Definition of Cumulants

In this appendix, we present the definitions of cumulants.
Consider a scalar random variable of zero mean, say *x*, whose characteristic
function is denoted by
:

(42) |

Expanding the logarithm of the characteristic function as a Taylor series, one obtains

where the are some constants. These constants are called the cumulants (of the distribution) of

(44) | |||

(45) | |||

(46) |

Of particular interest for us is the fourth-order cumulant, called kurtosis, which can be expressed as [88,109]

Kurtosis can be considered a measure of the non-Gaussianity of

Cumulants should be compared with (centered) moments. The *r*-th
moment of *x* is defined as
[88,109]. (For
simplicity, *x* was here
assumed to have zero mean, in which case the centered moments, or
moments about the mean, equal the non-centered moments, or moments
about zero). The moments may also be obtained from a Taylor expansion
that is otherwise identical to the one in (43), but no
logarithm is taken on the left side. Note that the first 3 moments
equal the first 3
cumulants. For *r*>3, however, this is no longer the case.

The mathematical simplicity of
the cumulant-based approach in ICA is due to certain linearity properties of
the cumulants [88]. For kurtosis, these can be formulated as
follows. If *x*_{1} and *x*_{2} are two independent
random variables, it holds
and
,
where
is a scalar.

Cumulants of several random variables
*x*_{1},...,*x*_{m} are defined similarly. The
cross-cumulant

for any set of indices
*i*_{j} is defined by the coefficient of
the term
*t*_{i1} *t*_{i2} ... *t*_{ik} in the Taylor expansion of the
logarithm of the characteristic function
of the vector
[88].