A Characterization of GEM Distributions

. Let (X n ) be a residual allocation model with i.i.d. residual fractions U n . For a random variable W with values in [0; 1] independent of (X n ) we define another sequence (Y n ) by setting (Y 1 ; Y 2 ; Y 3 ; Y 4 ; : : :) = ( (W; X 1 \Gamma W;X 2 ; X 3 ; : : :) if W ! X 1 ; (X 1 +X 2 ; X 3 ; X 4 ; X 5 ; : : :) if W X 1 : Under minor regularity assumptions we show that (X n ) and (Y n ) have the same probability law if and only if this law is a GEM distribution. In this case, the distribution of W and the U n 's is beta (1; `) for some ` ? 0. Keywords: GEM distribution, residual allocation model, Poisson-Dirichlet distribution, size-biased permutation, merge-and-split transformation. AMS 1991 Subject Classification: Primary 62E10. 1. The GEM distribution named after Griffiths, Engen and McCloskey is a probability law for the sequence (X n ) arising as a residual allocation model (RAM) X 1 = U 1 ; X n = (1 \Gamma U 1 ) (1 \Gamma U 2 ) : : : (1 \Gamma U n\Gamma1 ) U n ; n = 2; ..