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72 Characteristic Functions

Example 3.4 (Gamma distribution). Let X denote a real-valued random variable with an
absolutely continuous distribution with density function
β α α−1
f (x) = x exp(−βx), x > 0
(α)
where α and β are nonnegative constants. This is called a gamma distribution with param-
eters α and β.
The characteristic function of this distribution is given by
∞ β
α
ϕ(t) = exp(itx) x α−1 exp(−βx) dx
0 (α)
β α ∞ α−1 β α
= x exp(−(β − it)x) = , −∞ < t < ∞.
(α) 0 (β − it) α
The characteristic function has a number of useful properties that make it convenient
for deriving many important results. The main drawback of the characteristic function is
that it requires some results in complex analysis. However, the complex analysis required is
relatively straightforward and the advantages provided by the use of characteristic functions
far outweigh this minor inconvenience. For readers unfamiliar with complex analysis, a brief
summary is given in Appendix 2.

3.2 Basic Properties
Characteristic functions have a number of basic properties.

Theorem 3.1. Let ϕ(·) denote the characteristic function of a distribution on R. Then
(i) ϕ is a continuous function
(ii) |ϕ(t)|≤ 1 for all t ∈ R
(iii) Let X denote a real-valued random variable with characteristic function ϕ, let
a, b denote real-valued constants, and let Y = aX + b. Then ϕ Y , the characteristic
function of Y, is given by
ϕ Y (t) = exp(ibt)ϕ(at).
(iv) u is an even function and v is an odd function, where u and v are given by (3.1).

Proof. Note that

∞
|ϕ(t + h) − ϕ(t)|≤ | exp{ix(t + h)}− exp{itx}| dF X (x)
−∞
∞

≤ | exp{ixh}− 1| dF X (x).
−∞
Note that | exp{ixh}− 1| is a real-valued function bounded by 2. Hence, the continuity of
ϕ follows from the Dominated Convergence Theorem (see Appendix 1), using the fact that
exp(ixh)is continuous at h = 0. This establishes part (i).
Part (ii) follows from the fact that

∞
|ϕ(t)|≤ | exp{itx}| dF X (x) ≤ 1.
−∞
Parts (iii) and (iv) are immediate.