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.

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.