A Characterization of GEM Distributions
نویسنده
چکیده
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 deene another sequence (Y n) by setting 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. 1. The GEM distribution named after Griiths, Engen and McCloskey is a probability law for the sequence (X n) arising as a residual allocation model (RAM) for some 0 < < 1. The GEM distribution is closely connected with the famous Poisson-Dirichlet distribution and the Ewens sampling formula. It plays a key role in various sampling and combinatorial problems (see Chapter 41 of 1] for the history and background). A characterization result by McCloskey says that the GEM distribution is the only RAM which has i.i.d. residual fractions and is invariant under size-biased permutation. That is to say, if (X n) is regarded as a random discrete distribution over f1; 2; : : :g and N 1 ; N 2 ; : : : are diierent integers as they appear for the rst time in the course of sequential sampling from this distribution, then the probability law of (X Nn) coincides with that of
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