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15
Markov chains for exploring posterior distributions
 Annals of Statistics
, 1994
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Cited by 1049 (6 self)
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An almost sure invariance principle for random walks in a spacetime random environment
 PROBAB. THEORY RELAT. FIELDS
, 2005
"... We consider a discrete time random walk in a spacetime i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L 2 averaged ..."
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Cited by 31 (7 self)
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We consider a discrete time random walk in a spacetime i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L 2 averaged drift. We also state an a.s. invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.
A central limit theorem for random walks in random labyrinths
, 1999
"... Abstract.Abeam of light shines through the latticeZd, and is subjected to reflections determined by a random environment of mirrors at the vertices ofZd. The behaviour of the light ray is investigated under the hypothesis that the environment contains a strictly positive density of vertices at which ..."
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Cited by 4 (3 self)
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Abstract.Abeam of light shines through the latticeZd, and is subjected to reflections determined by a random environment of mirrors at the vertices ofZd. The behaviour of the light ray is investigated under the hypothesis that the environment contains a strictly positive density of vertices at which the light behaves in the manner of a random walk. When d ≥ 2 and the density of nontrivial reflectors is sufficiently small, the environment contains a.s. a unique infinite ‘interilluminating ’ class of vertices. Furthermore, when the light beam originates within this class, then its trajectory obeys a functional central limit theorem with a strictly positive diffusion constant. These facts are obtained using percolationtype arguments, together with the invariance principle proposed by Kipnis and Varadhan. 1.
On a directionally reinforced random walk
 Proceedings of the American Mathematical Society
, 2014
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ASYMPTOTIC VARIANCE OF FUNCTIONALS OF DISCRETE TIME MARKOV CHAINS VIA THE DRAZIN INVERSE Elect. Comm. in Probab. 12 (2007), 120–133 ELECTRONIC COMMUNICATIONS in PROBABILITY
, 2006
"... Drazin inverse; fundamental matrix; asymptotic variance We consider a ψirreducible, discretetime Markov chain on a general state space with transition kernel P. Under suitable conditions on the chain, kernels can be treated as bounded linear operators between spaces of functions or measures and th ..."
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Cited by 1 (0 self)
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Drazin inverse; fundamental matrix; asymptotic variance We consider a ψirreducible, discretetime Markov chain on a general state space with transition kernel P. Under suitable conditions on the chain, kernels can be treated as bounded linear operators between spaces of functions or measures and the Drazin inverse of the kernel operator I −P exists. The Drazin inverse provides a unifying framework for objects governing the chain. This framework is applied to derive a computational technique for the asymptotic variance in the central limit theorems of univariate and higherorder partial sums. Higherorder partial sums are treated as univariate sums on a ‘slidingwindow ’ chain. Our results are demonstrated on a simple AR(1) model and suggest a potential for computational simplification. 1
RELAXED SECTOR CONDITION
"... Inthis notewepresentanewsufficientcondition whichguarantees martingale approximation and central limit theorem à la Kipnis–Varadhan to hold for additive functionals of Markov processes. This condition, which we call the relaxed sector condition (RSC) generalizes the strong sector condition (SSC) and ..."
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Inthis notewepresentanewsufficientcondition whichguarantees martingale approximation and central limit theorem à la Kipnis–Varadhan to hold for additive functionals of Markov processes. This condition, which we call the relaxed sector condition (RSC) generalizes the strong sector condition (SSC) and the graded sector condition (GSC) in the case when the selfadjoint part of the infinitesimal generator acts diagonally in the grading. The main advantage being that the proof of the GSC in this case is more transparent and less computational than in the original versions. We also hope that the RSC may have direct applications where the earlier sector conditions do not apply. So far we do not have convincing examples in this direction. 1.
ISSN: 1083589X ELECTRONIC COMMUNICATIONS in PROBABILITY
"... We present a streamlined derivation of the theorem of M. Maxwell and M. Woodroofe [3], on martingale approximation of additive functionals of stationary Markov processes, from the nonreversible version of the KipnisVaradhan theorem. ..."
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We present a streamlined derivation of the theorem of M. Maxwell and M. Woodroofe [3], on martingale approximation of additive functionals of stationary Markov processes, from the nonreversible version of the KipnisVaradhan theorem.