scholarly journals Analysis of non-reversible Markov chains via similarity orbits

2020 ◽  
Vol 29 (4) ◽  
pp. 508-536
Author(s):  
Michael C. H. Choi ◽  
Pierre Patie
Keyword(s):  
Markov Chains ◽  
Sufficient Conditions ◽  
Transition Kernel ◽  
Discrete Observations ◽  
Similarity Orbit ◽  
Normal Transition ◽  
Depth Analysis ◽  
Reversible Markov Chains ◽  
By Products ◽  
Birth Death

AbstractIn this paper we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as that of birth–death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth–death chain. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and L2-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper we investigate a particular similarity orbit of reversible Markov kernels, which we call the pure birth orbit, and analyse various possibly non-reversible variants of classical birth–death processes in this orbit.

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.

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.

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&lt; λ1≤ λ2≤ ···} be the whole spectrum ofLin L2. Then ∑n≥1λn-1&lt; ∞ 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&lt; ∞ if and only ifα&gt; 0.

scholarly journals Uniqueness criteria for continuous-time Markov chains with general transition structures

2005 ◽  
Vol 37 (04) ◽  
pp. 1056-1074 ◽  
Author(s):  
Anyue Chen ◽  
Phil Pollett ◽  
Hanjun Zhang ◽  
Ben Cairns
Keyword(s):  
Markov Chains ◽  
Branching Processes ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Famous Result ◽  
Transition Structures ◽  
Systems Of Linear Equations ◽  
Necessary And Sufficient ◽  
Uniqueness Criteria ◽  
Birth Death

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

scholarly journals Uniqueness criteria for continuous-time Markov chains with general transition structures

2005 ◽  
Vol 37 (4) ◽  
pp. 1056-1074 ◽  
Author(s):  
Anyue Chen ◽  
Phil Pollett ◽  
Hanjun Zhang ◽  
Ben Cairns
Keyword(s):  
Markov Chains ◽  
Branching Processes ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Famous Result ◽  
Transition Structures ◽  
Systems Of Linear Equations ◽  
Necessary And Sufficient ◽  
Uniqueness Criteria ◽  
Birth Death

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

1996 ◽  
Vol 28 (01) ◽  
pp. 114-165 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Complex Analysis ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Transform Method ◽  
Technical Problems ◽  
The Matrix ◽  
Transient Case

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

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