Universität zu Köln
Winter semester 2016 / 2017
Institut für Mathematik
Lecturer: Prof. Dr. A. Drewitz
Assistant: L. Schmitz
To be handed in on November 30th before exercise class.
5th Homework Probability Theory II
(Martingale convergence and uniform integrability)
To abbreviate notation, in every exercise we assume the random variables to
be defined on a probability space (Ω, F, P).
Exercise 5.1
(3 points)
Let (Yn ) ⊂ L1 be a sequence of independent,
non-negative
random
variables
Qn
with E [Yn ] = 1 for all n. Define Xn := i=1 Yi .
a) Show that (Xn ) is a martingale w.r.t. its canonical filtration, which converges P-a.s. to a random variable X∞ ∈ L1 .
(1 p.)
b) Let P(Yn = 3/2) = 1/2 = P (Yn = 1/2). Show that X∞ is a degenerated
variable, i.e. X∞ = 0 P-a.s. and
"∞
#
∞
Y
Y
E
Yn 6=
E [Yn ] .
n=1
n=1
(2 p.)
Exercise 5.2
(7 points)
a) Let X ∈ L1 . Show that the family {E [X | G] : G is a sub-σ-algebra of F}
is uniformly integrable.
(2 p.)
b) Let (Xn ) ⊂ L1 be an i.i.d. sequence of random variables. Show with
n
the Strong Law of Large Numbers and Theorem 2.3.9., that X1 +...+X
n
1
converges in L to E [X].
(2 p.)
D
c) Let (Yn ) and Y be random variables with Yn −→ Y and assume the
family {Yn : n ∈ N} to be uniformly integrable. Show that Y ∈ L1 and
lim E [Yn ] = E [Y ] .
n→∞
(3 p.)
Because uniform convergence is very important, we are looking for more condtions:
Exercise 5.3
(5 points)
Let S ⊂ L be an L -bounded family. Show that the property of S to be
uniformly integrable is equivalent to each of the following conditions:
T
(i) For every sequence (An ) ⊂ F with A1 ⊃ A2 ⊃ . . . and ∞
n=1 An = ∅ we
have
lim sup E [|X|1An ] = 0;
1
1
n→∞
X∈S
(ii) for every sequence (Bn ) ⊂ F of pairwise disjoint sets we have
lim sup E [|X|1Bn ] = 0.
n→∞
X∈S
Hint: Lemma 2.3.4. might be helpful.
Exercise 5.4
(0 points)
Let (Xn ) ⊂ L1 be an i.i.d. sequence with E [Y1 ] = 0, E [Y12 ] = 1 and Z be
an N (0, 1)-distributed random variable which is independent of (Xn ). Define
Un := √1n (X1 + . . . + Xn ). Show that
lim E [|Un − Z|]
n→∞
exists and calculate the limit.
Hint: This is an application of Exercise 5.2 c).
Remark: Please just hand in solely exercises with positive rating. If you need
more than one sheet, please clip together the papers and write your group, your
name and the exercise on top of every sheet.
It is possible to hand in solutions in groups of at most two persons.
Total: 15
2