Regenerative Compositions in the Case of Slow Variation

For $S$ a subordinator and $\Pi_n$ an independent Poisson process of intensity $ne^{-x}, x>0,$ we are interested in the number $K_n$ of gaps in the range of $S$ that are hit by at least one point of $\Pi_n$. Extending previous studies in \cite{Bernoulli, GPYI, GPYII} we focus on the case when the tail of the L{\'e}vy measure of $S$ is slowly varying. We view $K_n$ as the terminal value of a random process ${\cal K}_n$, and provide an asymptotic analysis of the fluctuations of ${\cal K}_n$, as $n\to\infty$, for a wide spectrum of situations.