Lukacs's proportion-sum independence theorem

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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 for Eugene Lukacs.[1]

The theorem[edit]

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.


Suppose Y ii = 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.[2]


  1. ^ Lukacs, Eugene (1955). "A characterization of the gamma distribution". Annals of Mathematical Statistics. 26: 319–324. doi:10.1214/aoms/1177728549. 
  2. ^ Mosimann, James E. (1962). "On the compound multinomial distribution, the multivariate distribution, and correlation among proportions". Biometrika. 49 (1 and 2): 65–82. doi:10.1093/biomet/49.1-2.65. JSTOR 2333468.