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

$W=Y_1+Y_2\text{ and }P=\frac{Y_1}{Y_1+Y_2}$

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

$P_i=\frac{Y_i}{\sum _{i=1}^k{Y}_i}$

is independent of

$W=\sum _{i=1}^k{Y}_i$

if and only if all the Y _i have gamma distributions with the same scale parameter.

References

• Ng, W. N.; Tian, G-L; Tang, M-L (2011). Dirichlet and Related Distributions. John Wiley & Sons, Ltd.. ISBN 978-0-470-68819-9.  page 64. Lukacs's proportion-sum independence theorem and the corollary with a proof.