Asymptotic behavior and quasi-limiting distributions on time-fractional birth and death processes. (English) Zbl 1498.60340
Summary: 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 E. Orsingher and F. Polito [Bernoulli 17, No. 1, 114–137 (2011; Zbl 1284.60157)].
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60G18 | Self-similar stochastic processes |
Citations:
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