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

Thomas M. Michelitsch (), Federico Polito and Alejandro P. Riascos
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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

Mathematics, 2023, vol. 11, issue 2, 1-20

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.

Keywords: non-markovian random walk; telegraph (Cattaneo) process; generalized Sibuya distribution; discrete-time renewal process (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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