### Abstract

The probability density function (PDF) of the roughness, i.e., of the temporal variance, of [formula presented] noise signals is studied. Our starting point is the generalization of the model of Gaussian, time periodic, [formula presented] noise, discussed in our recent Letter [Phys. Rev. Lett. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions [Formula Presented] and [formula presented] distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any [formula presented] A simulation of the periodic process makes it possible to study also nonperiodic, thus experimentally more relevant, signals on relatively short intervals embedded in the full period. We find that for [formula presented] the scaled PDFs in both the periodic and the nonperiodic cases are Gaussian, but for [formula presented] they differ from the Gaussian and from each other. Both deviations increase with growing [formula presented] That conclusion, based on numerics, is reinforced by analytic results for [formula presented] and [formula presented] in the latter limit the scaling function of the PDF being finite for periodic signals, but developing a singularity for the aperiodic ones. Finally, an overview is given for the scaling of cumulants of the roughness and the various scaling regions in arbitrary dimensions. We suggest that our theoretical and numerical results open a different perspective on the data analysis of [formula presented] processes.

Original language | English |
---|---|

Number of pages | 1 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 65 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 1 2002 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

**Roughness distributions for [formula presented] signals.** / Antal, T.; Droz, M.; Györgyi, G.; Rácz, Z.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Roughness distributions for [formula presented] signals

AU - Antal, T.

AU - Droz, M.

AU - Györgyi, G.

AU - Rácz, Z.

PY - 2002/1/1

Y1 - 2002/1/1

N2 - The probability density function (PDF) of the roughness, i.e., of the temporal variance, of [formula presented] noise signals is studied. Our starting point is the generalization of the model of Gaussian, time periodic, [formula presented] noise, discussed in our recent Letter [Phys. Rev. Lett. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions [Formula Presented] and [formula presented] distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any [formula presented] A simulation of the periodic process makes it possible to study also nonperiodic, thus experimentally more relevant, signals on relatively short intervals embedded in the full period. We find that for [formula presented] the scaled PDFs in both the periodic and the nonperiodic cases are Gaussian, but for [formula presented] they differ from the Gaussian and from each other. Both deviations increase with growing [formula presented] That conclusion, based on numerics, is reinforced by analytic results for [formula presented] and [formula presented] in the latter limit the scaling function of the PDF being finite for periodic signals, but developing a singularity for the aperiodic ones. Finally, an overview is given for the scaling of cumulants of the roughness and the various scaling regions in arbitrary dimensions. We suggest that our theoretical and numerical results open a different perspective on the data analysis of [formula presented] processes.

AB - The probability density function (PDF) of the roughness, i.e., of the temporal variance, of [formula presented] noise signals is studied. Our starting point is the generalization of the model of Gaussian, time periodic, [formula presented] noise, discussed in our recent Letter [Phys. Rev. Lett. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions [Formula Presented] and [formula presented] distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any [formula presented] A simulation of the periodic process makes it possible to study also nonperiodic, thus experimentally more relevant, signals on relatively short intervals embedded in the full period. We find that for [formula presented] the scaled PDFs in both the periodic and the nonperiodic cases are Gaussian, but for [formula presented] they differ from the Gaussian and from each other. Both deviations increase with growing [formula presented] That conclusion, based on numerics, is reinforced by analytic results for [formula presented] and [formula presented] in the latter limit the scaling function of the PDF being finite for periodic signals, but developing a singularity for the aperiodic ones. Finally, an overview is given for the scaling of cumulants of the roughness and the various scaling regions in arbitrary dimensions. We suggest that our theoretical and numerical results open a different perspective on the data analysis of [formula presented] processes.

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U2 - 10.1103/PhysRevE.65.046140

DO - 10.1103/PhysRevE.65.046140

M3 - Article

VL - 65

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 4

ER -