Moderate parts in regenerative compositions: the case of regular variation

TL;DR

This paper studies the asymptotic distribution of block sizes in regenerative compositions derived from subordinators with regularly varying Lévy measures, revealing explicit thresholds and mixed Poisson limit laws.

Contribution

It provides explicit thresholds and limit distributions for block sizes in regenerative compositions with Lévy measures that are regularly varying at zero.

Findings

Number of blocks of size r(n) converges to a mixed Poisson distribution.

The threshold r(n) is regularly varying with index α/(α+1).

The mixing distribution is related to the exponential functional of the subordinator.

Abstract

A regenerative random composition of integer $n$ is constructed by allocating $n$ standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator $S$. Assuming that the L\'{e}vy measure of $S$ is infinite and regularly varying at zero of index $-\alpha$, $\alpha\in(0,\,1)$, we find an explicit threshold $r=r(n)$, such that the number $K_{n,\,r(n)}$ of blocks of size $r(n)$ converges in distribution without any normalization to a mixed Poisson distribution. The sequence $(r(n))$ turns out to be regularly varying with index $\alpha/(\alpha+1)$ and the mixing distribution is that of the exponential functional of $S$. The result is derived as a consequence of a general Poisson limit theorem for an infinite occupancy scheme with power-like decay of the frequencies. We also discuss asymptotic behavior of $K_{n,\,w(n)}$ in cases when $w(n)$ diverges but grows slower than $r(n)$. Our findings complement previously known strong laws of large numbers for $K_{n,\,r}$ in case of a fixed $r\in\mathbb{N}$. As a key tool we employ new Abelian theorems for Laplace--Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.