Using Renewal Processes to Generate Long-Range Dependence and High Variability

Taqqu, Murad S.; Levy, Joshua B.

doi:10.1007/978-1-4615-8162-8_3

Cited by 119 publications

(60 citation statements)

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“…In this class, each model deals with a specific underlying generic signal X -e.g., on-off processes (34), renewal processes (36,37), persistent random walks (38), and Ornstein-Uhlenbeck processes (39). The model established in this research is fundamentally different of the aforementioned superposition model: considering the randomization (via random transmission amplitudes and frequencies) of the superimposed signals rather than their stochastic scaling limits; considering arbitrary underlying generic signal processes rather than a specific one; and seeking amplitudinaluniversality and temporal-universality rather than setting as goal to obtain a fractional Brownian noise scaling limit.…”

Section: Section 4: Discussionmentioning

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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Section: Section 4: Discussionmentioning

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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“…In this paper we analyze a mechanism for creating a long-memory process, based on converting a static power law distribution into a random process with a power law autocorrelation function. Other examples of stochastic processes relating power laws to long-memory have been given by Mandelbrot [21] (analyzed by Taqqu and Levy [27]), and in the context of DNA sequences by Buldyrev et al [5].…”

Section: Introductionmentioning

confidence: 99%

Theory for long memory in supply and demand

2005

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Recent empirical studies have demonstrated long-memory in the signs of orders to buy or sell in financial markets [2,19]. We show how this can be caused by delays in market clearing. Under the common practice of order splitting, large orders are broken up into pieces and executed incrementally. If the size of such large orders is power law distributed, this gives rise to power law decaying autocorrelations in the signs of executed orders. More specifically, we show that if the cumulative distribution of large orders of volume v is proportional to v −α and the size of executed orders is constant, the autocorrelation of order signs as a function of the lag τ is asymptotically proportional to τ −(α−1) . This is a long-memory process when α < 2. With a few caveats, this gives a good match to the data. A version of the model also shows long-memory fluctuations in order execution rates, which may be relevant for explaining the long-memory of price diffusion rates. Contents

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“…They range from traditional queueing models to sophisticated on/o models 23,22,24], Markov modulated queues 25,26], shot noise models 31] and fractional Brownian motion 32,49,50].…”

Section: Background and Terminologymentioning

confidence: 99%

Untitled

1999

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This article reviews various models within the queueing framework which have been suggested for teletra c data. Such models aim to capture certain stylised features of the data, such as variability of arrival rates, heavy-tailedness of on-and o -periods and long-range dependence in teletra c transmission. Subexponential distributions constitute a large class of heavy-tailed distributions, and we investigate their (sometimes disastrous) in uence within teletra c models. We demonstrate some of the above e ects in an explorative data analysis of Munich Universities' intranet data.

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Using Renewal Processes to Generate Long-Range Dependence and High Variability

Cited by 119 publications

References 1 publication

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

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

Theory for long memory in supply and demand

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