Renewal, Modulation, and Superstatistics in Times Series

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Article discussing two different approaches, referred to as renewal and modulation, to generate time series with a nonexponential distribution of waiting times.

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13 p.

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Allegrini, Paolo; Barbi, Francesco; Grigolini, Paolo & Paradisi, Paolo April 27, 2006.

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Article discussing two different approaches, referred to as renewal and modulation, to generate time series with a nonexponential distribution of waiting times.

Physical Description

13 p.

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Copyright 2006 American Physical Society. The following article appeared in Physical Review E, 73:4; http://pre.aps.org/abstract/PRE/v73/i4/e046136

Abstract: We consider two different approaches, to which we refer to as renewal and modulation, to generate time series with a nonexponential distribution of waiting times. We show that different time series with the same waiting time distribution are not necessarily statistically equivalent, and might generate different physical properties. Renewal generates aging and anomalous scaling, while modulation yields no significant aging and either ordinary or anomalous diffusion, according to the dynamic prescription adopted. We show, in fact, that the physical realization of modulation generates two classes of events. The events of the first class are determined by the persistent use of the same exponential time scale for an extended lapse of time, and consequently are numerous; the events of the second class are identified with the abrupt changes from one to another exponential prescription, and consequently are rare. The events of the second class, although rare, determine the scaling of the diffusion process, and for this reason we term them as crucial events. According to the prescription adopted to produce modulation, the distribution density of the time distances between two consecutive crucial events might have, or not, a diverging second moment. In the former case the resulting diffusion process, although going through a transition regime very extended in time, will eventually become anomalous. In conclusion, modulation rather than ruling out the action of renewal events, produces crucial events hidden by clouds of exponential events, thereby setting the challenge for their identification.

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  • Physical Review E, 2006, College Park: American Physical Society

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  • Publication Title: Physical Review E
  • Volume: 73
  • Issue: 4
  • Pages: 13
  • Peer Reviewed: Yes

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UNT Scholarly Works

The Scholarly Works Collection is home to materials from the University of North Texas community's research, creative, and scholarly activities and serves as UNT's Open Access Repository. It brings together articles, papers, artwork, music, research data, reports, presentations, and other scholarly and creative products representing the expertise in our university community. Access to some items in this collection may be restricted.

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  • April 27, 2006

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  • Sept. 9, 2011, 2:01 p.m.

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  • May 23, 2014, 10:43 a.m.

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Allegrini, Paolo; Barbi, Francesco; Grigolini, Paolo & Paradisi, Paolo. Renewal, Modulation, and Superstatistics in Times Series, article, April 27, 2006; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc40400/: accessed May 29, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.