Regenerative compositions in the case of slow variation

For SS a subordinator and ΠnΠn an independent Poisson process of intensity ne−x,x>0, we are interested in the number KnKn of gaps in the range of SS that are hit by at least one point of ΠnΠn. Extending previous studies in [A.V. Gnedin, The Bernoulli sieve, Bernoulli 10 (2004) 79–96; A.V. Gnedin, J. Pitman, M. Yor, Asymptotic laws for compositions derived from transformed subordinators, Ann. Probab. 2006 (in press). http://arxiv.org/abs/math.PR/0403438, 2004; A.V. Gnedin, J. Pitman, M. Yor, Asymptotic laws for regenerative compositions: gamma subordinators and the like, Probab. Theory Related Fields (2006)] we focus on the case when the tail of the Lévy measure of SS is slowly varying. We view KnKn as the terminal value of a random process KnKn, and provide an asymptotic analysis of the fluctuations of KnKn, as n→∞n→∞, for a wide spectrum of situations.