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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 $S$. Assumingthat the L\'{e}vy measure of $S$ is infinite and regularly varying at zero ofindex $-\alpha$, $\alpha\in(0,\,1)$, we find an explicit threshold $r=r(n)$,such that the number $K_{n,\,r(n)}$ of blocks of size $r(n)$ converges indistribution without any normalization to a mixed Poisson distribution. Thesequence $(r(n))$ turns out to be regularly varying with index$\alpha/(\alpha+1)$ and the mixing distribution is that of the exponentialfunctional of $S$. The result is derived as a consequence of a general Poissonlimit theorem for an infinite occupancy scheme with power-like decay of thefrequencies. We also discuss asymptotic behavior of $K_{n,\,w(n)}$ in caseswhen $w(n)$ diverges but grows slower than $r(n)$. Our findings complementpreviously known strong laws of large numbers for $K_{n,\,r}$ in case of afixed $r\in\mathbb{N}$. As a key tool we employ new Abelian theorems forLaplace--Stiletjes transforms of regularly varying functions with the indexesof regular variation diverging to infinity.

Moderate parts in regenerative compositions: the case of regular variation