A note on a variation of Doeblin's condition for uniform ergodicity of Markov chains

Dorea, C. C. Y.; Pereira, André Grahl

doi:10.1007/s10474-006-0023-y

Cited by 17 publications

(18 citation statements)

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“…5, we find necessary and sufficient conditions for the weak ergodicity of nonhomogeneous discrete Markov chains (NDMC), which extend the results of [25,28] to an abstract scheme. Note that in [15] similar conditions were found for classical nonhomogeneous Markov processes to satisfy weak ergodicity. Applications of such kind of results to quadratic operators can be found in [8,30].…”

Section: Introductionsupporting

confidence: 62%

“…We remark that if X is an L 1 -space, then a similar result has been proved in [15]. In [36] the implications (ii)⇔ (iii) have been proved in case of infinite dimensional stochastic matrices [36].…”

Section: Remark 52supporting

confidence: 57%

“…We stress that Theorems 4.1, 4.6 and 4.7 cover all existing results in this direction (see [15,25]). Moreover, the obtained results are new even in case X * is a von Neumann algebra.…”

Section: Remark 48supporting

confidence: 53%

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Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Mukhamedov

2015

Positivity

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In the present paper, we define an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base , and study its properties. The defined coefficient is a generalization of the well-known the Dobrushin's ergodicity coefficient. By means of the ergodicity coefficient we provide uniform asymptotical stability conditions for nonhomogeneous discrete Markov chains (NDMC). These results are even new in case of von Neumann algebras. Moreover, we find necessary and sufficient conditions for the weak ergodicity of NDMC. Certain relations between uniform asymptotical stability and weak ergodicity are considered.

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Section: Introductionsupporting

confidence: 62%

Section: Remark 52supporting

confidence: 57%

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Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Mukhamedov

2015

Positivity

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“…We shall provide sufficient conditions for such processes to satisfy the weak ergodicity. Note that in [13] similar conditions were found for classical ones to satisfy weak ergodicity.…”

Section: Introductionsupporting

confidence: 66%

On Dobrushin Ergodicity Coefficient and weak ergodicity of Markov Chains on Jordan Algebras

Mukhamedov

2013

J. Phys.: Conf. Ser.

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Abstract. In this paper we study certain properties of Dobrushin's ergodicity coefficient for stochastic operators defined on non-associative L 1 -spaces associated with semi-finite JBWalgebras. Such results extends the well-known classical ones to a non-associative setting. This allows us to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. We provide a necessary and sufficient conditions for such processes to satisfy the weak ergodicity. IntroductionIt is known (see [23]) that the investigations of asymptotical behavior of iterations of Markov operators on commutative L 1 -spaces are very important. On the other hand, these investigations are related with several notions of ergodicity of L 1 -contractions of measure spaces. To the investigation of such ergodic properties of Markov operators were devoted lots of papers (see for example, [7,23]). On the other hand, such kind of operators were studied in noncommutative settings. Since, the study of quantum dynamical systems has had an impetuous growth in the last years, in view of natural applications to various field of mathematics and physics. It is then of interest to understand among the various ergodic properties, which ones survive and are meaningful by passing from the classical to the quantum case. Due to noncommutativity, the latter situation is much more complicated than the former. The reader is referred e.g. to [2,15,16,18,29] for further details relative to some differences between the classical and the quantum situations. It is therefore natural to study the possible generalizations to quantum case of the various ergodic properties known for classical dynamical systems. One of the generalizations of noncommutative algebras is Jordan algebras. Note that Jordan Banach algebras [3,6] are a non-associative real analogue of von Neumann algebras. The existence of exceptional JBW -algebras does not allow one to use the ideas and methods from von Neumann algebras. The motivation of these investigations arose in quantum statistical mechanics and quantum field theory (see, [9]). Mostly, in those investigations homogeneous Markov processes were considered. Many ergodic type theorems have been proved for Markov operators acting in nonassociative and noncommutative L p -spaces (see for example, [4, 5, 20, 21, 8, 18]) On the other hand, nonhomogeneous Markov processes with general state space have become a subject of interest due to their applications in many branches of mathematics and natural sciences. In many papers (see for example, [24,17,31]) the weak ergodicity of nonhomogeneous Markov process are given in terms of Dobrushin's ergodicity coefficient [12]. In [33] some

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“…iff ρ d (i, j) > t. QED Proof (of Proposition 4). The proof is based on Dorea & Pereira (2006). From Theorems 2 and 3, it is sufficient to show that there exists a probability µ t , an integer m t ≥ 1, and constants α t < 1/2 and β t > 0 such that for any…”

mentioning

confidence: 99%

Wage Dynamics and Peer Referrals

Boucher

Goussé

2015

SSRN Journal

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Abstract:We present a flexible model of wage dynamics where information about job openings is transmitted through social networks. The model is based on Calvó- Armengol & Jackson (2004, 2007 and extends their results outside the stationary distribution, and under observed and unobserved heterogeneity. We present an empirical application using the British Household Panel Survey by exploiting direct information about individual's social networks. We find that the distribution of job offers is positively affected by the employment status of an individual's friends, and that this relationship is stronger for women.

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A note on a variation of Doeblin's condition for uniform ergodicity of Markov chains

Abstract: We introduce a simple variation of Doeblin's condition, Condition D * , that assures the uniform ergodicity of a Markov chain. It is also shown that for non-homogeneous chains our conditions are equivalent to Dobrushin's weak ergodic coefficient.

Cited by 17 publications

References 4 publications

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

On Dobrushin Ergodicity Coefficient and weak ergodicity of Markov Chains on Jordan Algebras

Wage Dynamics and Peer Referrals

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