可交换Gibbs分拆与Stirling三角
Exchangeable Gibbs partitions and Stirling triangles
摘要：
对于两个非负且适当归一化的权重$\W=(\W_j)$和$\V=(\V_{n,k})$的集合，通过向通用分区$\{A_j, j\leq k\}$分配概率$\V_{n,k} \W_{|A_1|}... \W_{|A_k|}$来定义集合$\{1,...,n\}$的分区集合上的概率分布，其中$|A_j|$是$A_j$的元素数量。我们通过假设$[n]$的随机分区$Π_n$随着$n$的变化而一致来对权重施加约束，这意味着它们定义了所有自然数的集合的可交换分区。这意味着权重$\W$必须具有取决于单个参数$α\in [-\infty,1]$的非常特殊的形式。$α=1$的情况是平凡的，并且对于$α\neq 1$的每个值，可能的$\V$ -权重的集合是无限维的单纯形。我们确定的极值点的单纯形解决边界问题的广义斯特林三角形。特别是，我们表明，边界是离散的$-\infty\leqα<0$和连续的$0\leqα<1$。对于$α\leq 0$，极值对应于由$(α,θ)$索引的随机分区的Ewens-Pitman家族的成员，而对于$0<α<1$，极值是通过将$(α,θ)$ -分区以$Π_n$的区组数的渐近性为条件而获得的，因为$n$趋于无穷大。
英文摘要：
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 $\{A_j, j\leq k\}$ 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 $Π_n$ 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 $α\in [-\infty,1]$. The case $α=1$ is trivial, and for each value of $α\neq 1$ 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 $-\infty\leqα<0$ and continuous for $0\leqα<1$. For $α\leq 0$ the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by $(α,θ)$, while for $0<α<1$ the extremes are obtained by conditioning an $(α,θ)$-partition on the asymptotics of the number of blocks of $Π_n$ as $n$ tends to infinity.
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