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 Xn −P→ X, then g(Xn) −P→ g(X);
(ii): If Xn −a.s.→ X, then g(Xn) −a.s.→ g(X).
Q2: Prove the following statements:
(a) If Xn a.s.→ X and Yn a.s.→ Y then Xn + Yn a.s.→ X + Y . (b) If Xn →P X and Yn →P Y then Xn + Yn →P X + Y .
(c) It is not in general true that Xn + Yn →D X + Y if Xn →D X and Yn →D Y .
Q3: Let X1,X2,... be uncorrelated with EXi = µi and Var(Xi)/i → 0 as i → ∞. Let
, and µn = ESn/n. Show that Sn/n − µn → 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 Xn = |ξn| and . Find bn such that Sn/bn → 1 in probability.
Q5: Let pk = 1/(2kk(k + 1)), k = 1,2,···, and p0 = 1 − Pk≥1 pk. Notice that
∞
X k
2 pk = 1.
k=1
So, if we let X1,X2,... be i.i.d. with P(Xn = −1) = p0 and
P(Xn = 2k − 1) = pk, ∀k ≥ 1,
then EXn = 0. Let Sn = X1 + ... + Xn. Show that
Sn/(n/log2 n) → −1, in probability.
Q6: Suppose Xn are independent Poisson r.v.s with rate λn, i.e., ! for k = 0,1,2,.... Show that Sn/ESn → 1 a.s. if Pn λn = ∞.
Q7: Let Y1,Y2,... be i.i.d. Find necessary and sufficient conditions for
(i) Yn/n → 0 almost surely;
(ii) (maxm≤n Ym)/n → 0 almost surely; (iii) (maxm≤n Ym)/n → 0 in probability; (iv) Yn/n → 0 in probability.
Q8: Let Xi’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): Xi ∼ N(0,1)? If so, find the radius.
2
