# MATH 5411 – Advanced Probability I Homework 2

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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