Regenerative Compositions in the Case of Slow Variation

Abstract

For $S$ a subordinator and ${\Pi }_n$ an independent Poisson process of intensity $n{e}^{-x},x>0,$ 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{é}vy measure of $S$ is slowly varying. We view $K_n$ as the terminal value of a random process ${\mathcal{K}}_n$, and provide an asymptotic analysis of the fluctuations of ${\mathcal{K}}_n$, as $n\to \infty$, for a wide spectrum of situations.

Cited by in corpus (1)

• Self-similar and Markov composition structures