Cumulanten

Cumulants, known in Dutch as cumulanten, are a sequence of statistics associated with a random variable that summarize its distribution in a way that emphasizes shape and dependence properties. They are defined via a cumulant generating function and have properties that make them convenient for theoretical work, especially when combining independent random variables.

Let X be a real-valued random variable with moment generating function M_X(t) = E[e^{tX}] for t near 0. The cumulant generating function is K_X(t) = log M_X(t). The nth cumulant κ_n is the coefficient of t^n/n! in the Taylor expansion of K_X(t) around t = 0. Equivalently, cumulants are obtained from the moments by this logarithmic relationship. The first few cumulants have direct interpretations: κ_1 equals the mean E[X], κ_2 equals the variance Var(X). The third cumulant is the third central moment E[(X − E[X])^3], and the fourth cumulant equals E[(X − E[X])^4] − 3 Var(X)^2. In particular, for a normal distribution all cumulants κ_n with n ≥ 3 vanish.

A key property is additivity for independent variables: if X and Y are independent, then K_{X+Y}(t) = K_X(t) + K_Y(t), which implies κ_n(X+Y) = κ_n(X) + κ_n(Y) for all n. This makes cumulants especially useful in limit theorems and in expansions that approximate distributions, such as the Edgeworth expansion. Cumulants can also be defined for multiple variables, yielding joint cumulants that capture dependencies beyond covariance.

Cumulants provide a compact, interpretable alternative to moments for describing distributional features and for conducting probabilistic and statistical analyses. They are widely used in probability theory, statistics, and physics.