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Science China Mathematics

, Volume 59, Issue 2, pp 205–226 | Cite as

Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

  • Persi Diaconis
  • Laurent MicloEmail author
Articles Invited Articles

Abstract

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. Perron-Frobenius theory ensures the existence of a corresponding positive eigenvector ϕ. The goal of the paper is to give bounds on the amplitude max ϕ/min ϕ. Two approaches are proposed: One using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen by Diaconis and Miclo (2014).

Keywords

finite absorbing Markov process first Dirichlet eigenvector path method spectral estimates denumerable absorbing birth and death process entrance boundary 

MSC(2010)

60J27 15B48 15A42 05C50 60J80 47B36 

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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityCaliforniaUSA
  2. 2.Institut de Mathéatiques de ToulouseUniversité de Toulouse and CNRSUMRFrance

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