A [cumulant](https://en.wikipedia.org/wiki/Cumulant) is defined via the *cumulant generating function*
$$ g(t)\stackrel{\tiny def}{=} \sum\_{n=1}^\infty \kappa\_n \frac{t^n}{n},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants have some nice properties, including *additivity*- that for statistically independent variables $X$ and $Y$ we have
$$
g\_{X+Y}(t)=g\_X(t)+g\_Y(t)
$$
Additionally, in a multivariate setting, cumulants go to zero when variables are statistically independent, and so generalize [correlation](https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient) somehow. They are related to moments by [Moebius inversion](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula). They are a standard feature in undergraduate probability courses because they feature in a simple proof of the Central Limit Theorem (see for example [here](http://www.cs.toronto.edu/~yuvalf/CLT.pdf)).


So the cumulants are given by a formula and have a list of good properties. A cumulant is clearly a fundamental concept, but I'm having difficulty figuring out what it is actually measuring, and how it is more than just a computational convenience.



> 
> **Question**: What are cumulants actually measuring? What is their conceptual meaning? Are they measuring connectivity or cohesion of something?
> 


I apologise that this question is surely completely elementary. I'm in low dimensional topology, and I'm having difficulty wrapping my head around this elementary concept in probability; Google did not help much. I'm vaguely imagining that perhaps they are some kind of measure of "cohesion" of the probability distribution in some sense, but I have no idea how.



What is a cumulant really?
Cumulants have many other names depending on the context (statistics, quantum field theory, statistical mechanics,...): seminvariants, truncated correlation functions, connected correlation functions, Ursell functions...
I would say that the $n$-th cumulant $\langle X\_1,\ldots,X\_n\rangle^{T}$
of random variables $X\_1,\ldots,X\_n$ measures the interaction of the variables
which is genuinely of $n$-body type.
By interaction I mean the opposite of independence. Denoting the expectation by $\langle\cdot\rangle$ as in statistical mechanics, independence implies the factorization
$$
\langle X\_1\cdots X\_n\rangle=\langle X\_1\rangle\cdots\langle X\_n\rangle\ .
$$
If the variables are jointly Gaussian and centered then for instance
$$
\langle X\_1 X\_2 X\_3 X\_4\rangle=\langle X\_1 X\_2\rangle\langle X\_3 X\_4\rangle
+\langle X\_1 X\_3\rangle\langle X\_2 X\_4\rangle
+\langle X\_1 X\_4\rangle\langle X\_2 X\_3\rangle
$$
so the lack of factorization is due to $2$-body interactions: namely the absence of factorization
for $\langle X\_i X\_j\rangle$.
The $4$-th cumulant for variables with vanishing moments of odd order would be
the difference $LHS-RHS$ for the previous equation. Thus it would measure
the "interaction" between the four variables which is due to their conspiring all together
instead of being a consequence of conspiring in groups of two at a time.
For higher cumulants, the idea is the same.


Cumulant are definitely related to connectedness. For instance for variables whose joint probability density is a multiple of a Gaussian by a factor $\exp(-V)$ where $V$
is quartic, one can at least formally write moments as a sum of Feynman diagrams.
Cumulants are given by similar sums with the additional requirement that these diagrams or graphs must be connected.


Some references:


* Chapters 1 and 4 of the book ["Path Integrals in Quantum Mechanics"](https://books.google.com/books?id=MWQBAQAAQBAJ&source=gbs_navlinks_s) by Zinn-Justin.
* My article ["Feynman diagrams in algebraic combinatorics"](https://www.emis.de/journals/SLC/wpapers/s49abdess.html). Although it applies to a somewhat different context, it explains the combinatorics of Feynman diagrams in terms of Joyal's theory of species.
* The review ["Feynman diagrams for pedestrians and mathematicians"](https://arxiv.org/abs/math/0406251) by Polyak. However the discussion (e.g. in Section 4.5) of the issue of convergence does not give an accurate idea of this research area.


