Random partitions growth by appending parts: power weights case

We investigate a generalization of Ewens measures on integer partitions where parts of size $k$ have weights $\theta_k\ge 0$. The Ewens measure is a partial case of the constant sequence $\theta_k\equiv\theta>0$. In this paper we consider the case when partial sums $\theta_1+\dots+\theta_k$ have regular growth of index greater that $1$ as $k\to\infty$. We introduce a continuous time random partition growth process such that given it visits some partition of $n$, the random partition of $n$ it visits has the generalized Ewens distribution. In contrast to the often considered growth procedure, in which parts are increased by $1$ or a new part $1$ is added, in the growth process defined in the paper parts are added one by one and remain in the partition forever. The partition growth process is derived explicitly from a sequence of independent Poisson processes. This allows to establish strong laws of large numbers for some characteristics of the process and to determine its asymptotic behavior.