Moderate parts in regenerative compositions: the case of regular variation
Abstract
A regenerative random composition of integer n is constructed by allocating n standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator S. Assuming that the L\'evy measure of S is infinite and regularly varying at zero of index -α, α∈(0,\,1), we find an explicit threshold r=r(n), such that the number Kn,\,r(n) of blocks of size r(n) converges in distribution without any normalization to a mixed Poisson distribution. The sequence (r(n)) turns out to be regularly varying with index α/(α+1) and the mixing distribution is that of the exponential functional of S. The result is derived as a consequence of a general Poisson limit theorem for an infinite occupancy scheme with power-like decay of the frequencies. We also discuss asymptotic behavior of Kn,\,w(n) in cases when w(n) diverges but grows slower than r(n). Our findings complement previously known strong laws of large numbers for Kn,\,r in case of a fixed r∈N. As a key tool we employ new Abelian theorems for Laplace--Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.