Regenerative compositions in the case of slow variation: A renewal theory approach

Details Regenerative compositions in the case of slow variation: A renewal theory approach Regenerative compositions in the case of slow variation: A renewal theory approach

2011-09-27

arxiv.org/abs/1109.5985v2

Mathematics

Probability

21 pages; in the second version some corrections have been made in Section 4

Scientific paper

Regenerative composition structure is a coherent sequence of ordered partitions derived from the range of subordinator by a version of Kingman's paintbox correspondence. In this paper, we extend previous studies Barbour and Gnedin (2006), Gnedin, Iksanov and Marynych (2010) and Gnedin, Pitman and Yor (2006) on the asymptotics of the number of blocks $K_n$ in the composition of integer $n$, in the case when the L{\'e}vy measure of the subordinator has a property of slow variation at 0. Using tools from the renewal theory we identify the limit law of $K_n$ as either normal or other stable distribution depending on behavior of the L{\'e}vy measure at $\infty$. Limit distributions for the number of singleton blocks are obtained in terms of integrals of the Brownian motion or stable processes, respectively.

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