<p>From Wikipedia:</p>
<p>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.</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>There is also the concept of join cumulants described <a href="https://en.wikipedia.org/wiki/Cumulant#Joint_cumulants" rel="nofollow noreferrer">on the same page</a>.</p>
<p>Could anyone point me to a literature which discusses the computational complexity of higher order of cumulants?</p>

CC BY-SA 4.0
2022-11-12T23:15:07.807Z
Maybe not quite what you mean, but this paper extensively relies on estimating cumulants (of high order): https://arxiv.org/abs/1911.00911
Clement C.
CC BY-SA 4.0
2023-02-07T15:25:56.587Z
What kind of probability distributions do you consider? How do you represent the probability distribution as an input for the algorithm?
Emil Jeřábek
<st.statistics><statistical-physics>
Computational complexity of higher order cumulants
Omar Shehab
# Computational complexity of higher order cumulants

(Asked by: username_0 on 2022-11-12)

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](https://en.wikipedia.org/wiki/Cumulant#Joint_cumulants).

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

**Tags:** st.statistics, statistical-physics
> **Comments:**
> **username_1**: Maybe not quite what you mean, but this paper extensively relies on estimating cumulants (of high order): https://arxiv.org/abs/1911.00911  
> **username_2**: What kind of probability distributions do you consider? How do you represent the probability distribution as an input for the algorithm?  

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## 0 answers

