Random partitions growth by appending parts: the Cesàro convergent weights case

{\small Yakubovich Yu. V. Random partitions growth by appending parts: the Cesàro convergent weights case. We investigate a generalized Ewens measure on integer partitions when weights converge in Cesàro mean to a constant $\theta>0$, thus resembling the classical Ewens measure. We introduce a stochastic growth process on partitions which evolves by adding random parts one by one to the current partition. If the constructed process visits some partition of $n$ in its evolution, this random partition is distributed according to the generalized Ewens measure. The growth process is explicitly described in terms of certain independent Poisson processes. It has an interesting property that it produces parts in reversed size-biased order. Using this construction we show that the only non-trivial limit of scaled random partitions in this setting can be the Poisson–Dirichlet distribution with parameter $\theta$.