## Description

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

Sn =

Pn

i=1 Xi

, 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 + x

2

)]. Let Xn = |ξn|

and Sn =

Pn

i=1 Xi

. Find bn such that Sn/bn → 1 in probability.

Q5: Let pk = 1/(2kk(k + 1)), k = 1, 2, · · · , and p0 = 1 −

P

k≥1

pk. Notice that

X∞

k=1

2

k

pk = 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., P(Xn = k) = λ

k

n

e

−λn /k!

for k = 0, 1, 2, . . .. Show that Sn/ESn → 1 a.s. if P

n

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

X∞

n=0

Xnz

n

1

Is there any deterministic (almost surely) radius of convergence of the above series in the

following two cases (a): P(Xi = 1) = P(Xi = −1) = 1

2

, (b): Xi ∼ N(0, 1)? If so, find the

radius.

2