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76 Characteristic Functions

Corollary 3.1. Let X and Y denote real-value random variables with characteristic func-
tions ϕ X and ϕ Y ,respectively. X and Y have the same distribution if and only if
ϕ X (t) = ϕ Y (t), −∞ < t < ∞. (3.4)

Proof. Clearly, if X and Y have the same distribution then they have the same characteristic
function.
Now suppose that (3.4) holds; let F X denote the distribution function of X and let F Y
denote the distribution function of Y.It follows from Theorem 3.3 that if a and b are
continuity points of both F X and F Y , then
F X (b) − F X (a) = F Y (b) − F Y (a).

Let a n , n = 1, 2,..., denote a sequence of continuity points of both F X and F Y such
that a n diverges to −∞ as n →∞. Note that, since the points at which either F X or F Y is
not continuous is countable, such a sequence must exist. Then
F X (b) − F X (a n ) = F Y (b) − F Y (a n ), n = 1, 2,...

so that

F X (b) − F Y (b) = lim F X (a n ) − F Y (a n ) = 0.
n→∞
Hence, F X (b) and F Y (b) are equal for any point b that is a continuity point of both F X and
F Y .
Now suppose at least one of F X and F Y is not continuous at b. Let b n , n = 1, 2,...,
denote a sequence of continuity points of F X and F Y decreasing to b. Then

F X (b n ) = F Y (b n ), n = 1, 2,...
and, by the right-continuity of F X and F Y ,

F X (b) = F Y (b),
proving the result.

Characteristic function of a sum
The following result illustrates one of the main advantages of working with characteristic
functions rather than with distribution functions or density functions. The proof is straight-
forward and is left as an exercise.

Theorem 3.4. Let X and Y denote independent, real-valued random variables with char-
acteristic functions ϕ X and ϕ Y ,respectively. Let ϕ X+Y denote the characteristic function of
the random variable X + Y. Then

ϕ X+Y (t) = ϕ X (t)ϕ Y (t), t ∈ R.

The result given in Theorem 3.4 clearly extends to a sequence of n independent ran-
dom variables and, hence, gives one method for determining the distribution of a sum
X 1 +· · · + X n of random variables X 1 ,..., X n ; other methods will be discussed in
Chapter 7.