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Register a new userAsymptotic laws for regenerative compositions: gamma subordinators and the like
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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 clusters of $n$ uniform random points as they are separated by the points of $widetilde{cal R}$. We focus on the number of parts $K_n$ of the composition when $widetilde{cal R}$ is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for $K_n$ and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Levy measure is regularly varying at $0+$.
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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.
We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the $limsup$ and a law of the triple logarithm for the $liminf$. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424--445]. We apply our results to several queuing systems and a birth and death process.
We give a representation of the Gamma subordinator as a Krein functional of Brownian motion, using the known representations for stable subordinators and Esscher transforms. In particular, we have obtained Krein representations of the subordinators which govern the two parameter Poisson-Dirichlet family of distributions.
Asymptotic laws of records values have usually been investigated as limits in type. In this paper, we use functional representations of the tail of cumulative distribution functions in the extreme value domain of attraction to directly establish asymptotic laws of records value, not necessarily as limits in type. Results beyond the extreme value value domain are provided. Explicit asymptotic laws concerning very usual laws are listed as well. Some of these laws are expected to be used in fitting distribution
We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the one-sided stable $(Y/2)$ subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable $(1/2)$ subordinator$.$ The required time changes may be generated by simulating the requisite one-sided stable subordinator and throwing away some of the jumps as described in Rosinski (2001).
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