Description
Q1: Let A1, A2, · · · be a sequence of events. Define
Bn = ∪
∞
m=nAm, Cn = ∩
∞
m=nAm.
Clearly Cn ⊂ An ⊂ Bn. The sequences {Bn} and {Cn} are decreasing and increasing
respectively with limits
lim Bn = B = ∩nBn = ∩n ∪m≥n Am, lim Cn = C = ∪nCn = ∪n ∩m≥n Am.
The events B and C are denoted lim supn→∞ An and lim infn→∞ An, respectively. Show that
(a) B = {ω ∈ Ω : ω ∈ An for infinitely many values of n},
(b) C = {ω ∈ Ω : ω ∈ An for all but finitely many values of n},
We say that the sequence {An} converges to a limit A = lim An if B and C are the same set
A. Suppose that An → A and show that
(c) P(An) → P(A).
Q2: Let F be a σ-field, and let G, H ⊆ F be two sub σ-fields.
(i) Give one example which shows that G ∪ H is not a σ-field.
(ii) Prove that G ∩ H is a σ-field.
(iii) F1 ⊆ F2 ⊆ · · · is a sequence of sub σ-fields, prove that ∪
∞
i=1Fi
is a field. Give an
example to show that ∪
∞
i=1Fi
is not necessarily a σ-field.
Q3: Suppose that X and Y are random variables on (Ω, F, P) and let A ∈ F. We set
Z(ω) = X(ω) for all ω ∈ A and Z(ω) = Y (ω) for all ω ∈ Ac
. Prove that Z is a random
variable.
Q4: Prove the following two definitions of random vector are equivalent.
Def.1: X = (X1, . . . , Xd) : (Ω, F) → (R
d
, B(R
d
)) is a random vector if it is F-measurable.
Def.2: X = (X1, . . . , Xd) is a random vector if Xi
: (Ω, F) → (R, B(R)) is F-measurable
for all i = 1, . . . , d.
Q5: Prove the following reverse Fatou’s lemma: Let f1, f2, . . . be a sequence of Lebesgue
integrable functions on the probability space (Ω, F, P). Suppose that there exists a nonnegative integrable function g on Ω such that fn ≤ g for all n. Prove
lim sup
n→∞ Z
fndµ ≤
Z
lim sup
n→∞
fndµ.
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