Weird Sums of Random Variables

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.