# On absorption times and Dirichlet eigenvalues

ESAIM: Probability and Statistics (2010)

- Volume: 14, page 117-150
- ISSN: 1292-8100

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topMiclo, Laurent. "On absorption times and Dirichlet eigenvalues." ESAIM: Probability and Statistics 14 (2010): 117-150. <http://eudml.org/doc/250816>.

@article{Miclo2010,

abstract = {
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.
},

author = {Miclo, Laurent},

journal = {ESAIM: Probability and Statistics},

keywords = {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; irreducible reversible finite Markov chains; Dirichlet eigenvalues; strong random times},

language = {eng},

month = {5},

pages = {117-150},

publisher = {EDP Sciences},

title = {On absorption times and Dirichlet eigenvalues},

url = {http://eudml.org/doc/250816},

volume = {14},

year = {2010},

}

TY - JOUR

AU - Miclo, Laurent

TI - On absorption times and Dirichlet eigenvalues

JO - ESAIM: Probability and Statistics

DA - 2010/5//

PB - EDP Sciences

VL - 14

SP - 117

EP - 150

AB -
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.

LA - eng

KW - 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; irreducible reversible finite Markov chains; Dirichlet eigenvalues; strong random times

UR - http://eudml.org/doc/250816

ER -

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