Q1: Let A1,A2,··· be a sequence of events. Define
.
Clearly Cn ⊂ An ⊂ Bn. The sequences {Bn} and {Cn} are decreasing and increasing respectively with limits
limBn = B = ∩nBn = ∩n ∪m≥n Am, limCn = C = ∪nCn = ∪n ∩m≥n Am.
The events B and C are denoted limsupn→∞ An and liminfn→∞ 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 = limAn 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 is a field. Give an example to show that 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: )) 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
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