Abstract
In this chapter, we introduce the main problems on QSDs for irreducible finite Markov chains. In Sect. 3.1, we show that there is a unique QSD which is the normalized left Perron–Frobenius eigenvector of the jump rates matrix restricted to the allowed states. The right eigenvector is shown to be the asymptotic ratio of survival probabilities. In Sect. 3.2, it is proved that the trajectories that survive forever form a Markov chain which is an h-process of the original one with weights given by the right eigenvector. In Sect. 3.3, we give some computations when the jump rate matrix restricted to the allowed states is symmetric. In Sect. 3.5, we compute the unique QSD for the random walk with barriers. In Sect. 3.6, we prove a Central Limit Theorem for the chain conditioned to survive when time is discrete.
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References
Darroch, J. N., & Seneta, E. (1965). On quasi-stationary distributions in absorbing discrete-time finite Markov chains. Journal of Applied Probability, 2, 88–100.
Seneta, E. (1981). Springer series in statistics. Non-negative matrices and Markov chains. New York: Springer.
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Collet, P., Martínez, S., San Martín, J. (2013). Markov Chains on Finite Spaces. In: Quasi-Stationary Distributions. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33131-2_3
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DOI: https://doi.org/10.1007/978-3-642-33131-2_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33130-5
Online ISBN: 978-3-642-33131-2
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