From Wikipedia:

In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.

The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the n-th-order cumulant of their sum is equal to the sum of their n-th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.

There is also the concept of join cumulants described on the same page.

Could anyone point me to a literature which discusses the computational complexity of higher order of cumulants?

  • 1
    $\begingroup$ Maybe not quite what you mean, but this paper extensively relies on estimating cumulants (of high order): arxiv.org/abs/1911.00911 $\endgroup$
    – Clement C.
    Nov 12, 2022 at 23:15
  • 3
    $\begingroup$ What kind of probability distributions do you consider? How do you represent the probability distribution as an input for the algorithm? $\endgroup$ Feb 7 at 15:25


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