The eigentime identity for continuous-time ergodic Markov chains

2004 ◽  
Vol 41 (04) ◽  
pp. 1071-1080 ◽  
Author(s):  
Yong-Hua Mao
Keyword(s):  
Markov Chains ◽  
Essential Spectrum ◽  
Continuous Time ◽  
Spectral Gap ◽  
Markov Generator ◽  
Reversible Markov Chains ◽  
Explicit Formulae ◽  
Birth Death

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generatorL. When the essential spectrum is empty, let {0 = λ0< λ1≤ λ2≤ ···} be the whole spectrum ofLin L2. Then ∑n≥1λn-1< ∞ implies the existence of the spectral gapαofLin L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑n≥1λn-1< ∞ if and only ifα> 0.

The eigentime identity for continuous-time ergodic Markov chains

2004 ◽  
Vol 41 (4) ◽  
pp. 1071-1080 ◽  
Author(s):  
Yong-Hua Mao
Keyword(s):  
Markov Chains ◽  
Essential Spectrum ◽  
Continuous Time ◽  
Spectral Gap ◽  
Markov Generator ◽  
Reversible Markov Chains ◽  
Explicit Formulae ◽  
Birth Death

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generator L. When the essential spectrum is empty, let {0 = λ0 < λ1 ≤ λ2 ≤ ···} be the whole spectrum of L in L2. Then ∑n≥1 λn-1 < ∞ implies the existence of the spectral gap α of L in L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑n≥1 λn-1 < ∞ if and only if α > 0.

Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

1998 ◽  
Vol 35 (3) ◽  
pp. 545-556 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Hazard Rate ◽  
First Passage Time ◽  
Death Process ◽  
Continuous Time Markov Chain ◽  
Reversed Hazard Rate ◽  
The Right ◽  
Birth Death

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

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 Level Crossing Ordering of Skip-Free-to-the-Right Continuous-Time Markov Chains

2005 ◽  
Vol 42 (1) ◽  
pp. 52-60 ◽  
Author(s):  
Fátima Ferreira ◽  
António Pacheco
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Level Crossing ◽  
Continuous Time Markov Chains ◽  
The Right ◽  
Birth Death

As proposed by Irle and Gani in 2001, a process X is said to be slower in level crossing than a process Y if it takes X stochastically longer to exceed any given level than it does Y. In this paper, we extend a result of Irle (2003), relative to the level crossing ordering of uniformizable skip-free-to-the-right continuous-time Markov chains, to derive a new set of sufficient conditions for the level crossing ordering of these processes. We apply our findings to birth-death processes with and without catastrophes, and M/M/s/c systems.

scholarly journals Level Crossing Ordering of Skip-Free-to-the-Right Continuous-Time Markov Chains

2005 ◽  
Vol 42 (01) ◽  
pp. 52-60 ◽  
Author(s):  
Fátima Ferreira ◽  
António Pacheco
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Level Crossing ◽  
Continuous Time Markov Chains ◽  
The Right ◽  
Birth Death

As proposed by Irle and Gani in 2001, a processXis said to be slower in level crossing than a processYif it takesXstochastically longer to exceed any given level than it doesY. In this paper, we extend a result of Irle (2003), relative to the level crossing ordering of uniformizable skip-free-to-the-right continuous-time Markov chains, to derive a new set of sufficient conditions for the level crossing ordering of these processes. We apply our findings to birth-death processes with and without catastrophes, and M/M/s/csystems.

Some Problems in Mathematical Genealogy

1975 ◽  
Vol 12 (S1) ◽  
pp. 325-345 ◽  
Author(s):  
David G. Kendall
Keyword(s):  
Markov Chains ◽  
Social Mobility ◽  
Present State ◽  
Continuous Time ◽  
General Review ◽  
Continuous Time Markov Chains ◽  
Immigration Process ◽  
Countable State ◽  
Birth Death

After a general review of symmetric reversibility for countable-state continuous-time Markov chains the author shows that the birth-death-and-immigration process is symmetrically reversible and further that it remains so even when the description of the present state is refined to include a list of the sizes of all ‘families’ alive at the epoch in question. This result can be useful in genealogy because the operational direction of time there is the negative one. In view of the symmetric reversibility, some of the questions which face the genealogist can be answered without further calculation by quoting known results for the process with the usual (‘forward’ instead of ‘backward’) direction of time.Further topics discussed include social mobility matrices, surname statistics, and Colin Rogers' ‘problem of the Spruces’.

Taboo rate and hitting time distribution of continuous-time reversible Markov chains

2021 ◽  
Vol 169 ◽  
pp. 108969
Author(s):  
Xuyan Xiang ◽  
Haiqin Fu ◽  
Jieming Zhou ◽  
Yingchun Deng ◽  
Xiangqun Yang
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Time Distribution ◽  
Hitting Time ◽  
Reversible Markov Chains

scholarly journals Event-Chain Monte Carlo: Foundations, Applications, and Prospects

2021 ◽  
Vol 9 ◽  
Author(s):  
Werner Krauth
Keyword(s):  
Aqueous Solution ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Real World ◽  
Continuous Time ◽  
Statistical Physics ◽  
Reversible Markov Chains ◽  
Sampling Problem ◽  
Review Reports

This review treats the mathematical and algorithmic foundations of non-reversible Markov chains in the context of event-chain Monte Carlo (ECMC), a continuous-time lifted Markov chain that employs the factorized Metropolis algorithm. It analyzes a number of model applications and then reviews the formulation as well as the performance of ECMC in key models in statistical physics. Finally, the review reports on an ongoing initiative to apply ECMC to the sampling problem in molecular simulation, i.e., to real-world models of peptides, proteins, and polymers in aqueous solution.

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