Yesterday, while trying to build a counter-example for a research problem, I started playing with the idea of designing a random variable, not by picking a vanilla distribution, but by selecting its moments myself. That is, if I pick the moments myself, can I find a distribution that has them? More formally, if I provide , is there a : probability measure such that
Turns out this is quite the rabbit hole; not only is this an old problem, but it has already been attacked by mathematicians like Hausdorff and Stieltjes. It is called the moment problem, and, in full generality, deals with the construction of Borel measures that have specific moments.
Such a sequence is called completely monotonic, and this idea can be generalized to functions.