Lukacs's proportion-sum independence theorem

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.^[1]

The theorem

If Y₁ and Y₂ are non-degenerate, independent random variables, then the random variables

${\displaystyle W=Y_{1}+Y_{2}{\text{ and }}P={\frac {Y_{1}}{Y_{1}+Y_{2}}}}$

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

${\displaystyle P_{i}={\frac {Y_{i}}{\sum _{i=1}^{k}Y_{i}}}}$

is independent of

${\displaystyle W=\sum _{i=1}^{k}Y_{i}}$

if and only if all the Y _i have gamma distributions with the same scale parameter.^[2]