Cumulants are quantities that are extremely useful for characterizing probability distributions, particularly peaked ones. They have many applications in probability and statistics, physics, and other fields. In particular, they are origin-independent (except the first one) and if you convolute distributions, their corresponding cumulants just add.
The first two cumulants describe the position (mean) and squared width (variance) and in some sense the higher order cumulants describe the deviations in shape of the distribution from a Gaussian (Normal) distribution. The third cumulant relates to the skewness and the fourth relates to the “kurtosis”, measuring the weights in the tails, relative to a Gaussian. C2 and C3 are equal to the central moments (the moments if the origin is taken at the centroid of the distribution) but the higher order cumulants are not equal to their corresponding central moments.
An interesting review of the subject can be found here.
In 1983 I derived some simple recurrence relations for univariate distributions which are useful and I think should better known.
Here is the citation: (Bunker, G. (1983). Application of the ratio method of EXAFS analysis to disordered systems. Nuclear Instruments and Methods in Physics Research, 207(3), 437-444.)
The relations generate expressions for the cumulants as a function of power moments, and also the reverse. Low order expressions are available everywhere – higher order ones, not so much.
They are easily implemented in Mathematica which I have done.
A brief description can be found here (pdf)
Mathematica notebook to generate expressions for cumulants in terms of moments
Mathematica notebook to generate expressions for moments in terms of cumulants
If you want to download actual expressions you can do so here.
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