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Intertwining of birth-and-death processes. (English) Zbl 1221.60125

Author’s abstract: It has been known for a long time that, for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues, ordered from high to low, and whose death rates are zero, in such a way that the latter process is always ahead of the former, and both arrive at the same time at the given level. In this note, we extend their methods by constructing a third process, whose birth rates are the negatives of the eigenvalues ordered from low to high and whose death rates are zero, which always lags behind the original process and also arrives at the same time.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J27 Continuous-time Markov processes on discrete state spaces
15A18 Eigenvalues, singular values, and eigenvectors
37A30 Ergodic theorems, spectral theory, Markov operators
60G40 Stopping times; optimal stopping problems; gambling theory
60J35 Transition functions, generators and resolvents
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References:

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