  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.
