In the case of many random variables we have a similar generating function giving the various moments. 
,
Here we have
The cumulant expansion in this case is then given by equating coefficients of the 
in
where 
indicates 
, and 
in 
consists of 
indices, etc.
The first four cumulants are [27]
In general, all distinct terms where each variable appears exactly once are included, and the coefficient of any term is 
where n is the number of averages in the term.