36,606 research outputs found
Quantitative convergence rates for sub-geometric Markov chains
We provide explicit expressions for the constants involved in the
characterisation of ergodicity of sub-geometric Markov chains. The constants
are determined in terms of those appearing in the assumed drift and one-step
minorisation conditions. The result is fundamental for the study of some
algorithms where uniform bounds for these constants are needed for a family of
Markov kernels. Our result accommodates also some classes of inhomogeneous
chains.Comment: 14 page
Quantitative bounds on convergence of time-inhomogeneous Markov chains
Convergence rates of Markov chains have been widely studied in recent years.
In particular, quantitative bounds on convergence rates have been studied in
various forms by Meyn and Tweedie [Ann. Appl. Probab. 4 (1994) 981-1101],
Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566], Roberts and Tweedie
[Stochastic Process. Appl. 80 (1999) 211-229], Jones and Hobert [Statist. Sci.
16 (2001) 312-334] and Fort [Ph.D. thesis (2001) Univ. Paris VI]. In this
paper, we extend a result of Rosenthal [J. Amer. Statist. Assoc. 90 (1995)
558-566] that concerns quantitative convergence rates for time-homogeneous
Markov chains. Our extension allows us to consider f-total variation distance
(instead of total variation) and time-inhomogeneous Markov chains. We apply our
results to simulated annealing.Comment: Published at http://dx.doi.org/10.1214/105051604000000620 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
CONVERGENCE OF MARKOV CHAIN APPROXIMATIONS TO STOCHASTIC REACTION DIFFUSION EQUATIONS
In the context of simulating the transport of a chemical or bacterial contaminant through a moving sheet of water, we extend a well established method of approximating reaction-diffusion equations with Markov chains by allowing convection, certain Poisson measure driving sources and a larger class of reaction functions. Our alterations also feature dramatically slower Markov chain state change rates often yielding a ten to one hundred fold simulation speed increase over the previous version of the method as evidenced in our computer implementations. On a weighted L2 Hilbert space chosen to symmetrize the elliptic operator, we consider existence of and convergence to pathwise unique mild solutions of our stochastic reaction-diffusion equation. Our main convergence result, a quenched law of large numbers, establishes convergence in probability of our Markov chain approximations for each fixed path of our driving Poisson measure source. As a consequence, we also obtain the annealed law of large numbers establishing convergence in probability of our Markov chains to the solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains.
On improved convergence conditions and bounds for Markov chains
Improved rates of convergence for ergodic Markov chains and relaxed
conditions for them, as well as analogous convergence results for
non-homogeneous Markov chains are studied. The setting from the previous works
is extended. Examples are provided where the new bounds are better and where
they give the same convergence rate as in the classical Markov -- Dobrushin
inequality.Comment: 33 pages, 27 reference
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