The limiting behavior of some infinitely divisible exponential dispersion models

Bar‐Lev, Shaul K.; Letac, Gérard

doi:10.1016/j.spl.2010.08.013

Cited by 6 publications

(5 citation statements)

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“…The physics community will also be familiar with distributions like this that exhibit heavy (or power law) tails such as the Pareto and Lévy distributions. Whereas these distributions qualify as EDMs [32], in general, they do not possess finite second moments, and thus they would not express Taylor's law.…”

Section: Statistical Convergence Explains Both Taylor's Law and 1/mentioning

confidence: 99%

Tweedie convergence: A mathematical basis for Taylor's power law,1/fnoise, and multifractality

Kendal

Jørgensen

2011

Phys. Rev. E

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Plants and animals of a given species tend to cluster within their habitats in accordance with a power function between their mean density and the variance. This relationship, Taylor's power law, has been variously explained by ecologists in terms of animal behavior, interspecies interactions, demographic effects, etc., all without consensus. Taylor's law also manifests within a wide range of other biological and physical processes, sometimes being referred to as fluctuation scaling and attributed to effects of the second law of thermodynamics. 1/f noise refers to power spectra that have an approximately inverse dependence on frequency. Like Taylor's law these spectra manifest from a wide range of biological and physical processes, without general agreement as to cause. One contemporary paradigm for 1/f noise has been based on the physics of self-organized criticality. We show here that Taylor's law (when derived from sequential data using the method of expanding bins) implies 1/f noise, and that both phenomena can be explained by a central limit-like effect that establishes the class of Tweedie exponential dispersion models as foci for this convergence. These Tweedie models are probabilistic models characterized by closure under additive and reproductive convolution as well as under scale transformation, and consequently manifest a variance to mean power function. We provide examples of Taylor's law, 1/f noise, and multifractality within the eigenvalue deviations of the Gaussian unitary and orthogonal ensembles, and show that these deviations conform to the Tweedie compound Poisson distribution. The Tweedie convergence theorem provides a unified mathematical explanation for the origin of Taylor's law and 1/f noise applicable to a wide range of biological, physical, and mathematical processes, as well as to multifractality.

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Section: Statistical Convergence Explains Both Taylor's Law and 1/mentioning

confidence: 99%

Tweedie convergence: A mathematical basis for Taylor's power law,1/fnoise, and multifractality

Kendal

Jørgensen

2011

Phys. Rev. E

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“…Example 4 (cf. [1,2]). Let the density of Y t be given by f t (x) = e −x x −1 tI t (x), where I t is the modified Bessel function of order one.…”

Section: Explicit Examplesmentioning

confidence: 99%

“…where F t * denotes the distribution with LST ( ∞ 0 e −su dF (u)) t . The class {F (θ) t , t ≥ 0, θ ≥ 0} is called an exponential dispersion model (see [2]).…”

Section: A Generalizationmentioning

confidence: 99%

“…In Section 3, we present necessary and sufficient criteria for the convergence Y −t t d → Y * for subordinators in terms of their characteristics. We also provide an alternative proof of the result of [2]. Section 4 presents several applications.…”

Section: Introductionmentioning

confidence: 96%

“…In [2], it is proved that for a subclass of exponential dispersion models (cf. [7]) generated by an infinitely divisible probability measure µ on [0, ∞) and associated with an unbounded Lévy measure ν satisfying ν((x, ∞)) ∼ −γ log x as x → 0, the limit F * is a Pareto type law supported on [1, ∞).…”

Section: Introductionmentioning

confidence: 99%

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On the small-time behavior of subordinators

Bar‐Lev¹,

Löpker²,

Stadje³

2012

Bernoulli

Self Cite

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We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show for L\'{e}vy subordinators that the Pareto law on $[1,\infty)$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ363 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Asymptotic behavior of some cone-valued infinitely divisible stochastic process through projection

Rejeb

Masmoudi

2022

Lith Math J

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The limiting behavior of some infinitely divisible exponential dispersion models

Abstract: International audienc

Cited by 6 publications

References 5 publications

Tweedie convergence: A mathematical basis for Taylor's power law,1/fnoise, and multifractality

Tweedie convergence: A mathematical basis for Taylor's power law,1/fnoise, and multifractality

On the small-time behavior of subordinators

Asymptotic behavior of some cone-valued infinitely divisible stochastic process through projection

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