Characterization and sufficient conditions for normed ergodicity of Markov chains

2004 ◽  
Vol 36 (1) ◽  
pp. 227-242 ◽  
Author(s):  
A. A. Borovkov ◽  
A. Hordijk
Keyword(s):  
Markov Chain ◽  
Lyapunov Function ◽  
Markov Chains ◽  
Sufficient Conditions ◽  
Operator Norm ◽  
Sufficient Condition ◽  
Uniform Integrability ◽  
Exponential Ergodicity ◽  
Step Transition

Normed ergodicity is a type of strong ergodicity for which convergence of the nth step transition operator to the stationary operator holds in the operator norm. We derive a new characterization of normed ergodicity and we clarify its relation with exponential ergodicity. The existence of a Lyapunov function together with two conditions on the uniform integrability of the increments of the Markov chain is shown to be a sufficient condition for normed ergodicity. Conversely, the sufficient conditions are also almost necessary.

Characterization and sufficient conditions for normed ergodicity of Markov chains

2004 ◽  
Vol 36 (01) ◽  
pp. 227-242 ◽  
Author(s):  
A. A. Borovkov ◽  
A. Hordijk
Keyword(s):  
Markov Chain ◽  
Lyapunov Function ◽  
Markov Chains ◽  
Sufficient Conditions ◽  
Operator Norm ◽  
Sufficient Condition ◽  
Uniform Integrability ◽  
Exponential Ergodicity ◽  
Step Transition

Normed ergodicity is a type of strong ergodicity for which convergence of thenth step transition operator to the stationary operator holds in the operator norm. We derive a new characterization of normed ergodicity and we clarify its relation with exponential ergodicity. The existence of a Lyapunov function together with two conditions on the uniform integrability of the increments of the Markov chain is shown to be a sufficient condition for normed ergodicity. Conversely, the sufficient conditions are also almost necessary.

Algebraic characterization of infinite Markov chains where movement to the right is limited to one step

1977 ◽  
Vol 14 (04) ◽  
pp. 740-747 ◽  
Author(s):  
Ester Samuel-Cahn ◽  
Shmuel Zamir
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Algebraic Characterization ◽  
Transient States ◽  
One Step ◽  
The Right

We consider an infinite Markov chain with states E 0, E 1, …, such that E 1, E 2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E 0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.

Algebraic characterization of infinite Markov chains where movement to the right is limited to one step

1977 ◽  
Vol 14 (4) ◽  
pp. 740-747 ◽  
Author(s):  
Ester Samuel-Cahn ◽  
Shmuel Zamir
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Algebraic Characterization ◽  
Transient States ◽  
One Step ◽  
The Right

We consider an infinite Markov chain with states E0, E1, …, such that E1, E2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.

On a Class of Generalized Baker's Transformations

1993 ◽  
Vol 45 (3) ◽  
pp. 638-649 ◽  
Author(s):  
M. Rahe
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Sufficient Conditions ◽  
Conditional Entropy ◽  
Sufficient Condition ◽  
Difference Sequence ◽  
Hölder Continuous ◽  
The Past ◽  
Summable Sequence ◽  
Necessary And Sufficient

AbstractLet f define a baker's transformation (Tf, Pf). We find necessary and sufficient conditions on f for (Tf, Pf) to be an N(ω)-step random Markov chain. Using this result, we give a simplified proof of Bose's results on Holder continuous baker's transformations where f is bounded away from zero and one. We extend Bose's results to show that, for the class of baker's transformations which are random Markov chains where TV has finite expectation, a sufficient condition for weak Bernoullicity is that the Lebesgue measure λ{x f(x) = 0 or f(x) = 1} = 0. We also examine random Markov chains satisfying a strictly weaker condition, those for which the differences between the entropy of the process and the conditional entropy given the past to time n form a summable sequence; and we show that a similar result holds. A condition is given on/ which is weaker than Holder continuity, but which implies that the entropy difference sequence is summable. Finally, a particular baker's transformation is exhibited as an easy example of a weakly Bernoulli transformation on which the supremum of the measure of atoms indexed by n-strings decays only as the reciprocal of n.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

Strong ergodicity for continuous-time, non-homogeneous Markov chains

1982 ◽  
Vol 19 (3) ◽  
pp. 692-694 ◽  
Author(s):  
Mark Scott ◽  
Barry C. Arnold ◽  
Dean L. Isaacson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Homogeneous Markov Chain ◽  
Strong Ergodicity

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.

Exponential ergodicity in derived Markov chains

1968 ◽  
Vol 5 (03) ◽  
pp. 669-678 ◽  
Author(s):  
Jozef L. Teugels
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Exponential Ergodicity ◽  
Stationary Markov Chain ◽  
Continuous Time Markov Chains ◽  
General Proposition

A general proposition is proved stating that the exponential ergodicity of a stationary Markov chain is preserved for derived Markov chains as defined by Cohen [2], [3]. An application to a certain type of continuous time Markov chains is included.

Single-shelf library-type Markov chains with infinitely many books

1977 ◽  
Vol 14 (02) ◽  
pp. 298-308 ◽  
Author(s):  
Peter R. Nelson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Natural Order ◽  
Order Book ◽  
Exit Boundary ◽  
One Step ◽  
Step Transition ◽  
The Right ◽  
Positive Recurrent

In a single-shelf library having infinitely many books B 1 , B 2 , …, the probability of selecting each book is assumed known. Books are removed one at a time and replaced in position k prior to the next removal. Books are moved either to the right or the left as is necessary to vacate position k. Those arrangements of books where after some finite position all the books are in natural order (book i occupies position i) are considered as states in an infinite Markov chain. When k > 1, we show that the chain can never be positive recurrent. When k = 1, we find the limits of ratios of one-step transition probabilities; and when k = 1 and the chain is transient, we find the Martin exit boundary.

Some sufficient conditions for non-ergodicity of markov chains

1985 ◽  
Vol 22 (01) ◽  
pp. 138-147 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Lyapunov Functions ◽  
Sufficient Conditions

Some sufficient conditions for non-ergodicity are given for a Markov chain with denumerable state space. These conditions generalize Foster's results, in that unbounded Lyapunov functions are considered. Our criteria directly extend the conditions obtained in Kaplan (1979), in the sense that a class of Lyapunov functions is studied. Applications are presented through some examples; in particular, sufficient conditions for non-ergodicity of a multidimensional Markov chain are given.

Lumpability and marginalisability for continuous-time Markov chains

1993 ◽  
Vol 30 (3) ◽  
pp. 518-528 ◽  
Author(s):  
Frank Ball ◽  
Geoffrey F. Yeo
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Ion Channel ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Continuous Time Markov Chains ◽  
Probabilistic Proof ◽  
Necessary And Sufficient ◽  
Mutually Independent

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

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