Page 83 - Elements of Distribution Theory

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

3

Characteristic Functions

3.1 Introduction
The properties of a random variable may be described by its distribution function or, in some
cases, by its density or frequency function. In Section 1.8 it was shown that expectations of
the form E[g(X)] for all bounded, continuous, real-valued functions g completely determine
the distribution of X (Theorem 1.11). However, the entire set of all bounded, continuous
real-valued functions is not needed to characterize the distribution of a random variable in
this way.
Let X denote a real-valued random variable with distribution function F.For each t ∈ R,
let g t denote a function on the range of X such that E[|g t (X)|] < ∞. Then the function

∞
W(t) = E[g t (X)] = g t (X) dF(x), t ∈ R,
−∞
gives the expected values of all functions of the form g t .If the set of functions G =
{g t : t ∈ R} is chosen appropriately, then function W will completely characterize the dis-
tribution of X, and certain features of F will be reflected in W.Infact, we have already
seen one simple example of this with the distribution function, in which g t (x) = I {x≤t} .
A function such as W is called an integral transform of F; the properties of an integral
transform will depend on the properties of the class of functions G.In this chapter, we
consider a particular integral transform, the characteristic function.Two other integral
transforms, the Laplace transform and the moment-generating function, are discussed in
Chapter 4.
The characteristic function of the distribution of a random variable X,or, more simply,
the characteristic function of X,is defined as
∞

ϕ(t) ≡ ϕ X (t) = E[exp(it X)] ≡ exp(itx) dF(x), −∞ < t < ∞,
−∞
where exp(itx)isa complex number; writing

exp(itx) = cos(tx) + i sin(tx),

as described in Appendix 2, we may write ϕ(t) = u(t) + iv(t) where

∞ ∞
u(t) = cos(tx) dF(x) and v(t) = sin(tx) dF(x). (3.1)
−∞ −∞
69