ys=k) = e−ssk
k!
Compute the chf of Ys. Prove
Ys−s
√s=⇒N(0,1) as s −→ ∞
1
