Cumulants

Cumulants are statistical measures related to the moments of a probability distribution but provide additional insights, particularly in the context of the independence and interactions among random variables. Cumulants offer an alternative to moments for characterizing distributions and are particularly valuable in simplifying calculations in probability theory and statistics, especially in the context of the central limit theorem and in the field of statistical physics.

Definition and Relation to Moments

The cumulants of a random variable $X$ can be defined through the cumulant-generating function (CGF), which is the logarithm of the moment-generating function (MGF). If $M_X(t)=E[e^{tX}]$ is the MGF of $X$, then the CGF, $K_X(t)$, is given by: $K_X(t)=\log M_X(t)$

The $n$-th cumulant ${\kappa }_n$ of the random variable $X$ can then be obtained by differentiating $K_X(t)$ $n$ times with respect to $t$ and evaluating at $t=0$: ${\kappa }_n={\frac{d^n}{d{t}^n}{K}_X(t)|}_{t=0}$

First Four Cumulants

1. First Cumulant (${\kappa }_1$): This is equal to the mean of the distribution ($\mu$). ${\kappa }_1=\mu$

2. Second Cumulant (${\kappa }_2$): This represents the variance of the distribution (${\sigma }^2$). ${\kappa }_2={\sigma }^2$

3. Third Cumulant (${\kappa }_3$): This is related to the skewness of the distribution and is proportional to the third central moment. ${\kappa }_3=E[(X-\mu {)}^3]$

4. Fourth Cumulant (${\kappa }_4$): This measures the excess kurtosis (kurtosis minus 3), and provides insight into the tail behavior relative to a normal distribution. ${\kappa }_4=E[(X-\mu {)}^4]-3{\sigma }^4$

Properties of Cumulants

• Additivity: For independent random variables $X$ and $Y$, the $n$-th cumulant of their sum is the sum of their $n$-th cumulants: ${\kappa }_n(X+Y)={\kappa }_n(X)+{\kappa }_n(Y)$ This property is particularly useful for analyzing sums of random variables and plays a critical role in the proof of the central limit theorem.

• Interpretability: While higher moments can become increasingly difficult to interpret, cumulants maintain a more direct interpretability in terms of distribution shape characteristics like asymmetry (skewness) and tail thickness (kurtosis).

• Sensitivity to Tail: Higher cumulants (beyond the fourth) are increasingly sensitive to the tails of the distribution, making them useful in risk analysis and in fields dealing with extreme values.

Cumulants play a significant role in statistical theory and applications, such as in signal processing where they are used to analyze and manage signals characterized by non-Gaussian noise. In theoretical statistics, cumulants are essential in the study of approximations and limit theorems, offering a way to succinctly express complex probabilistic relationships.