Summary: Math­571. 6­th Homework. Due Thursday, April 26, 2007.
1. Let (t) be the ch.f. of a r.v. X. Show that if |(t)| = 1 for each t in a neighborhood of
zero, then X is degenerate.
2. Show that (t) = 1
1+t2 is a ch.f.
3. Prove that for each > 2, (t) = e-|t|
is not a ch.f.
4. Let {Xn,j : 1 j kn, n 1} be a triangular array of is rowwise independent r.v.'s.
Suppose that the distribution of Xn,j is uniform in the interval (-j, j +n). Find constants
an, bn such that Sn-bn
an
d
N(0, 1). You need to prove that Sn-bn
an
d
N(0, 1).
5. Let {Xj} be a sequence of independent r.v.'s. Suppose that:
(i) There exists a constant M such that |Xj| M a.s. for each j 1.
(ii)
j=1 Var(Xj) = .

Source: Arcones, Miguel A. - Department of Mathematical Sciences, State University of New York at Binghamton

Collections: Mathematics