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 Y1 and Y2 are non-degenerate, independent random variables, then the random variables
are independently distributed if and only if both Y1 and Y2 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.