Intertwining of birth-and-death processes

Swart, Jan M.. "Intertwining of birth-and-death processes." Kybernetika 47.1 (2011): 1-14. <http://eudml.org/doc/196582>.

@article{Swart2011,
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.},
author = {Swart, Jan M.},
journal = {Kybernetika},
keywords = {intertwining of Markov processes; birth and death process; averaged Markov process; first passage time; coupling; eigenvalues; eigenvalues; intertwining of Markov processes; birth and death process; averaged Markov process; first passage time; coupling},
language = {eng},
number = {1},
pages = {1-14},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Intertwining of birth-and-death processes},
url = {http://eudml.org/doc/196582},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Swart, Jan M.
TI - Intertwining of birth-and-death processes
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 1
SP - 1
EP - 14
AB - 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.
LA - eng
KW - intertwining of Markov processes; birth and death process; averaged Markov process; first passage time; coupling; eigenvalues; eigenvalues; intertwining of Markov processes; birth and death process; averaged Markov process; first passage time; coupling
UR - http://eudml.org/doc/196582
ER -