cumulants

Moments answer 'what is the average of X^k', but they describe a single distribution and become tangled when you add variables. Cumulants are an alternative set of descriptive numbers — kappa_1, kappa_2, kappa_3, ... — engineered to have one beautiful property that moments lack: under addition of independent variables, cumulants simply add. They are the shape-summaries built for combining.

Concretely, the cumulants are the coefficients in the power series of the cumulant generating function: K_X(t) = log E[e^(tX)] = kappa_1 t + kappa_2 t^2/2! + kappa_3 t^3/3! + ... The first few are reassuringly familiar quantities wearing new names: kappa_1 = E[X] is the mean, kappa_2 = Var(X) is the variance, kappa_3 = E[(X - mu)^3] is the third central moment (which, divided by sigma^3, gives skewness), and kappa_4 relates to kurtosis. The headline property is additivity: if X and Y are independent, then kappa_n(X + Y) = kappa_n(X) + kappa_n(Y) for every n. So the mean of a sum is the sum of means, the variance of a sum is the sum of variances, and the same clean adding continues all the way up the ladder.

Cumulants matter most because they make the normal distribution's special role transparent. The normal is exactly the distribution whose cumulants beyond the second are all zero, so kappa_3, kappa_4, kappa_5, ... measure precisely how far a distribution departs from normality. When you average many independent variables and rescale, the higher cumulants of the average shrink toward zero while the first two survive — and a distribution with only the first two cumulants is the normal. That is the cumulant's-eye view of why the central limit theorem is true. The usual caveat holds: cumulants past a certain order may not exist if the corresponding moments diverge.