Regenerative compositions in the case of slow variation

For S a subordinator and Π 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 Π n . 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 S is slowly varying. We view K n as the terminal value of a random process K n , and provide an asymptotic analysis of the fluctuations of K n , as n → ∞ , for a wide spectrum of situations.