Theory of Statistics - George Mason University

46 1 Probability Theory
Euler’s formula also provides an alternate expression for the CF that is
sometimes useful:
ϕX(t) = E cos t T X + iE sin t T X . (1.89)
Although the CF always exists, it may not have an explicit representation.
The CF for the lognormal distribution, for example, cannot be represented
explicitly, but can be approximated to any tolerance by a divergent series
(exercise).
Note that the integration in the expectation operator defining the CF is
not complex integration; we interpret it as ordinary Lebesgue integration in
the real dummy variable in the PDF (or dF). Hence, if the MGF is finite
for all t such that |t| < ɛ for some ɛ > 0, then the CF can be obtained by
replacing t in ψX(t) by it. Note also that the MGF may be defined only in a
neighborhood of 0, but the CF is defined over IR.
There are some properties of the characteristic function that are immediate
from the definition:
ϕX(−t) = ϕX(t) (1.90)
and
ϕX(0) = 1. (1.91)
The CF is real if the distribution is symmetric about 0. We see this from
equation (1.89).
The CF is bounded:
|ϕX(t)| ≤ 1. (1.92)
We see this property by first observing that
E e itTX
≤ E
e it T X
,
and then by using Euler’s formula on eitT
x
.
The CF and the moments of the distribution if they exist are closely related,
as we will see below. Another useful property provides a bound on the
difference of the CF at any point and a partial series in the moments in terms
of expected absolute moments.
ϕX(t)
n (it)
−
k
k! E(Xk
n
2|tX|
) ≤ E min ,
n!
|tX|n+1
, (1.93)
(n + 1)!
k=0
This property follows immediately from inequality 0.0.71 on page 656.
Another slightly less obvious fact is
Theorem 1.22
The CF is uniformly continuous on IR.
Theory of Statistics c○2000–2013 James E. Gentle