On Two Recent Papers on Ergodicity in Nonhomogeneous Markov Chains

Iosifescu, Marius

doi:10.1214/aoms/1177692411

Cited by 31 publications

(34 citation statements)

References 5 publications

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“…For nonhomogeneous Markov processes with countable state space, investigation of the general conditions of weak ergodicity leads to the definition of a special subclass of regular matrices. In many papers (see for example, [6,11,15,18]) the weak ergodicity of nonhomogeneous Markov process are given in terms of Dobrushin's ergodicity coefficient [1]. In general case, one may consider several kinds of convergence [10].…”

Section: Introductionmentioning

confidence: 99%

On L 1-weak ergodicity of nonhomogeneous discrete Markov processes and its applications

Mukhamedov

2012

Rev Mat Complut

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Abstract. In the present paper we investigate the L 1 -weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the L 1 -weak ergodicity is weaker than well-known weak ergodicity. We provide a necessary and sufficient condition for such processes to satisfy the L 1 -weak ergodicity. Moreover, we apply the obtained results to establish L 1 -weak ergodicity of discrete time quadratic stochastic processes. As an application of the main result, certain concrete examples are also provided.

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Section: Introductionmentioning

confidence: 99%

On L 1-weak ergodicity of nonhomogeneous discrete Markov processes and its applications

Mukhamedov

2012

Rev Mat Complut

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“…Corollary 3.9 (sections 3.1 and 4.3 in [85], pages 1733-1734 in [41]). If S ∈ R n×n is a stochastic matrix, then τ 1 (S) = 1 − min ij n k=1 min{s ik , s jk }.…”

Section: Explicit Expressionsmentioning

confidence: 99%

Ergodicity Coefficients Defined by Vector Norms

Ipsen¹,

Selee²

2011

SIAM J. Matrix Anal. & Appl.

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Ergodicity coefficients for stochastic matrices determine inclusion regions for subdominant eigenvalues; estimate the sensitivity of the stationary distribution to changes in the matrix; and bound the convergence rate of methods for computing the stationary distribution. We survey results for ergodicity coefficients that are defined by p-norms, for stochastic matrices as well as for general real or complex matrices. We express ergodicity coefficients in the one-, two-, and infinitynorms as norms of projected matrices, and we bound coefficients in any p-norm by norms of deflated matrices. We show that two-norm ergodicity coefficients of a matrix A are closely related to the singular values of A. In particular, the singular values determine the extreme values of the coefficients. We show that ergodicity coefficients can determine inclusion regions for subdominant eigenvalues of complex matrices, and that the tightness of these regions depends on the departure of the matrix from normality. In the special case of normal matrices, two-norm ergodicity coefficients turn out to be Lehmann bounds.

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“…in the case if Σ is a compact set and the map P (u) = P u : Σ → L is continuous. Then (we refer to [5,6] for the properties of Dobrushin's coefficient δ(P )): It follows from the definition and (3.1) that δ(P w ) ≤ M(1 − δ) |w|/r , for any w ∈ Σ * and some M > 0. Maass and Sontag used a strong Doeblin's condition to prove the computational power of noisy neural networks [8].…”

Section: The Reduction Lemma and Quasi-compact Mcsmentioning

confidence: 99%

On probabilistic analog automata

Ben-Hur

Roitershtein

Siegelmann

2004

Theoretical Computer Science

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We consider probabilistic automata on a general state space and study their computational power. The model is based on the concept of language recognition by probabilistic automata due to Rabin [12] and models of analog computation in a noisy environment suggested by Maass and Orponen [7], and Maass and Sontag [8]. Our main result is a generalization of Rabin's reduction theorem that implies that under very mild conditions, the computational power of the automaton is limited to regular languages.

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On Two Recent Papers on Ergodicity in Nonhomogeneous Markov Chains

Cited by 31 publications

References 5 publications

On L 1-weak ergodicity of nonhomogeneous discrete Markov processes and its applications

On L 1-weak ergodicity of nonhomogeneous discrete Markov processes and its applications

Ergodicity Coefficients Defined by Vector Norms

On probabilistic analog automata

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