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 and P = Y 1 Y 1 + Y 2 {\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
P i = Y i ∑ i = 1 k Y i {\displaystyle P_{i}={\frac {Y_{i}}{\sum _{i=1}^{k}Y_{i}}}}
is independent of
W = ∑ i = 1 k Y i {\displaystyle 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.