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Exam Measure Theoretic Probability
Korteweg-de Vries Instituut voor wiskunde, UvA
Lecturer: Sonja Cox
Date/time: Wednesday, January 7th 2015, 14:00-17:00
Student name:
Student number:
Student affiliation (university):
Exercise
Max points
Points scored
(1st correction)
Points scored
(2nd correction)
1
4
2
12
3
14
4
16
5
8
total
54
NOTE: Unless indicated otherwise, all results (theorems/propositions/lemmas)
in the MTP lecture notes may be used to answer the exam questions. Please indicate
what results you are using.
Exercise 1 (4pt)
Let (Xn )∞
variables
n=1 be a sequence of independent, identically distributed random
Q
on a probability space (Ω, F, P). Assume EX1 = 0. Prove: limn→∞ nk=1 eXk /n = 1
a.s.
Exercise 2 (12pt)
For A ⊂ B([−1, 1]) define −A = {x ∈ R : − x ∈ A}. Let λ denote the Lebesgue
measure on B([−1, 1]). Define
A = {A ∈ B([−1, 1]) : λ(A) = λ(−A)}.
i. (3pt) Prove that A is a d-system.
ii. (1 pt) Prove that A = B([−1, 1]).
1
iii. (4pt) Prove: if f :R[−1, 1] → R is a simple function and f (x) = −f (−x) for all
1
x ∈ [−1, 1], then −1 f (x) dx = 0.
iv. (4pt) Prove: if f : [−1,
R 1 1] → R is Lebesgue integrable and f (x) = −f (−x) for
all x ∈ [−1, 1], then −1 f (x) dx = 0.
Exercise 3 (14pt)
Let (Xn )∞
n=1 be a sequence of independent, identically distributed random variables
on a probability space (Ω, F, P) satisfying EX1 = 0 and EX12 = 1. Define F0 =
{∅, Ω} and for n ∈ N define Fn = σ(X1 , . . . , Xn ) and
Mn =
n
X
1
(X1
k
· . . . · Xk )
k=1
(i.e., M1 = X1 , M2 = X1 + 21 X1 X2 , M3 = X1 + 12 X1 X2 + 13 X1 X2 X3 ).
i. (4pt) Calculate EMn2 , n ∈ N.
∞
ii. (4pt) Prove that (Mn )∞
n=1 is an (Fn )n=1 -martingale.
iii. (4pt) Provide hM i.
iv. (2pt) Is (Mn )∞
n=1 uniformly integrable?
Exercise 4 (18pt)
Let (Ω, F, P) be a probability space. Let An,j ∈ F, n ∈ N0 , j ∈ {1, 2, 3, . . . , 2n }, be
Sn
such that for all n ∈ N0 it holds that 2j=1 An,j = Ω and
∀i, j ∈ {1, 2, 3, . . . , 2n }, i 6= j : An,i ∩ An,j = ∅ and An,i = An+1,2i−1 ∪ An+1,2i . (1)
For n ∈ N0 define
Fn = σ({An,j : j ∈ {1, 2, 3, . . . , 2n }).
i. (2pt) Prove that (Fn )∞
n=0 is a filtration.
ii. (3pt) Let µ : F → [0, ∞) be a probability measure on (Ω, F). Assume:
If P(A) = 0 for some A ∈ F, then µ(A) = 0.
(2)
For n ∈ N0 define
( µ(A
Mn (ω) =
n,j )
,
P(An,j )
ω ∈ An,j , P(An,j ) 6= 0;
0,
otherwise.
Prove that for all n ∈ N0 and all j ∈ {1, 2, 3, . . . , 2n } it holds that
Z
Z
Mn+1 dP =
Mn dP.
An,j
An,j
2
(3)
(4)
∞
iii. (1pt) Explain why it follows that (Mn )∞
n=1 is (Fn )n=0 -martingale.
R
iv. (2pt) Prove that for all A ∈ Fn it holds that µ(A) = A Mn dP.
v. (4pt) Assumption (2) is equivalent to:
∀ε > 0 ∃δ > 0 : if P(A) < δ for some A ∈ F, then µ(A) < ε.
(5)
Prove that (Mn )∞
n=1 is uniformly integrable.
S
vi. (4pt) Define F∞ =R σ( n∈N Fn ). Prove that there exists a M∞ ∈ L1 (Ω, F∞ , P)
such that µ(A) = A M∞ dP for all A ∈ F∞ .
Exercise 5 (10pt)
Let (Xn )∞
n=1 be a sequence of independent, identically distributed random variables
1
on a probability space (Ω, F, P) satisfying P(X1 = 1) = P(X
Pn 1 = −1) = 2 . For n ∈ N
1
let φn : R → C denote the characteristic function of √n k=1 Xk .
In this exercise you may not use the Central Limit Theorem.
You may use that by l’Hopital’s rule we have that
lim
x↓0
log(cos(ux))
−u sin(ux)
−u2 cos(ux)
u2
=
lim
=
lim
=
−
.
x↓0 2x cos(ux)
x↓0 2(cos(ux) − x sin(ux))
x2
2
i. (4pt) Prove that φn (u) = (cos(n−1/2 u))n for u ∈ R.
ii. (4pt) Let γ : Ω → R be a standard Gaussian random variable and let φγ : R →
2
R denotePits characteristic function, i.e., φγ (u) = e−u /2 for all u ∈ R. Prove
that √1n nk=1 Xk → γ weakly as n → ∞.
Exercise
Let (Xn )∞
n=1 be a sequence of independent, identically distributed random variables
on a probability space (Ω, F, P) satisfying P(X1 = 1) =PP(X1 = −1) = 21 . For
n ∈ N and k ∈ {1, . . . , n} let ak,n ∈ [0, ∞) be such that nj=1 a2n,j = 1 and define
ξn,k = an,k Xk .
i. Formulate the Lindeberg condition for (ξn,k )n∈N,k∈{1,...,n} .
P
ii. Provide explicit values of an,k , n ∈ N, k ∈ {1, . . . , n}, such that nk=1 ξn,k → γ
weakly as n → ∞, where γ is a standard normal Gaussian random variable.
Exercise
Let (S, S) be a measurable space and let µ, ν : S → [0, ∞) be finite measures on
(S, S). Assume that for all A ∈ S it holds that µ(A) = 0 implies ν(A) = 0. Prove
that for all ε > 0 there exists a δ > 0 such that if µ(A) < δ, then ν(A) < ε .
3
Exercise
Let (Ω, F, P) be a probability space and let An ∈ F, n ∈ N. Assume F =
σ({An : n ∈ N}). For n ∈ N define Fn = σ({Ak : k ∈ {1, . . . , n}}). Let B ∈ F.
i. Prove that limn→∞ E[1B |Fn ] = 1B a.s.
ii. Does it hold that limn→∞ E|E[1B |Fn ] − 1B | = 0? Motivate your answer.
Exercise
Let (Xn )∞
n=1 be a sequence of independent, identically distributed random variables
on a probability space (Ω, F, P) satisfying E(X12 ) < ∞. Let Yn → Y weakly as
n → ∞. Prove that for all a, b ∈ R, a < b, it holds that lim inf n→∞ P(Yn ∈ (a, b)) >
P(Y ∈ (a, b).
4