If the means and variances of a sequence of random variables converge, and all semiinvariants (cumulants) of sufficiently high order tend to zero, then the variables converge in distribution to a normal distribution. Thus no information is needed on the remaining (finitely many) semiinvariants. This is applied to give a new criterion for asymptotic normality of sums of dependent variables. An example is included where this criterion is applied to the number of induced subgraphs of a particular type in a random graph.
"Normal Convergence by Higher Semiinvariants with Applications to Sums of Dependent Random Variables and Random Graphs." Ann. Probab. 16 (1) 305 - 312, January, 1988. https://doi.org/10.1214/aop/1176991903