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 $\hat{f}(t)$:

$$\hat{f}(t)=E\{\exp(itx)\}.$$ (42)

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

$$\log \hat{f}(t)=\kappa_1(it)+\kappa_2(it)^2/2+...+\kappa(it)^r/r!+...$$ (43)

where the $\kappa_r$ are some constants. These constants are called the cumulants (of the distribution) of x. In particular, the first three cumulants (for zero-mean variables) have simple expressions:

$$\kappa_1=E\{x\}=0$$ (44)

$$\kappa_2=E\{x^2\}$$ (45)

$$\kappa_3=E\{x^3\}$$ (46)

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

$$\:\mbox{kurt}\:(x)=E\{x^4\}-3(E\{x^2\})^2$$ (47)

Kurtosis can be considered a measure of the non-Gaussianity of x. For a Gaussian random variable, kurtosis is zero; it is typically positive for distributions with heavy tails and a peak at zero, and negative for flatter densities with lighter tails. Distributions of positive (resp. negative) kurtosis are thus called super-Gaussian (resp. sub-Gaussian).

Cumulants should be compared with (centered) moments. The r-th moment of x is defined as $E\{x^r\}$ [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 x₁ and x₂ are two independent random variables, it holds $\:\mbox{kurt}\:(x_1+x_2) = \:\mbox{kurt}\:(x_1) + \:\mbox{kurt}\:(x_2)$and $\:\mbox{kurt}\:(\alpha x_1) = \alpha^4 \:\mbox{kurt}\:(x_1)$, where $\alpha$ is a scalar.

Cumulants of several random variables x₁,...,x_m are defined similarly. The cross-cumulant
$\mbox{cum}\:(x_{i_1},x_{i_2},..,x_{i_k})$ 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 $\hat{f}(t)=E\{\exp(i\sum t_j x_j)\}$ of the vector ${\bf x}=(x_1,...,x_m)^T$ [88].