Geodesic
Intertwining of birth-and-death processes
Kybernetika, Tome 47 (2011) no. 1, pp. 1-14 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 15A18, 37A30, 60G40, 60J27, 60J35, 60J80
Keywords: intertwining of Markov processes; birth and death process; averaged Markov process; first passage time; coupling; eigenvalues 
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Swart, Jan M. Intertwining of birth-and-death processes. Kybernetika, Tome 47 (2011) no. 1, pp. 1-14. http://geodesic.mathdoc.fr/item/KYB_2011_47_1_a0/

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