Alternative characterization of ergodicity for doubly stochastic chains

Behrouz Touri; Angelia Nedic

doi:10.1109/CDC.2011.6161372

Alternative characterization of ergodicity for doubly stochastic chains

Behrouz Touri, Angelia Nedic

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

10 Scopus citations

Abstract

In this paper we discuss the ergodicity of stochastic and doubly stochastic chains. We define absolute infinite flow property and show that this property is necessary for ergodicity of any stochastic chain. The proof is constructive and makes use of a rotational transformation, which we introduce and study. We then focus on doubly stochastic chains for which we prove that the absolute infinite flow property and ergodicity are equivalent. The proof of this result makes use of a special decomposition of a doubly stochastic matrix, as given by Birkhoff-von Neumann theorem. Finally, we show that a backward product of doubly stochastic matrices is convergent up to a permutation sequence and, as a result, the set of accumulation points of such a product is finite.

Original language	English (US)
Title of host publication	2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	5371-5376
Number of pages	6
ISBN (Print)	9781612848006
DOIs	https://doi.org/10.1109/CDC.2011.6161372
State	Published - 2011
Externally published	Yes
Event	2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 - Orlando, FL, United States Duration: Dec 12 2011 → Dec 15 2011

Publication series

Name	Proceedings of the IEEE Conference on Decision and Control
ISSN (Print)	0743-1546
ISSN (Electronic)	2576-2370

Other

Other	2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011
Country/Territory	United States
City	Orlando, FL
Period	12/12/11 → 12/15/11

ASJC Scopus subject areas

Control and Systems Engineering
Modeling and Simulation
Control and Optimization

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10.1109/CDC.2011.6161372

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Touri, B., & Nedic, A. (2011). Alternative characterization of ergodicity for doubly stochastic chains. In 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 (pp. 5371-5376). Article 6161372 (Proceedings of the IEEE Conference on Decision and Control). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2011.6161372

Alternative characterization of ergodicity for doubly stochastic chains. / Touri, Behrouz; Nedic, Angelia.
2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011. Institute of Electrical and Electronics Engineers Inc., 2011. p. 5371-5376 6161372 (Proceedings of the IEEE Conference on Decision and Control).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Touri, B & Nedic, A 2011, Alternative characterization of ergodicity for doubly stochastic chains. in 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011., 6161372, Proceedings of the IEEE Conference on Decision and Control, Institute of Electrical and Electronics Engineers Inc., pp. 5371-5376, 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011, Orlando, FL, United States, 12/12/11. https://doi.org/10.1109/CDC.2011.6161372

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AB - In this paper we discuss the ergodicity of stochastic and doubly stochastic chains. We define absolute infinite flow property and show that this property is necessary for ergodicity of any stochastic chain. The proof is constructive and makes use of a rotational transformation, which we introduce and study. We then focus on doubly stochastic chains for which we prove that the absolute infinite flow property and ergodicity are equivalent. The proof of this result makes use of a special decomposition of a doubly stochastic matrix, as given by Birkhoff-von Neumann theorem. Finally, we show that a backward product of doubly stochastic matrices is convergent up to a permutation sequence and, as a result, the set of accumulation points of such a product is finite.

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Alternative characterization of ergodicity for doubly stochastic chains

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