16,553 research outputs found
Intertwining of birth-and-death processes
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.Comment: 12 pages. 1 figure. Some typoes corrected and minor change
Intertwining and commutation relations for birth-death processes
Given a birth-death process on with semigroup
and a discrete gradient depending on a positive weight , we
establish intertwining relations of the form
, where is the Feynman-Kac
semigroup with potential of another birth-death process. We provide
applications when is nonnegative and uniformly bounded from below,
including Lipschitz contraction and Wasserstein curvature, various functional
inequalities, and stochastic orderings. Our analysis is naturally connected to
the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death
processes. The proofs are remarkably simple and rely on interpolation,
commutation, and convexity.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ433 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Abrupt Convergence and Escape Behavior for Birth and Death Chains
We link two phenomena concerning the asymptotical behavior of stochastic
processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape
behavior usually associated to exit from metastability. The former is
characterized by convergence at asymptotically deterministic times, while the
convergence times for the latter are exponentially distributed. We compare and
study both phenomena for discrete-time birth-and-death chains on Z with drift
towards zero. In particular, this includes energy-driven evolutions with energy
functions in the form of a single well. Under suitable drift hypotheses, we
show that there is both an abrupt convergence towards zero and escape behavior
in the other direction. Furthermore, as the evolutions are reversible, the law
of the final escape trajectory coincides with the time reverse of the law of
cut-off paths. Thus, for evolutions defined by one-dimensional energy wells
with sufficiently steep walls, cut-off and escape behavior are related by time
inversion.Comment: 2 figure
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