Question 1: Suppose that Ω = [0, 1] is the unit interval, and is the set of all subsets A such that either A or AC is countable (i.e., finite or countably infinite), and Pis defined by P(A) = 0 if A is countable, and P(A) = 1 if AC is countable. Answer and prove in details:
(a) Is F an algebra?
(b) Is F a σ-algebra?
(c) Is P finitely additive?
(d) Is P countably additive on F, meaning that if A1, A2, ... ∈ F are disjoint, and if it happens that UnAn ∈ F, then P(UnAn) = ∑nP(An)?
(e) Is P uncountably additive?
Question 2: Let X1, X2, ... be a sequence of independent random variables on a probability space (Ω, F, P).
(a) Show that n=1Σ∞ P(|Xn| > n) < ∞ implies lim supn |Xn/n| ≤ 1 a.s.
(b) Suppose X1, X2, ... be independent. Prove that supn Xn < ∞ a.s. if and only if n=1Σ∞ P(Xn > c) < ∞, for some c ∈ R.
Question 3: Let X1, X2, ... be a sequence independent of r.v.s. and a ∈ R. Denote Sn = k=1ΣnXk.
(a) Which of the following events are tail events, i.e., which sets belong to G = n=1∩∞ σ(Xn, Xn+1, ...)? Explain why in details.
(i) {limn→∞ Xn = a}
(ii) {limn→∞ Sn = a}
(iii) k=1Σ∞ Xk converges}
(b) Prove that there exists c, c' ∈ R ∪ {-∞, ∞} such that limn sup Xn = c a.s. and limn inf Xn = c' a.s.
Question 4: Let X1, X2, ... be some random variables on a probability space (Ω, F, P).
(a) Prove that Xn → +∞ a.s. if and only if ∀c > 0, P(Xn < c i.o.) = 0.
(b) Suppose Xn ≥ 0, ∀n ∈ N, and suppose there exists a random variable X such that Xn ≤ X, ∀n ∈ N, and E[|X|] < ∞. Prove that Xn → X in probability, as n → ∞, implies Xn → X in L1, as n → ∞.
Question 5: Let X1, X2, ... be a sequence of i.i.d. r.v.s. Show that E[|X1|] = ∞ implies lim supn |Sn|/n = ∞ a.s., where Sn = k=1Σn Xk.
Question 6: Let p ≥ 1 and c ∈ R. Prove:
(a) In Xn → c a.s. as n → ∞, then Sn → c a.s., as n → ∞.
(b) Give an example for independent sequence of random variables Xn that Xn →0 in probability, but Sn/n →~ 0 in probability as n → ∞.
Question 7: Let X1, X2, ... be independent random variables with (Var(Xn))/n → 0, as n → ∞. Prove
(Sn - E[Sn])/n → 0,
in L2 and in probability, as n → ∞.
Question 8: Let X1, X2, ... be i.i.d. r.v.s. with E[X1] = 0 and V ar(X1) = 1, and Sn = k=1Σn Xk. Let φ(n) = √(2n log log n), and remember that the law of iterated logarithm implies that
limn sup Sn/φ(n) = 1 a.s.,
that is equivalent to: ∀∈ > 0,
P(Sn/φ(n) ≥ 1 + ∈ i.o) = 0 and P(Sn/φ(n) ≥ 1 - ∈ i.o) = 1.
(a) Use a very basic property of lim sup and lim inf to conclude
limn inf sn/φ(n) = -1 a.s.
(b) From the law of the iterated logarithm and part (a) show that for any ∈ > 0, with probability 1
-(1 + s)φ(n) < Sn < (1 + ∈)φ(n) almost always (a.a.)
(c) Use part (b) to show that for any sequence of real numbers (an)n∈N with φ(n)/an → 0, as n → ∞, we get Sn/an → 0 a.s.
Remark: Observe that an = n gives us the SLLN with the finite second moment condition. In fact, Sn/φ(n) does not converge a.s. However, you DO NOT need to prove this remark.
Question 9: Let X1, X2, ... be independent random variables on (Ω, F, P) with E[Xk] = 0 and V ar(Xk) = σk2, for any k ∈ N. Show that Σn=1∞ σ2n/n2< ∞ implies
(Σk=1n Xk)/n → 0 a.s.
Hint: Apply Kronecker's Lemma.