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Abstract
A regenerative composition structure is a sequence of ordered partitions derived from the range of a subordinator by a natural sampling procedure. In this paper, we extend previous studies on the asymptotics of the number of blocks $K_n$ in the composition of integer $n$, in the case when the Levy measure of the subordinator has a property of slow variation at $0$. Using tools from the renewal theory the limit laws for $K_n$ are obtained in terms of integrals involving the Brownian motion or stable processes. In other words, the limit laws are either normal or other stable distributions, depending on the behavior of the tail of Levy measure at $\infty$. Similar results are also derived for the number of singleton blocks.
Highlights
Let S := (S(t))t≥0 be a subordinator with S(0) = 0, zero drift, no killing and a nonzero Levy measure ν on R+
The closed range R of the process S is a regenerative subset of R+ of zero Lebesgue measure
The range R splits the positive halfline in infinitely many disjoint component intervals that form an open set (0, ∞) \ R
Summary
Kn and Kn, may be normalized by the same constant (no centering required) to entail convergence to multiples of the same random variable, which may be represented as the exponential functional of a subordinator [13] This case is relatively easy, because the number of occupied gaps within the partial range in [S(t1), S(t2)] is of the same order of growth, as n → ∞, for every time interval 0 ≤ t1 < t2 ≤ ∞. Normal limits for Kn for wider families of slowly varying functions were obtained in [1] under the assumption that the subordinator has finite variance and the Laplace exponent of S satisfies certain smoothness and growth conditions It turned that the case of slow variation required a further division, with qualitatively different scaling functions in each subcase [1]. Our approach to the limit laws of Kn, could be extended to Kn,r for all r ≥ 1, but to avoid technical complications we do not pursue this extension here, as our main focus is the development of the new method
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