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).
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Diaconis, P., Miclo, L. Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks. Sci. China Math. 59, 205–226 (2016). https://doi.org/10.1007/s11425-015-5085-2
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DOI: https://doi.org/10.1007/s11425-015-5085-2
Keywords
- finite absorbing Markov process
- first Dirichlet eigenvector
- path method
- spectral estimates
- denumerable absorbing birth and death process
- entrance boundary