I was reading Feller, and found that sums of random variables have some weird properties. Univariate distributions form a semigroup under sum in the following sense. Let *F* and *G* be distributions, and *X* and *Y* be independent random variables with those distributions. Then *X + Y* gives you a new univariate distribution (given by the convolution).

I’d sort-of assumed that it would be a cancellative semigroup: if *X + Y ~ X’ + Y*, then *X ~ X’* (for independent *X, X’, Y*), but Feller has a counterexample. He also has a related example. Let *X ~ X’* and *Y ~ Y’*. Suppose *X + X’ ~ Y + Y’*. Then *X ~ Y*, right? Wrong.

In both cases, the critical property you need is for the characteristic function to have a zero. If you restrict to distributions whose characteristic functions are never zero, then you get a cancellative semigroup. Many nice distributions have this property, including all infinitely divisible distributions.