1,267 research outputs found
Subgeometric ergodicity of Markov chains
The goal of this paper is to give a short and self contained proof of general
bounds for subgeometric rates of convergence, under practical conditions. The
main result whose proof, based on coupling, provides an intuitive understanding
of the results of Nummelin and Tuominen (1983) and Tuominen and Tweedie (1994).
To obtain practical rates, a very general drift condition, recently introduced
in Douc et al (2004) is used
Exponential convergence rate of ruin probabilities for level-dependent L\'evy-driven risk processes
We explicitly find the rate of exponential long-term convergence for the ruin
probability in a level-dependent L\'evy-driven risk model, as time goes to
infinity. Siegmund duality allows to reduce the pro blem to long-term
convergence of a reflected jump-diffusion to its stationary distribution, which
is handled via Lyapunov functions.Comment: 20 pages, 5 figure
Renewal theory and computable convergence rates for geometrically ergodic Markov chains
We give computable bounds on the rate of convergence of the transition
probabilities to the stationary distribution for a certain class of
geometrically ergodic Markov chains. Our results are different from earlier
estimates of Meyn and Tweedie, and from estimates using coupling, although we
start from essentially the same assumptions of a drift condition toward a
``small set.'' The estimates show a noticeable improvement on existing results
if the Markov chain is reversible with respect to its stationary distribution,
and especially so if the chain is also positive. The method of proof uses the
first-entrance-last-exit decomposition, together with new quantitative versions
of a result of Kendall from discrete renewal theory.Comment: Published at http://dx.doi.org/10.1214/105051604000000710 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Strong Stationary Duality for M\"obius Monotone Markov Chains: Unreliable Networks
For Markov chains with a partially ordered finite state space we show strong
stationary duality under the condition of M\"obius monotonicity of the chain.
We show relations of M\"obius monotonicity to other definitions of monotone
chains. We give examples of dual chains in this context which have transitions
only upwards. We illustrate general theory by an analysis of nonsymmetric
random walks on the cube with an application to networks of queues
Speeding up the FMMR perfect sampling algorithm: A case study revisited
In a previous paper by the second author,two Markov chain Monte Carlo perfect
sampling algorithms -- one called coupling from the past (CFTP) and the other
(FMMR) based on rejection sampling -- are compared using as a case study the
move-to-front (MTF) self-organizing list chain. Here we revisit that case study
and, in particular, exploit the dependence of FMMR on the user-chosen initial
state. We give a stochastic monotonicity result for the running time of FMMR
applied to MTF and thus identify the initial state that gives the
stochastically smallest running time; by contrast, the initial state used in
the previous study gives the stochastically largest running time. By changing
from worst choice to best choice of initial state we achieve remarkable speedup
of FMMR for MTF; for example, we reduce the running time (as measured in Markov
chain steps) from exponential in the length n of the list nearly down to n when
the items in the list are requested according to a geometric distribution. For
this same example, the running time for CFTP grows exponentially in n.Comment: 19 pages. See also http://www.mts.jhu.edu/~fill/ and
http://www.mathcs.carleton.edu/faculty/bdobrow/. Submitted for publication in
May, 200
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