Lukacs's proportion-sum independence theorem
Introduction In statistics, '''Lukacs's proportion-sum independence theorem''' is a result that is used when studying proportions, in particular the Dirichlet distribution. It is named after Eugene Lukacs.
The theorem If ''Y''1 and ''Y''2 are non-degenerate, independent random variables, then the random variables: are independently distributed if and only if both ''Y''1 and ''Y''2 have gamma distributions with the same scale parameter.===Corollary===Suppose ''Y'' ''i'', ''i'' = 1, ..., ''k'' be non-degenerate, independent, positive random variables. Then each of ''k'' − 1 random variables:is independent of: if and only if all the ''Y'' ''i'' have gamma distributions with the same scale parameter.
References *  page 64.Lukacs's proportion-sum independence theorem and the corollary with a proof.