Regenerative compositions in the case of slow variation: A renewal theory approach

A regenerative composition structure is a sequence of ordered partitions derived from the range of a subordinator by a natural sampling procedure. In this paper, we extend previous studies Barbour and Gnedin (2006), Gnedin, Iksanov and Marynych (2010) and Gnedin, Pitman and Yor (2006) on the asymptotics of the number of blocks $K_n$ in the composition of integer $n$, in the case when the L{\'e}vy measure of the subordinator has a property of slow variation at 0. Using tools from the renewal theory the limit laws for $K_n$ are obtained in terms of integrals involving the Brownian motion or stable processes. In other words, the limit laws are either normal or other stable distributions, depending on the behavior of the tail of L{\'e}vy measure at $\infty$. Similar results are also derived for the number of singleton blocks.