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82 Characteristic Functions

d
Corollary 3.3. Let X denote a random vector taking values in R and let X = (X 1 , X 2 )
where X 1 takes values in R d 1 and X 2 takes values in R . Let ϕ denote the characteris-
d 2
tic function of X, let ϕ 1 denote the characteristic function of X 1 , and let ϕ 2 denote the
characteristic function of X 2 . Then X 1 and X 2 are independent if and only if

d 2
ϕ(t) = ϕ 1 (t 1 )ϕ(t 2 ) for all t = (t 1 , t 2 ), t 1 ∈ R , t 2 ∈ R .
d 1
Example 3.11 (Multinomial distribution). Consider a multinomial distribution with
m
parameters n and (θ 1 ,...,θ m ), θ j = 1. Recall that this is a discrete distribution with
j=1
frequency function

n
x 1 x 2
p(x 1 ,..., x m ) = θ θ ··· θ m x m
1
2
x 1 , x 2 ,..., x m
m
for x j = 0, 1,..., n, j = 1,..., m, such that x j = n; see Example 2.2.
j=1
The characteristic function of the distribution is given by

ϕ(t) = exp(it 1 x 1 +· · · + it m x m )p(x 1 ,..., x m )
X
where the sum is over all

m

m
(x 1 ,..., x m ) ∈ X = (x 1 ,..., x m ) ∈{0, 1,...} : x j = n .
j=1
Hence,

n
x 1
ϕ(t) = [exp(it 1 )θ 1 ] ··· [exp(it m )θ m ] x m
x 1 , x 2 ,..., x m
X
n

m
n

= exp(it j )θ j
x 1 , x 2 ,..., x m
j=1 X
x 1 x m

exp(it 1 )θ 1 exp(it m )θ m
× m ··· m
j=1 exp(it j )θ j j=1 exp(it j )θ j
n

m

= exp(it j )θ j
j=1
where t = (t 1 ,..., t m ).
Using Theorems 3.6 and 3.7, it follows immediately from this result that the sum of r
independent identically distributed multinomial random variables with n = 1 has a multi-
nomial distribution with n = r.
3.3 Further Properties of Characteristic Functions
We have seen that the characteristic function of a random variable completely determines
its distribution. Thus, it is not surprising that properties of the distribution of X are reflected
in the properties of its characteristic function. In this section, we consider several of these
properties for the case in which X is a real-valued random variable.