Reference no: EM132462187

Assignment

Problem: 1. Suppose that Ω = [0, 1] is the unit interval, and is the set of all subsets A such that either A or A^C is countable (i.e., finite or countably infinite), and Pis defined by P(A) = 0 if A is countable, and P(A) = 1 if A^C 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 U_nA_n ∈ F, then P(U_nA_n) = ∑_nP(A_n)?
(e) Is P uncountably additive?

Problem: 2. Let X₁, X₂, ... be a sequence of independent random variables on a probability space (Ω, F, P).
(a) Show that _n=1Σ^∞ P(|X_n| > n) < ∞ implies lim sup_n |X_n/n| ≤ 1 a.s.

(b) Suppose X₁, X₂, ... be independent. Prove that sup_n X_n < ∞ a.s. if and only if _n=1Σ^∞ P(X_n > c) < ∞, for some c ∈ R.

Problem: 3. Let X₁, X₂, ... be a sequence independent of r.v.s. and a ∈ R. Denote S_n = _k=1ΣⁿXk.

(a) Which of the following events are tail events, i.e., which sets belong to G = _n=1∩^∞ σ(X_n, X_n+1, ...)? Explain why in details.

(i) { lim_n→∞ X_n = a}
(ii) { lim_n→∞ S_n = a}

(iii) _k=1Σ^∞ X_k converges}

(b) Prove that there exists c, c' ∈ R ∪ {-∞, ∞} such that lim_n sup X_n = c a.s. and lim_n inf X_n = c' a.s.

Problem: 4. Let X₁, X₂, ... be some random variables on a probability space (Ω, F, P).

(a) Prove that X_n → +∞ a.s. if and only if ∀_c > 0, P(X_n < c i.o.) = 0.

(b) Suppose X_n ≥ 0, ∀n ∈ N, and suppose there exists a random variable X such that X_n ≤ X, ∀_n ∈ N, and E[|X|] < ∞. Prove that X_n → X in probability, as n → ∞, implies Xn → X in L¹, as n → ∞.

Problem: 5. Let X₁, X₂, ... be a sequence of i.i.d. r.v.s. Show that E[|X₁|] = ∞ implies lim sup_n |Sn|/n = ∞ a.s., where S_n =  _k=1Σⁿ X_k.

Problem: 6. Let p ≥ 1 and c ∈ R. Prove:

(a) In X_n → c a.s. as n → ∞, then S_n → c a.s., as n → ∞.

(b) Give an example for independent sequence of random variables X_n that X_n →0 in probability, but S_n/n →~ 0 in probability as n → ∞.

Problem: 7. Let X₁, X₂, ... be independent random variables with (Var(X_n))/n → 0, as n → ∞. Prove

(S_n - E[S_n])/n → 0,

in L² and in probability, as n → ∞.

Problem: 8. Let X₁, X₂, ... be i.i.d. r.v.s. with E[X₁] = 0 and V ar(X₁) = 1, and S_n = _k=1Σⁿ X_k. Let φ(n) = √(2n log log n), and remember that the law of iterated logarithm implies that

lim_n sup S_n/φ(n) = 1 a.s.,

that is equivalent to: ∀_∈ > 0,

P(S_n/φ(n) ≥ 1 + ∈ i.o) = 0 and P(S_n/φ(n) ≥ 1 - ∈ i.o) = 1.

(a) Use a very basic property of lim sup and lim inf to conclude

lim_n inf s_n/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) < S_n < (1 + ∈)φ(n) almost always (a.a.)

(c) Use part (b) to show that for any sequence of real numbers (a_n)_n∈N with φ(n)/a_n → 0, as n → ∞, we get S_n/a_n → 0 a.s.

Remark: Observe that an = n gives us the SLLN with the finite second moment condition. In fact, S_n/φ(n) does not converge a.s. However, you DO NOT need to prove this remark.

Problem: 9. Let X₁, X₂, ... be independent random variables on (Ω, F, P) with E[X_k] = 0 and V ar(X_k) = σ_k², for any k ∈ N. Show that Σ_n=1^∞ σ²n/n²< ∞ implies

(Σ_k=1ⁿ X_k)/n → 0 a.s.

Hint: Apply Kronecker's Lemma.