cumulant

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The cumulant of two random variables X\, and Y\, is defined to be the connected part of the expected value of their product:

\kappa(X,Y) = E(XY) - E(X)E(Y)\,,

sometimes written as

\langle X Y \rangle_c = \langle X Y \rangle - \langle X\rangle\langle Y \rangle\,

More generally, one subtracts off all the disconnected parts which are enumerated by all the proper partitions of \{1,2,\dots,n\}\, (which excludes the partition \{1,2,\dots,n\}\, itself):

\kappa(X_1, X_2, \dots, X_n) = E(X_1 X_2 \dots X_n) \,- \sum_\begin{matrix}\{S_k\}\in \mathrm{proper}\\\mathrm{partitions\,\,of}\\\{1, 2,\dots, n\} \end{matrix} E\left( \prod_{i \in S_1} A_i \right)\dots E\left( \prod_{i \in S_k} A_i \right)\,
on to linked cluster theorem
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