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There's few examples of distributions whose cumulants can be computed relatively easily. Normal random variables are of course the easiest: the first cumulant is the mean and the second is the variance. All other cumulants vanish. Another relatively simple class of examples are exponential distributions. The cumulants of the $\mbox{Exp}(\lambda)$ distribution are given by $\kappa_j=\lambda^{-j}(j-1)!$. I've recently become interested in the following question. What are examples of distributions with all cumulants positive? Is the property of positive cumulants equivalent to a seemingly unrelated property of the underlying distribution? I am also interested in examples of distributions whose odd cumulants are negative and even cumulants are positive. The only example I can think of so far is when you take a normal with a negative mean.

Andrew
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  • I had the same question in my mind. The function $A(\eta)$ in the definition of exponential families are called cumulant functions (https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/other-readings/chapter8.pdf). While the density only requires that $exp(-A(\eta))$ be positive and the condition for regularity of the families is that $A(\eta) < \infty$, I was wondering if A can be negative as it is logarithm. – Mewbacca Feb 02 '25 at 19:53

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