cumulants

Cumulants are a set of statistics that describe the shape of a probability distribution. They are defined as the derivatives at zero of the cumulant generating function, which is the logarithm of the moment generating function. For a random variable X with M_X(t) = E[e^{tX}] finite near t = 0, the cumulant generating function is K_X(t) = log M_X(t). The nth cumulant, κ_n, is κ_n = K_X^(n)(0). The first cumulant is the mean μ = E[X], the second is the variance Var(X). The third cumulant equals the third central moment μ_3 = E[(X−μ)^3], and the fourth cumulant equals μ_4 − 3μ_2^2, where μ_j are central moments. In general, cumulants are polynomials in the moments and can be related via Bell polynomials.

Independence and scaling: If X and Y are independent, κ_n(X+Y) = κ_n(X) + κ_n(Y) for all n. For a constant a, κ_n(aX) = a^n κ_n(X).

Common distributions: For a normal distribution N(μ, σ^2), κ_1 = μ, κ_2 = σ^2, and κ_n = 0 for n ≥ 3. For a Poisson(λ) distribution, κ_n = λ for all n ≥ 1.

Uses: Cumulants provide a compact way to characterize distribution shape, quantify deviations from normality, and are foundational in asymptotic expansions such as the Edgeworth expansion, as well as in certain estimation methods.