Lorenzian analysis of infinite poissonian populations and the phenomena of Paretian ubiquity

Eliazar, Iddo

doi:10.4108/icst.valuetools.2008.48

Cited by 1 publication

(1 citation statement)

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“…In recent years, we applied Poissonian randomizations in various topics in statistical physics-obtaining results which are unattainable by IID randomizations. Examples include nonlinear shot noise systems (18), fractality in the context of random populations (19)(20)(21), and statistical resilience of random populations to the action of random perturbations (22).…”

Section: Section 2: the Stationary Superposition Modelmentioning

confidence: 99%

A unified and universal explanation for Lévy laws and 1/f noises

Eliazar

Klafter

2009

Proc. Natl. Acad. Sci. U.S.A.

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Lévy laws and 1/f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals. The signals are considered to follow, statistically, a common-yet arbitrarygeneric signal pattern which may be either stationary or dissipative. Each signal is considered to have its own random transmission amplitude and frequency. We characterize the amplitudefrequency randomizations which render the system output's stationary law and power-spectrum universal-i.e., independent of the underlying generic signal pattern. The classes of universal stationary laws and power spectra are shown to coincide, respectively, with the classes of Lévy laws and 1/f noises-thus providing a unified and universal explanation for the ubiquity of these classes of "anomalous statistics" in various fields of science and engineering.anomalous statistics | universality | Poissonian randomizations | shot noise A nomalous statistics are ubiquitously observed in many fields of science and engineering, and their exploration has drawn major interest in recent years by both experimentalists and theoreticians (1-3). In the context of stationary stochastic processes and signals, one can observe either amplitudinal or temporal anomalous statistics.Amplitudinal anomalous statistics are manifested by wide process fluctuations, and are referred to as the Noah effect (4). Quantitatively, amplitudinal anomalous statistics are characterized by "heavy-tailed" stationary laws (5)-stationary laws whose distribution tails follow asymptotic power-law decay. The quintessential proxy of amplitudinal anomalous statistics is the class of Lévy laws (6, 7). This class of probability laws emerges from the Central Limit Theorem as the universal scaling limits of sums of independent and identically distributed (IID) random variables with infinite variance (8, 9).Temporal anomalous statistics are manifested by long process memory and are referred to as the Joseph effect (4). Quantitatively, temporal anomalous statistics are characterized by longrange correlations (10, 11)-autocorrelation functions following asymptotic power-law decay. The quintessential proxy of temporal anomalous statistics is the class of 1/f noises (12-14)-stationary processes with power-law power spectra.Stationary stochastic processes and signals are prevalent across many fields of science and engineering. Examples of such processes and signals include the intensity of solar luminosity, the sales of a consumer product, and transmissions sent through communication channels. In many systems, a very large collection of microscopic stationary stochastic inputs are aggregated up to form a macroscopic stationary system output. In the context of the aforementioned examples, consider, respectively, the luminosity of a galaxy, the sales of a large department store, and transmissions sent through a central communication router.In this research, we consider systems whose outputs are superpositions of many stationary stochastic in...

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