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110 Moments and Cumulants

as in the case of a real-valued random variable, δ is known as the radius of convergence
of M X .
Many of the properties of a moment-generating function for a real-valued random vari-
able extend to the vector case. Several of these properties are given in the following theorem;
the proof is left as an exercise.

Theorem 4.12. Let X and Y denote d-dimensional random vectors with moment-generating
functions M X and M Y , and radii of convergence δ X and δ Y ,respectively.
d
(i) Let A denote an m × d matrix of real numbers and let b denote an element of R .
Then M AX+b , the moment-generating function of AX + b, satisfies
T T
M AX+b (t) = exp{t b}M X (A t), ||t|| <δ A
for some δ A > 0, possibly depending on A.
(ii) If X and Y are independent then M X+Y , the moment-generating function of X + Y,
exists and is given by
M X+Y (t) = M X (t)M Y (t), ||t|| < min(δ X ,δ Y ).

(iii) X and Y have the same distribution if and only if there exists a δ> 0 such that
M X (t) = M Y (t) for all ||t|| <δ.

Asisthecasewiththecharacteristicfunction,thefollowingresultshowsthatthemoment-
generating function can be used to establish the independence of two random vectors; the
proof is left as an exercise.

d
Corollary 4.2. 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 M denote the moment-
d 2
generating function of X with radius of convergence δ, let M 1 denote the moment-generating
function of X 1 with radius of convergence δ 1 , and let M 2 denote the moment-generating
function of X 2 with radius of convergence δ 2 .
X 1 and X 2 are independent if and only if there exists a δ 0 > 0 such that for all t = (t 1 , t 2 ),
t 1 ∈ R ,t 2 ∈ R , ||t|| <δ 0 ,
d 2
d 1
M(t) = M 1 (t 1 )M(t 2 ).

4.4 Cumulants
Although moments provide a convenient summary of the properties of a random variable,
they are not always easy to work with. For instance, let X denote a real-valued random
variable and let a, b denote constants. Then the relationship between the moments of X and
those of aX + b can be quite complicated. Similarly, if Y is a real-valued random variable
such that X and Y are independent, then the moments of X + Y do not have a simple
relationship to the moments of X and Y.
Suppose that X and Y have moment-generating functions M X and M Y , respectively.
Some insight into the properties described above can be gained by viewing moments of
a random variable as derivatives of its moment-generating function at 0, rather than as
integrals with respect to a distribution function. Since the moment-generating function of