[SOLVED] MATH5411 Homework 2

Q1: Let X,X₁,X₂,… be a sequence of random variables defined on the same probability space. Further, let g : R → R be a coninuous function. Prove the following continuous mapping theorem

(i): If X_n −^P→ X, then g(X_n) −^P→ g(X);

(ii): If X_n −^a.s.→ X, then g(X_n) −^a.s.→ g(X).

Q2: Prove the following statements:

(a) If X_n a.s.→ X and Y_n ^a.s.→ Y then X_n + Y_n ^a.s.→ X + Y . (b) If X_n →^P X and Y_n →^P Y then X_n + Y_n →^P X + Y .

(c) It is not in general true that X_n + Y_n →^D X + Y if X_n →^D X and Y_n →^D Y .

Q3: Let X₁,X₂,… be uncorrelated with EX_i = µ_i and Var(X_i)/i → 0 as i → ∞. Let

, and µ_n = ES_n/n. Show that S_n/n − µ_n → 0 in mean square and thus in

probability.

Q4: Let ξ₁,ξ₂,…. be i.i.d Cauchy r.v.s. with common density 1/[π(1 + x²)]. Let X_n = |ξ_n| and. Find b_n such that S_n/b_n → 1 in probability.

Q5: Let p_k = 1/(2^kk(k + 1)), k = 1,2,···, and p₀ = 1 − ^P_k≥1 p_k. Notice that

∞

X k

2 p_k = 1.

k=1

So, if we let X₁,X₂,… be i.i.d. with P(X_n = −1) = p₀ and

P(X_n = 2^k − 1) = p_k,            ∀k ≥ 1,

then EX_n = 0. Let S_n = X₁ + … + X_n. Show that

S_n/(n/log₂ n) → −1,           in probability.

Q6: Suppose X_n are independent Poisson r.v.s with rate λ_n, i.e., ! for k = 0,1,2,…. Show that S_n/ES_n → 1 a.s. if ^P_n λ_n = ∞.

Q7: Let Y₁,Y₂,… be i.i.d. Find necessary and sufficient conditions for

• Y_n/n → 0 almost surely;
• (max_m≤n Y_m)/n → 0 almost surely; (iii) (max_m≤n Y_m)/n → 0 in probability; (iv) Y_n/n → 0 in probability.

Q8: Let X_i’s be i.i.d. random variables. Consider the random power series

1

Is there any deterministic (almost surely) radius of convergence of the above series in the following two cases (a): , (b): X_i ∼ N(0,1)? If so, find the radius.

2