ON RANDOM TOPOLOGICAL MARKOV CHAINS WITH BIG IMAGES AND PREIMAGES

We introduce a relative notion of the "big images and preimages"-property for random topological Markov chains. This condition then implies that a relative version of the Ruelle–Perron–Frobenius theorem holds with respect to summable and locally Hölder continuous potentials.

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Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-02-03120-3 ◽

2002 ◽

Vol 355 (1) ◽

pp. 373-405 ◽

Cited By ~ 25

Author(s):

Richard F. Bass ◽

Edwin A. Perkins

Keyword(s):

Differential Equations ◽

Markov Chains ◽

Stochastic Differential Equations ◽

Hölder Continuous

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On a Class of Generalized Baker's Transformations

Canadian Journal of Mathematics ◽

10.4153/cjm-1993-035-0 ◽

1993 ◽

Vol 45 (3) ◽

pp. 638-649 ◽

Cited By ~ 4

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.

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A NOTE ON RÉNYI'S ENTROPY RATE FOR TIME-INHOMOGENEOUS MARKOV CHAINS

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996481800044x ◽

2018 ◽

Vol 33 (4) ◽

pp. 579-590

Author(s):

Wenxi Li ◽

Zhongzhi Wang

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Entropy Rate ◽

Transition Matrices ◽

Frobenius Theorem ◽

Primitive Matrix ◽

Inhomogeneous Markov Chain ◽

Renyi’S Entropy ◽

Renyi’S Divergence

AbstractIn this note, we use the Perron–Frobenius theorem to obtain the Rényi's entropy rate for a time-inhomogeneous Markov chain whose transition matrices converge to a primitive matrix. As direct corollaries, we also obtain the Rényi's entropy rate for asymptotic circular Markov chain and the Rényi's divergence rate between two time-inhomogeneous Markov chains.

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The Perron–Frobenius theorem – a proof with the use of Markov chains

Journal of Mathematical Sciences ◽

10.1007/s10958-009-9347-9 ◽

2009 ◽

Vol 157 (5) ◽

pp. 675-680

Author(s):

Yu. A. Al’pin ◽

V. S. Al’pina

Keyword(s):

Markov Chains ◽

Frobenius Theorem

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IMPRECISE MARKOV CHAINS AND THEIR LIMIT BEHAVIOR

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964809990039 ◽

2009 ◽

Vol 23 (4) ◽

pp. 597-635 ◽

Cited By ~ 41

Author(s):

Gert de Cooman ◽

Filip Hermans ◽

Erik Quaeghebeur

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Transition Probabilities ◽

Finite Markov Chain ◽

Initial State ◽

Frobenius Theorem ◽

Credal Set ◽

Credal Sets ◽

Definition Of ◽

Uncertainty Models

When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-calledcredal setsthat these probabilities are known or believed to belong to and by allowing the probabilities to vary over such sets. This leads to the definition of animprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-calledlowerandupper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at timenevolves asn→∞: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalization of the classical Perron–Frobenius theorem to imprecise Markov chains.

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