Moderate parts in regenerative compositions: the case of regular variation

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 $-alpha$, $alphain(0,,1)$, 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 $rinmathbb{N}$. 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.