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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 13/1999

Alexander GNEDIN and
Sergei KEROV
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A CHARACTERIZATION OF GEM DISTRIBUTIONS

This preprint was accepted September 11, 1999

Contact:
` S. V. Kerov `

ABSTRACT:
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
$$
(Y_1,Y_2,Y_3,Y_4,\ldots) =
\left\{
\aligned
&(W,X_1-W ,X_2,X_3,\ldots)\text{ \ \ \ if \ } W < X_1, \\
&(X_1+X_2,X_3,X_4,X_5,\ldots) \text{ \ if \ } W \ge X_1.
\endaligned\right.
$$
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 $(1,\theta)$ for some $\theta>0$

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