Exchangeable Gibbs partitions and Stirling triangles.
(2004)Abstract
For two collections of nonnegative and suitably normalised weights
$\W=(\W_j)$ and $\V=(\V_n,k)$, a probability distribution on the set of
partitions of the set $1,...,n$ is defined by assigning to a generic
partition $A_j, jk$ the probability $\V_n,k \W_|A_1|...
\W_|A_k|$, where $|A_j|$ is the number of elements of $A_j$. We impose
constraints on the weights by assuming that the resulting random partitions
$\Pi_n$ of $n$ are consistent as $n$ varies, meaning that they define an
exchangeable partition of the set of all natural numbers. This implies that the
weights $\W$ must be of a very special form depending on a single parameter
$\alpha-ınfty,1$. The case $\alpha=1$ is trivial, and for each value of
$\alpha1$ the set of possible $\V$-weights is an infinite-dimensional
simplex. We identify the extreme points of the simplex by solving the boundary
problem for a generalised Stirling triangle. In particular, we show that the
boundary is discrete for $-ınftyłeq\alpha<0$ and continuous for
$0łeq\alpha<1$. For $\alpha0$ the extremes correspond to the members of
the Ewens-Pitman family of random partitions indexed by $(\alpha,þeta)$,
while for $0<\alpha<1$ the extremes are obtained by conditioning an
$(\alpha,þeta)$-partition on the asymptotics of the number of blocks of
$\Pi_n$ as $n$ tends to infinity.