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Theory of Probability and Mathematical Statistics

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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.

References [Enhancements On Off] (What's this?)

  • 1. Kai Lai Chung, Markov chains with stationary transition probabilities, Die Grundlehren der mathematischen Wissenschaften, Bd. 104, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0116388
  • 2. N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375
  • 3. N. V. Kartashov, Calculation of the spectral ergodicity exponent for the birth and death process, Ukraïn. Mat. Zh. 52 (2000), no. 7, 889–897 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 52 (2000), no. 7, 1018–1028 (2001). MR 1817318,
  • 4. S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1993. MR 1287609
  • 5. Vladimir M. Zolotarev, Modern theory of summation of random variables, Modern Probability and Statistics, VSP, Utrecht, 1997. MR 1640024
  • 6. A. M. Zubkov, Inequalities for transition probabilities with taboos and their applications, Math. USSR Sb. 37 (1980), no. 4, 451-488.

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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

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