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Modelling Heavy Tailed Phenomena Using a LogNormal Distribution Having a Numerically Verifiable Infinite Variance

Author

Listed:
  • Marco Cococcioni

    (Department of Information Engineering, L.go Lucio Lazzarino, 1-56122 Pisa, Italy
    These authors contributed equally to this work.)

  • Francesco Fiorini

    (Department of Information Engineering, L.go Lucio Lazzarino, 1-56122 Pisa, Italy
    These authors contributed equally to this work.)

  • Michele Pagano

    (Department of Information Engineering, L.go Lucio Lazzarino, 1-56122 Pisa, Italy
    These authors contributed equally to this work.)

Abstract

One-sided heavy tailed distributions have been used in many engineering applications, ranging from teletraffic modelling to financial engineering. In practice, the most interesting heavy tailed distributions are those having a finite mean and a diverging variance. The LogNormal distribution is sometimes discarded from modelling heavy tailed phenomena because it has a finite variance, even when it seems the most appropriate one to fit the data. In this work we provide for the first time a LogNormal distribution having a finite mean and a variance which converges to a well-defined infinite value. This is possible thanks to the use of Non-Standard Analysis. In particular, we have been able to obtain a Non-Standard LogNormal distribution, for which it is possible to numerically and experimentally verify whether the expected mean and variance of a set of generated pseudo-random numbers agree with the theoretical ones. Moreover, such a check would be much more cumbersome (and sometimes even impossible) when considering heavy tailed distributions in the traditional framework of standard analysis.

Suggested Citation

  • Marco Cococcioni & Francesco Fiorini & Michele Pagano, 2023. "Modelling Heavy Tailed Phenomena Using a LogNormal Distribution Having a Numerically Verifiable Infinite Variance," Mathematics, MDPI, vol. 11(7), pages 1-16, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1758-:d:1117696
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    References listed on IDEAS

    as
    1. Michele Leonardo Bianchi & Stoyan V Stoyanov & Gian Luca Tassinari & Frank J Fabozzi & Sergio M Focardi, 2019. "Handbook of Heavy-Tailed Distributions in Asset Management and Risk Management," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 11118, September.
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    Cited by:

    1. Karol I. Santoro & Diego I. Gallardo & Osvaldo Venegas & Isaac E. Cortés & Héctor W. Gómez, 2023. "A Heavy-Tailed Distribution Based on the Lomax–Rayleigh Distribution with Applications to Medical Data," Mathematics, MDPI, vol. 11(22), pages 1-15, November.

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