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

TL;DR

This paper investigates the asymptotic behavior of the number of blocks in regenerative compositions derived from subordinators with slowly varying Lévy measures, using renewal theory to establish limit laws involving Brownian motion and stable processes.

Contribution

It extends previous work by deriving new limit laws for block counts in regenerative compositions with slowly varying Lévy measures, employing renewal theory techniques.

Findings

Limit laws for block counts are either normal or stable distributions.

Results depend on the tail behavior of the Lévy measure.

Derived asymptotics for singleton blocks.

Abstract

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.