Stat 553: Asymptotic Tools
Midterm
Fall 2011
SOLUTIONS
Problem 1. For n = 1, 2, . . ., suppose that Xn is distributed as a Bernoulli(pn ) random variable. That is,
P (Xn = 1) = 1 P (Xn = 0) = pn .
P
n
(a) [3 points] Suppose that pn = 1/n. Prove that X n 0
Assignment 10
Exercise 4.8 Prove that (4.13) implies both (4.12) and (4.14) (the forward half of
the Lindeberg-Feller Theorem). Use the following steps:
(a) Prove that for any complex numbers a1 , . . . , an and b1 , . . . , bn with |ai | 1
and |bi | 1,
n
Assignment 9
Exercise 5.5 Let Xn binomial(n, p), where p (0, 1) is unknown. Obtain condence intervals for p in two dierent ways:
d
(a) Since n(Xn /n p) N [0, p(1 p)], the variance of the limiting distribution
P
depends only on p. Use the fact that Xn /n p
Assignment 8
Exercise 4.7 Use the Cramr-Wold Theorem along with the univariate Central Limit
e
Theorem (from Example 2.12) to prove Theorem 4.9.
Sketch of solution: This proof is actually given in the course notes, just
before Exercise 4.8: Let X Nk (0, )
Assignment 7
Exercise 4.3 Use the Continuity Theorem to prove the Cramr-Wold Theorem, Thee
orem 4.12.
d
Hint: a Xn a X implies that a
Xn (1)
a
X (1).
Sketch of solution: As we pointed out in class, the only tricky part of the
d
d
Cramr-Wold Theorem is sh
Assignment 6
Exercise 1.41 Kolmogorovs inequality is a strengthening of Chebyshevs inequality
for a sum of independent random variables: If X1 , . . . , Xn are independent random
variables, dene
k
(Xi E Xi )
Sk =
i=1
to be the centered k th partial sum fo
Assignment 5
a.s.
Exercise 3.2 The diagram at the end of this section suggests that neither Xn X
qm
nor Xn X implies the other. Construct two counterexamples, one to show that
qm
qm
a.s.
Xn X does not imply Xn X and the other to show that Xn X does not
a.
Assignment 4
Exercise 2.10 The goal of this Exercise is to construct an example of an independent
P
sequence X1 , X2 , . . . with E Xi = such that X n but Var X n does not
converge to 0. There are numerous ways we could proceed, but let us suppose
that fo
Assignment 3
Exercise 1.43 The complex plane C consists of all points x + iy , where x and y are
real numbers and i = 1. The elegant result known as Eulers formula relates
the points on the unit circle to the complex exponential function:
expcfw_it = cos
Assignment 2
Exercise 1.20 According to the result of Exercise 1.16, the limit (1.21) implies that
the relative dierence between n=1 (1/i) and log n goes to zero. But this does
i
not imply that the dierence itself goes to zero (in general, the dierence ma
Assignment 1
Exercise 1.1 Assume that an a and bn b, where a and b are real numbers.
(a) Prove that an bn ab
Hint: Show that |an bn ab| |(an a)(bn b)| + |a(bn b)| + |b(an a)| using
the triangle inequality.
(b) Prove that if b = 0, an /bn a/b.
Sketch of so
Assignment 11
Exercise 7.1 In this problem, we explore an example in which the MLE is not consistent. Suppose that for (0, 1), X is a continuous random variable with
density
f (x) = g (x) +
1
h
()
x
()
,
(7.4)
where () > 0 for all , g (x) = I cfw_1 < x