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Category: MATH 5411

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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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