Salins
MA 780 Practice Midterm
March 2023
Instructions:
• Explain all of your steps.
• There is no need to simplify arithmetic (unless stated otherwise).
• You may refer to the textbook, recorded lectures, homework assignments, and lecture notes.
• You may not use any resources other than the textbook, recorded lectures, homework assignments, and lecture notes.
• You may not use a calculator or calculator website.
• You may not collaborate with anybody.
• Do not cheat.
Salins
MA 780 Practice Midterm
March 2023
1. Let Xk be i.i.d. random variables satisfying P(Xk = −1) = 12 , P(Xk =
1) = 12 . Let ak be a sequence of positive real numbers that converges
to 0. Prove that
∞
X
ak Xk converges with probability 1
k=1
if and only if
∞
X
a2k < +∞.
k=1
Show all of your work and explain each step.
Salins
MA 780 Practice Midterm
March 2023
α
2. Let Xk be independent random variables
Pn such that P(Xk = k ) =
1
and P(Xk = 0) = 1 − kα . Let Sn = k=1 Xk .
(a) Prove that when α < 1,
(b) Prove that when α > 1,
(c) Prove that when α = 1,
Sn
n
Sn
n
Sn
n
converges to 1 almost surely.
converges to 0 almost surely.
does not converge.
1
kα
Salins
MA 780 Practice Midterm
March 2023
3. Let Xk be i.i.d. random variables with E(Xk ) = 0 and Var(Xk ) = σ 2 .
(a) Prove that
Pn
2
k=1 Xk
σ2n
converges to 1 almost surely.
(b) Prove that
Pn
Xk
pPk=1
converges in distribution to N (0, 1).
n
2
k=1 Xk
Salins
MA 780 Practice Midterm
March 2023
4. Let Xk be independent random variables such that P(Xk = 1) =
P(Xk = 0) = 1 − k1 . Prove that
Pn
k=1 Xk − log(n)
p
converges in distribution to N (0, 1).
log(n)
Remember to use the fact that
Pn
lim
n→∞
1
k=1 k
log(n)
= 1.
1
k
and