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Quantitative and qualitative limits for exponential asymptotics of hitting times for birth-and-death chains in a scheme of series

Author: N. V. Kartashov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 45-56
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
Published electronically: January 26, 2015
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Abstract: We consider a time-homogeneous discrete birth-and-death Markov chain $ (X_{t})$ and investigate the asymptotics of the hitting time $ \tau _n=\inf (t\geq 1\colon X_{t}\geq n)$ as well as the chain position before this time in the scheme of series as $ n\to \infty $. In our case, one-step probabilities of the chain vary simultaneously with $ n$. The proofs are based on the explicit two-sided inequalities with numerical bounds for the survival probability $ \mathsf {P}(\tau _n>t)$. These inequalities can be used also for the pre-limit finite-time schemes. We have applied the results obtained for constructing the uniform asymptotic representations of the corresponding risk function.

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Additional Information

N. V. Kartashov
Affiliation: Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, 01601 Kyiv, Ukraine

Keywords: Birth-and-death Markov chains, hitting times, exponential asymptotics
Received by editor(s): September 4, 2013
Published electronically: January 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society