Suppose that we have a real-valued discrete random variable, whose probability distribution has finite support on some set $S$ of real numbers. Then if $N = |S|$, we know that we can construct the entire distribution from the first $N$ raw moments, as described in this paper:
The transformation is a simple Vandermonde matrix that converts from moments to probabilities.
Suppose that we instead want to use the L-moments. Is there an analogous result where we can completely reconstruct the distribution using only the first $N$ L-moments, and if so, what does the resulting matrix look like?
To be specific, I am looking for the basis specified by the matrix that solves this problem in the discrete, finite case. For the classical (raw) moments, the basis is the monomials up to order $N$.
I know that Bernstein polynomials are often mentioned in connection with the L-moments, although I’m not sure if this helps here.
I also understand that we can use the L-moments to reconstruct the quantile function, but I’m not sure how many L-moments are needed to reconstruct the entire thing, nor how this translates into a basis for the discrete finite probability distribution.