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Theory of Probability and Mathematical Statistics

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Ehrenfest-Brillouin-type correlated continuous time random walk and fractional Jacobi diffusion


Authors: N. N. Leonenko, I. Papić, A. Sikorskii and N. Šuvak
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 99 (2018).
Journal: Theor. Probability and Math. Statist. 99 (2019), 137-147
MSC (2010): Primary 47D07, 60J10, 60J60, 60G22, 60G50
DOI: https://doi.org/10.1090/tpms/1086
Published electronically: February 27, 2020
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Abstract: Continuous time random walks (CTRWs) have random waiting times between particle jumps. Based on the Ehrenfest-Brillouin-type model motivated by economics, we define the correlated CTRW that converges to the fractional Jacobi diffusion $ Y(E(t))$, $ t\ge 0$, defined as a time change of Jacobi diffusion process $ Y(t)$ to the inverse $ E(t)$ of the standard stable subordinator. In the CTRW considered in this paper, the jumps are correlated so that in the limit the outer process $ Y(t)$ is not a Lévy process but a diffusion process with non-independent increments. The waiting times between jumps are selected from the domain of attraction of a stable law, so that the correlated CTRWs with these waiting times converge to $ Y(E(t))$.


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Additional Information

N. N. Leonenko
Affiliation: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF244AG, United Kingdom
Email: LeonenkoN@cardiff.ac.uk

I. Papić
Affiliation: Department of Mathematics, J.J. Strossmayer University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia
Email: ipapic@mathos.hr

A. Sikorskii
Affiliation: Departments of Psychiatry and Statistics and Probability, Michigan State University, 909 Fee Road, East Lansing, Michigan 48824
Email: sikorska@stt.msu.edu

N. Šuvak
Affiliation: Department of Mathematics, J.J. Strossmayer University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia
Email: nsuvak@mathos.hr

DOI: https://doi.org/10.1090/tpms/1086
Keywords: Correlated continuous time random walk, Ehrenfest--Brillouin Markov chain, fractional diffusion, Jacobi diffusion, Pearson diffusion
Received by editor(s): July 20, 2018
Published electronically: February 27, 2020
Article copyright: © Copyright 2020 American Mathematical Society