# When can a set of numbers be the moments of a random variable?

Suppose that I have a set of known measurable scalar-valued functions $f_{1},\ldots,f_{K}$.

Associated with these functions, I also have a set of known real numbers $a_{1},\ldots,a_{K}$.

Under what conditions on $f_{1},\ldots,f_{K}$ and $a_{1},\ldots,a_{K}$ does there exist a random variable $X$ such that $E[f_{k}(X)] = a_{k}$ for all $k = 1,\ldots,K$?

Can you provide a reference to the literature where such a result is established?

If the general case is too difficult, suppose that $f_{k}(x) = x^{k}$ for all $k$.

As an example, suppose that $K = 2$, $f_{1}(x) = x$, and $f_{2}(x) = x^{2}$.

Then if $a_{1} = 0$ and $a_{2} = 1$, I know that such an $X$ exists---for example take $X \sim N(0,1)$.

On the other hand, if $a_{1} = 2$ and $a_{2} = 1$, then no such $X$ could exist, for if it did we would have $Var(X) = 1 - 2^{2} = -3 < 0$.