Volume 14, 2010
|Page(s)||117 - 150|
|Published online||10 May 2010|
On absorption times and Dirichlet eigenvalues
Laboratoire d'Analyse, Topologie, Probabilités, UMR 6632, Université de Provence – CNRS, CNRS, France
Corresponding author: email@example.com
Revised: 9 July 2008
This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chain which is irreducible and reversible outside the absorbing point. This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues. In the discrete time setting, if the underlying Dirichlet eigenvalues (namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point) are nonnegative, we show that T is distributed as a mixture of sums of independent geometric laws whose parameters are successive Dirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law. This result leads to a probabilistic interpretation of the spectrum, in terms of strong random times and local equilibria through a simple intertwining relation. Next this study is extended to the continuous time framework, where geometric laws have to be replaced by exponential distributions having the (opposite) Dirichlet eigenvalues of the generator as parameters. Returning to the discrete time setting we consider the influence of negative eigenvalues which are given another probabilistic meaning. These results generalize results of Karlin and McGregor and Keilson for birth and death chains.
Mathematics Subject Classification: 60J10 / 60J27 / 37A30 / 31C25 / 60J80
Key words: Irreducible and reversible subMarkovian matrices / exit or absorption times / Dirichlet eigenvalues / mixtures / geometric laws / exponential distributions / strong random times / local equilibria / intertwining / birth and death chains and processes
© EDP Sciences, SMAI, 2010
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.