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Register a new userModerate parts in regenerative compositions: the case of regular variation
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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.
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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 $0+$.
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 dependence between two adjacent cycles, and mention two further extensions of the second definition. We mention the connection of regenerative processes to the single-server queue, to multi-server queues and more generally to Harris ergodic Markov chains and processes. In the main theorem, we pay some attention to the conditions under which a limiting distribution exists and provide references that should serve as a starting point for the interested reader.
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 different directions, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.
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 $L_t := sup{h in mathbb{R} : h - alpha(t-s) le X_s text{ for all } s le t} = inf{X_s + alpha(t-s) : s le t}$, $t in mathbb{R}$, and $M_t := sup { h in mathbb{R} : h - alpha|t-s| leq X_s text{ for all } s in mathbb{R} } = inf {X_s + alpha |t-s| : s in mathbb{R}}$, $t in mathbb{R}$. The corresponding contact sets are the random sets $mathcal{H}_alpha := { t in mathbb{R} : X_{t}wedge X_{t-} = L_t}$ and $mathcal{Z}_alpha := { t in mathbb{R} : X_{t}wedge X_{t-} = M_t}$. For a fixed $alpha>mathbb{E}[X_1]$ (resp. $alpha>|mathbb{E}[X_1]|$) the set $mathcal{H}_alpha$ (resp. $mathcal{Z}_alpha$) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections $(mathcal{H}_{alpha})_{alpha > mathbb{E}[X_1]}$ and $(mathcal{Z}_{alpha})_{alpha > |mathbb{E}[X_1]|}$ are increasing in $alpha$ and the regeneration property is compatible with these inclusions in that each family is a continuum of embedded regenerative sets in the sense of Bertoin. We show that $(sup{t < 0 : t in mathcal{H}_alpha})_{alpha > mathbb{E}[X_1]}$ is a c`adl`ag, nondecreasing, pure jump process with independent increments and determine the intensity measure of the associated Poisson process of jumps. We obtain a similar result for $(sup{t < 0 : t in mathcal{Z}_alpha})_{alpha > |beta|}$ when $(X_t)_{t in mathbb{R}}$ is a (two-sided) Brownian motion with drift $beta$.
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