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Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times

Author

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  • Thomas M. Michelitsch

    (Sorbonne Université, Institut Jean le Rond d’Alembert CNRS UMR 7190, 4 Place Jussieu, CEDEX 05, 75252 Paris, France)

  • Federico Polito

    (Department of Mathematics “Giuseppe Peano”, University of Torino, 10123 Torino, Italy)

  • Alejandro P. Riascos

    (Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México 01000, Mexico)

Abstract

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the ‘generalized Sibuya distribution’ (GSD), is such that the moments are finite up to a certain order r ≤ m − 1 ( m ≥ 1 ) and diverging for orders r ≥ m capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case ( m = 1 ). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order m of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.

Suggested Citation

  • Thomas M. Michelitsch & Federico Polito & Alejandro P. Riascos, 2023. "Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times," Mathematics, MDPI, vol. 11(2), pages 1-20, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:2:p:471-:d:1037116
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    References listed on IDEAS

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    1. Michelitsch, Thomas M. & Polito, Federico & Riascos, Alejandro P., 2021. "On discrete time Prabhakar-generalized fractional Poisson processes and related stochastic dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    2. Metzler, Ralf & Compte, Albert, 1999. "Stochastic foundation of normal and anomalous Cattaneo-type transport," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 268(3), pages 454-468.
    3. Alessandro Gregorio & Stefano Iacus, 2008. "Parametric estimation for the standard and geometric telegraph process observed at discrete times," Statistical Inference for Stochastic Processes, Springer, vol. 11(3), pages 249-263, October.
    4. Tomasz J. Kozubowski & Krzysztof Podgórski, 2018. "A generalized Sibuya distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 855-887, August.
    5. Bogachev, Leonid & Ratanov, Nikita, 2011. "Occupation time distributions for the telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1816-1844, August.
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