2.5 The Cumulant Generating Function
Definition 2.5.1. The cumulant generating function of \(X\) is \[K_X(t) = \log M_X(t),\] and the cumulants \(\kappa _1,\kappa _2,\dots \) are its coefficients: \[K_X(t) = \sum _{k=1}^{\infty }\kappa _k\,\frac {t^{k}}{k!} .\]
Proof. Differentiating \(K_X = \log M_X\), \[K_X'(t) = \frac {M_X'(t)}{M_X(t)},\qquad K_X''(t) = \frac {M_X''(t)M_X(t) - \left [M_X'(t)\right ]^{2}}{\left [M_X(t)\right ]^{2}} .\] Since \(M_X(0)=1\), \(M_X'(0)=E(X)\) and \(M_X''(0)=E\left (X^{2}\right )\), \[K_X'(0) = E(X),\qquad K_X''(0) = E\left (X^{2}\right ) - \left [E(X)\right ]^{2} = \Var (X).\] □
Remark. The point of taking a logarithm is what it does to sums. For independent \(X\) and \(Y\) the moment generating functions multiply, so the cumulant generating functions add: \[K_{X+Y}(t) = \log \left [M_X(t)M_Y(t)\right ] = K_X(t) + K_Y(t).\] Every cumulant is therefore additive over independent summands. For \(\kappa _1\) and \(\kappa _2\) this is the familiar statement that means and variances add; the content of the definition is that the same holds at every order, which is not true of the ordinary moments — \(E\left [(X+Y)^{3}\right ]\) is not \(E\left (X^{3}\right )+E\left (Y^{3}\right )\).
The third and fourth cumulants give the standard shape measures, \[\text {skewness} = \frac {\kappa _3}{\kappa _2^{3/2}},\qquad \text {excess kurtosis} = \frac {\kappa _4}{\kappa _2^{2}} .\] For the normal distribution \(K_X(t)=\mu t + \frac 12\sigma ^{2}t^{2}\) exactly, so every cumulant beyond the second is zero — which is the cleanest statement of what makes the normal distribution special, and why both shape measures are calibrated to vanish there.
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