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.

For instance, the Hausdorff moment problem is exactly like~(1), but the measure needs to be supported in . That is,

Hausdorff resolved this in 1921. If you want your sequence to be the moments of a (Borel) measure, then for every , has to satisfy

is the difference operator, think discrete version of the derivative of a function,

Such a sequence is called **completely monotonic**, and this idea can be generalized to functions.

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