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Asymptotic behavior and quasi-limiting distributions on time-fractional birth and death processes

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  11 November 2022

Jorge Littin Curinao Open the ORCID record for Jorge Littin Curinao [Opens in a new window]
Jorge Littin Curinao*
Affiliation:
Universidad Católica del Norte
*
*Postal address: Departamento de Matemáticas, Universidad Católica del Norte, Angamos 0610, Antofagasta, Chile. Email address: jlittin@ucn.cl
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Abstract

In this article we provide new results for the asymptotic behavior of a time-fractional birth and death process $N_{\alpha}(t)$ , whose transition probabilities $\mathbb{P}[N_{\alpha}(t)=\,j\mid N_{\alpha}(0)=i]$ are governed by a time-fractional system of differential equations, under the condition that it is not killed. More specifically, we prove that the concepts of quasi-limiting distribution and quasi-stationary distribution do not coincide, which is a consequence of the long-memory nature of the process. In addition, exact formulas for the quasi-limiting distribution and its rate convergence are presented. In the first sections, we revisit the two equivalent characterizations for this process: the first one is a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Finally, we apply our main theorems to the linear model originally introduced by Orsingher and Polito [23].

Keywords

Fractional processesquasi-limiting distributionsinverse stable subordinators

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60G18: Self-similar processes 60K15: Markov renewal processes, semi-Markov processes
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1199 - 1227
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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