Cumulant

The cumulants of a random variable are defined by expanding the logarithm of its characteristic function. Let be the characteristic function, defined as the Fourier transform of the probability density function using Fourier transform parameters ,

			(1)
			(2)

The cumulants are then defined by

(3)

(Abramowitz and Stegun 1972, p. 928). Taking the Maclaurin series gives

(4)

where are raw moments, so

			(5)
			(6)
			(7)
			(8)
			(9)

These transformations can be given by CumulantToRaw[n] in the Mathematica application package mathStatica.

In terms of the central moments ,

			(10)
			(11)
			(12)
			(13)
			(14)

where is the mean and is the variance. These transformations can be given by CumulantToCentral[n].

When only finite absolute moments through positive integer order are assumed, the cumulants through that order are defined by the corresponding derivatives of at , without requiring convergence of the full series. For real independent random variables and with these finite absolute moments, the cumulants satisfy (Kenney and Keeping 1951, pp. 78-79). Their characteristic functions multiply, so the logarithms chosen continuously from the value 0 at add near the origin. Thus cumulants are additive under convolution.

In contrast, for independent random variables with finite fourth moments, the fourth central moment satisfies . The mixed term shows why the fourth central moment is not generally additive.

Multivariate cumulants can be expressed in terms of raw moments, e.g.,

			(15)
			(16)

and central moments, e.g.,

			(17)
			(18)
			(19)
			(20)
			(21)

using CumulantToRaw[m, n, ...] and CumulantToCentral[m, n, ...], respectively.

The k-statistics are unbiased estimators of the cumulants.