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On times to quasi-stationarity for birth and death processes. (English) Zbl 1186.60086

Authors’ abstract: The purpose of this paper is to present a probabilistic proof of the well-known result stating that the time needed by a continuous-time finite birth and death process for going from the left end to the right end of its state space is a sum of independent exponential variables whose parameters are the negatives of the eigenvalues of the underlying generator when the right end is treated as an absorbing state. The exponential variables appear as fastest strong quasi-stationary times for successive dual processes associated to the original absorbed process. As an aftermath, we get an interesting probabilistic representation of the time marginal laws of the process in terms of “local equilibria”.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J27 Continuous-time Markov processes on discrete state spaces
60J35 Transition functions, generators and resolvents
37A30 Ergodic theorems, spectral theory, Markov operators
15A18 Eigenvalues, singular values, and eigenvectors
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