A Characterization of GEM Distributions

Abstract

Let (Xn) be a residual allocation model with i.i.d. residual fractions Un: For W a random variable with values in [0; 1] and independent of (Xn), we define another sequence (Yn) by setting(formula here)Under minor regularity assumptions we show that (Xn) and (Yn) have the same probability law if and only if this law is a GEM distribution. In this case, the distribution of W and the Uns is Beta(1; θ) for some θ > 0.