On the Lyapunov Foster criterion and Poincare inequality for Reversible Markov Chains

Amirhossein Taghvaei; Prashant G. Mehta

doi:10.1109/TAC.2021.3089643

On the Lyapunov Foster criterion and Poincare inequality for Reversible Markov Chains

Research output: Contribution to journal › Article › peer-review

Abstract

This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincare inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.

Original language	English (US)
Pages (from-to)	2605-2609
Number of pages	5
Journal	IEEE Transactions on Automatic Control
Volume	67
Issue number	5
DOIs	https://doi.org/10.1109/TAC.2021.3089643
State	Accepted/In press - 2021

Keywords

Convergence
Eigenvalues and eigenfunctions
Kernel
Lyapunov methods
Markov processes
Mechanical variables measurement
Stability criteria

ASJC Scopus subject areas

Control and Systems Engineering
Computer Science Applications
Electrical and Electronic Engineering

Online availability

10.1109/TAC.2021.3089643

Cite this

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abstract = "This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincare inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.",

keywords = "Convergence, Eigenvalues and eigenfunctions, Kernel, Lyapunov methods, Markov processes, Mechanical variables measurement, Stability criteria",

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AU - Taghvaei, Amirhossein

AU - Mehta, Prashant G.

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PY - 2021

Y1 - 2021

N2 - This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincare inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.

AB - This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincare inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.

KW - Convergence

KW - Eigenvalues and eigenfunctions

KW - Kernel

KW - Lyapunov methods

KW - Markov processes

KW - Mechanical variables measurement

KW - Stability criteria

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JO - IEEE Transactions on Automatic Control

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On the Lyapunov Foster criterion and Poincare inequality for Reversible Markov Chains

Abstract

Keywords

ASJC Scopus subject areas

Online availability

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