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Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes. (English) Zbl 0856.60080

The authors analyze the discrete-time counterpart of the first author [Adv. Appl. Probab. 17, 514-530 (1985; Zbl 0597.60080) and ibid. 23, No. 4, 683-700 (1991; Zbl 0736.60076)] which discuss exponential ergodicity and quasi-stationarity, respectively, in continuous-time birth-death processes. They obtain bounds for the decay parameter, conditions for geometric ergodicity and expressions for quasi-stationary distributions. The analysis is based on the spectral representation for the \(n\)-step transition probabilities of a birth-death process developed by S. Karlin and J. McGregor [Ill. J. Math. 3, 66-81 (1959; Zbl 0104.11804)].

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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