Abstract
In this chapter, we study QSDs on birth-and-death chains. In Sect. 5.1, we describe the quantities relevant for this study. The main tools for describing the QSDs are the Karlin–McGregor decomposition and the dual process, which are supplied in Sect. 5.2.
Among the main results of the chapter is the description of the set of QSDs done in Theorem 5.4 of Sect. 5.3: exponential killing implies that there exist QSDs; if ∞ is a natural boundary, there is a continuum of QSDs; when ∞ is an entrance boundary, there is a unique one. In Theorem 5.8 of Sect. 5.4, it is shown that the extremal QSD is a quasi-limiting distribution. The process of survival trajectories is studied in Sect. 5.5 and its classification is given in Theorem 5.12. In Sect. 5.6, it is proved that the QSDs are stochastically ordered, the extremal QSD is the minimal one. In Sect. 5.7, explicit conditions on the birth-and-death parameters ensuring discrete spectrum are supplied. Some examples of killed birth-and-death chains can be found in Sect. 5.9. In the last Sect. 5.10, we study QSDs for a population model where each individual is characterized by some trait, and it can die or gives birth to a clonal or a mutated individual.
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Collet, P., Martínez, S., San Martín, J. (2013). Birth-and-Death Chains. In: Quasi-Stationary Distributions. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33131-2_5
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