Page 107 - Elements of Distribution Theory

P1: JZP
052184472Xc03 CUNY148/Severini May 24, 2005 2:34

3.5 Suggestions for Further Reading 93

3.22 Let the random variable Z be defined as in Exercise 3.5.
(a) Suppose that, for each j = 1, 2, the distribution of X j is symmetric about 0. Is the distri-
bution of Z symmetric about 0?
(b) Suppose that X 1 and X 2 each have a lattice distribution. Does Z have a lattice distribution?
3.23 Let X denote a real-valued random variable with an absolutely continuous distribution. Suppose
that the distribution of X is symmetric about 0. Show that there exists a density function p for
the distribution satisfying
p(x) = p(−x) for all x.
3.24 Let X denote a d-dimensional random vector with characteristic function ϕ. Show that X has a
d
degenerate distribution if and only if there exists an a ∈ R such that
d
T
|ϕ(a t)|= 1 for all t ∈ R .
3.25 Prove Theorem 3.10.

3.5 Suggestions for Further Reading
A comprehensive reference for characteristic functions is Lukacs (1960); see also Billingsley (1995,
Section 26), Feller (1971, Chapter XV), Karr (1993, Chapter 6), Port (1994, Chapter 51), Stuart
and Ord (1994, Chapter 4). See Apostol (1974, Chapter 11) for further details regarding Fourier
transforms. Lattice distributions are discussed in detail in Feller (1968). Symmetrization is discussed
in Feller (1971, Section V.5).