scholarly journals Coupling methods for random topological Markov chains

2015 ◽  
Vol 37 (3) ◽  
pp. 971-994 ◽  
Author(s):  
MANUEL STADLBAUER
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Shift Spaces ◽  
Transfer Operators ◽  
Coupling Methods ◽  
Coupling Techniques

We apply coupling techniques in order to prove that the transfer operators associated with random topological Markov chains and non-stationary shift spaces with the big images and preimages property have a spectral gap.

Conductance bounds on the L 2 convergence rate of Metropolis algorithms on unbounded state spaces

2004 ◽  
Vol 36 (01) ◽  
pp. 243-266
Author(s):  
Søren F. Jarner ◽  
Wai Kong Yuen
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Spectral Gap ◽  
Unbounded Domains ◽  
Decomposition Theorem ◽  
Metropolis Algorithm ◽  
The State ◽  
Bounded Domains ◽  
Target Density ◽  
State Spaces

In this paper we derive bounds on the conductance and hence on the spectral gap of a Metropolis algorithm with a monotone, log-concave target density on an interval of ℝ. We show that the minimal conductance set has measure ½ and we use this characterization to bound the conductance in terms of the conductance of the algorithm restricted to a smaller domain. Whereas previous work on conductance has resulted in good bounds for Markov chains on bounded domains, this is the first conductance bound applicable to unbounded domains. We then show how this result can be combined with the state-decomposition theorem of Madras and Randall (2002) to bound the spectral gap of Metropolis algorithms with target distributions with monotone, log-concave tails on ℝ.

Asymptotic and monotonicity properties of some repairable systems

1998 ◽  
Vol 30 (4) ◽  
pp. 1089-1110 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Repairable Systems ◽  
Lifetime Distributions ◽  
Failure Times ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Monotonicity Properties ◽  
Heavy Tailed

The paper studies a model of repairable systems which is flexible enough to incorporate the standard imperfect repair and many other models from the literature. Palm stationarity of virtual ages, inter-failure times and degrees of repair is studied. A Loynes-type scheme and Harris recurrent Markov chains combined with coupling methods are used. Results on the weak total variation and moment convergences are obtained and illustrated by examples with IFR, DFR, heavy-tailed and light-tailed lifetime distributions. Some convergences obtained are monotone and/or at a geometric rate.

The spectral gap and perturbation bounds for reversible continuous-time Markov chains

2004 ◽  
Vol 41 (4) ◽  
pp. 1219-1222 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Spectral Gap ◽  
Complete Information ◽  
Stationary States ◽  
Perturbation Bounds ◽  
Continuous Time Markov Chains ◽  
Distribution Vector

We show that, for reversible continuous-time Markov chains, the closeness of the nonzero eigenvalues of the generator to zero provides complete information about the sensitivity of the distribution vector to perturbations of the generator. Our results hold for both the transient and the stationary states.

scholarly journals Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matrices

2016 ◽  
Vol 53 (3) ◽  
pp. 946-952
Author(s):  
Loï Hervé ◽  
James Ledoux
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Spectral Radius ◽  
Spectral Gap ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Geometric Ergodicity ◽  
Transition Matrices ◽  
Essential Spectral Radius ◽  
Band Transition

AbstractWe analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius ress(P|𝓁²(𝜋)) of P|𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−NNlim supi→+∞(P(i,i+m)P*(i+m,i)1∕2<1. Moreover, ress(P|𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

Uniform coupling of non-homogeneous Markov chains

1975 ◽  
Vol 12 (04) ◽  
pp. 753-762 ◽  
Author(s):  
David Griffeath
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Weak Ergodicity ◽  
Coupling Techniques

The Markov-Dobrushin condition for (weak) ergodicity of non-homogeneous discrete-time Markov chains, and an analogous criterion for continuous chains, are derived by means of coupling techniques.

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