The cumulant generating function of a random variable is the natural logarithm of its moment generating function.
The cumulant generating function is often used because it facilitates some calculations. In particular, its derivatives at zero, called cumulants, have interesting relations with moments and central moments.
Remember that the
moment
generating function (mgf) of a random variable
is defined
as
provided
that the expected
value exists and is finite for all
belonging to a closed interval
,
with
.
The mgf has the property that its derivatives at zero are equal to the
moments of
:
The existence of the mgf guarantees that the moments (hence the derivatives at
zero) exist and are finite for every
.
The cumulant generating function (cgf) is defined as follows.
Definition
Suppose that a random variable
possesses a moment generating function
.
Then, the
function
is
the cumulant generating function of
.
Note that the cgf is well-defined since
is strictly positive for any
.
Since the mgf completely characterizes the distribution of a random variable and the natural logarithm is a one-to-one function, also the cumulant generating function completely characterizes the distribution of a random variable.
The derivatives of the cgf at zero are called cumulants.
The
-th
cumulant
is
The first cumulant is equal to the expected
value:
The first derivative of the cgf
isSince
we
have
The second cumulant is equal to the
variance:
The second derivative of the cgf
isWhen
we evaluate it at
,
we
get
The third cumulant is equal to the third
central
moment:
The third derivative of the cgf
isWhen
we evaluate it at
,
we
get
But
Therefore,
The relation of higher cumulants to moments and central moments is more complicated.
When
is a
random vector, the
joint
moment generating function of
is defined
as
provided
that the expected value exists and is finite for all
real vectors
belonging to a closed rectangle
:
with
for all
.
The joint mgf has the property
that
The existence of the mgf guarantees the existence and finiteness of the cross-moments on the left-hand side of the equation.
Having reviewed the basic properties of the joint mgf, we are ready to define the joint cumulant generating function.
Definition
Suppose that a random vector
possesses a joint moment generating function
.
Then, the
function
is
the joint cumulant generating function of
.
The definition is basically the same given for random variables.
The partial derivatives of the joint cgf at zero are called cross-cumulants (or joint cumulants).
Cross-cumulants are denoted as
follows:
First-order cross-cumulants are equal to the expected values of the entries of
:
The first partial derivative of the joint
cgf with respect to
is
Since
we
have
Second-order joint cumulants are equal to the
covariances between the
entries of
:
The first partial derivative of the joint
cgf with respect to
is
By
taking the second derivative with respect to
,
we
obtain
Since
we
have
Please cite as:
Taboga, Marco (2021). "Cumulant generating function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/cumulant-generating-function.
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