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Fractional calculus approach to the statistical characterization of random variables and vectors

Author

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  • Cottone, Giulio
  • Di Paola, Mario
  • Metzler, Ralf

Abstract

Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. Applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided, showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis.

Suggested Citation

  • Cottone, Giulio & Di Paola, Mario & Metzler, Ralf, 2010. "Fractional calculus approach to the statistical characterization of random variables and vectors," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(5), pages 909-920.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:5:p:909-920
    DOI: 10.1016/j.physa.2009.11.018
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    References listed on IDEAS

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    1. Marta C. González & César A. Hidalgo & Albert-László Barabási, 2009. "Understanding individual human mobility patterns," Nature, Nature, vol. 458(7235), pages 238-238, March.
    2. D. Brockmann & L. Hufnagel & T. Geisel, 2006. "The scaling laws of human travel," Nature, Nature, vol. 439(7075), pages 462-465, January.
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    Cited by:

    1. Alotta, G. & Di Paola, M., 2015. "Probabilistic characterization of nonlinear systems under α-stable white noise via complex fractional moments," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 420(C), pages 265-276.
    2. Xu, Jun & Song, Jinheng & Yu, Quanfu & Kong, Fan, 2023. "Generalized distribution reconstruction based on the inversion of characteristic function curve for structural reliability analysis," Reliability Engineering and System Safety, Elsevier, vol. 229(C).

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