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,
Exercise 5.5 Let Xn binomial(n, p), where p (0, 1) is unknown. Obtain condence intervals for p in two dierent ways:
(a) Since n(Xn /n p) N [0, p(1 p)], the variance of the limiting distribution
depends only on p. Use the fact that Xn /n p
Exercise 4.7 Use the Cramr-Wold Theorem along with the univariate Central Limit
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, )
Exercise 4.3 Use the Continuity Theorem to prove the Cramr-Wold Theorem, Thee
Hint: a Xn a X implies that a
Sketch of solution: As we pointed out in class, the only tricky part of the
Cramr-Wold Theorem is sh
Exercise 1.41 Kolmogorovs inequality is a strengthening of Chebyshevs inequality
for a sum of independent random variables: If X1 , . . . , Xn are independent random
(Xi E Xi )
to be the centered k th partial sum fo
Exercise 3.2 The diagram at the end of this section suggests that neither Xn X
nor Xn X implies the other. Construct two counterexamples, one to show that
Xn X does not imply Xn X and the other to show that Xn X does not
Exercise 2.10 The goal of this Exercise is to construct an example of an independent
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
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
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
not imply that the dierence itself goes to zero (in general, the dierence ma
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
Stat 553: Asymptotic Tools
This midterm is worth 15 points. You have 60 minutes.
Problem 1. Let X1 , X2 , . . . be independent and identically distributed from a beta(1, ) distribution for
some > 0.
The beta(1, ) distribution
Stat 553: Asymptotic Tools
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 .
(a) [3 points] Suppose that pn = 1/n. Prove that X n 0