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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{e}vy measure of $S$ is infinite and regularly varying at zero of index $-alpha$, $alphain(0,,1)$, we find an explicit threshold $r=r(n)$, such that the number $K_{n,,r(n)}$ 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 $alpha/(alpha+1)$ 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 $K_{n,,w(n)}$ in cases when $w(n)$ diverges but grows slower than $r(n)$. Our findings complement previously known strong laws of large numbers for $K_{n,,r}$ in case of a fixed $rinmathbb{N}$. 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.
For $widetilde{cal R} = 1 - exp(- {cal R})$ a random closed set obtained by exponential transformation of the closed range ${cal R}$ of a subordinator, a regenerative composition of generic positive integer $n$ is defined by recording the sizes of cl
We survey the connections between extreme-value theory and regular variation, in one and higher dimensions, from the algebraic point of view of our recent work on Popa groups.
We review the theory of regenerative processes, which are processes that can be intuitively seen as comprising of i.i.d. cycles. Although we focus on the classical definition, we present a more general definition that allows for some form of dependen
We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in differ
Given a two-sided real-valued Levy process $(X_t)_{t in mathbb{R}}$, define processes $(L_t)_{t in mathbb{R}}$ and $(M_t)_{t in mathbb{R}}$ by $L_t := sup{h in mathbb{R} : h - alpha(t-s) le X_s text{ for all } s le t} = inf{X_s + alpha(t-s) : s le t}