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On absorption times and Dirichlet eigenvalues

Published online by Cambridge University Press:  10 May 2010

Laurent Miclo
Laurent Miclo*
Affiliation:
Laboratoire d'Analyse, Topologie, Probabilités, UMR 6632, Université de Provence – CNRS, CNRS, France
*
[email protected]
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Abstract

This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chainwhich 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 ofsums of independent geometric laws whose parameters are successiveDirichlet 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,wheregeometric 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 ofnegative eigenvalues which are given another probabilistic meaning.These results generalize results of Karlin and McGregor and Keilson for birth and death chains.

Keywords

Irreducible and reversible subMarkovian matricesexit or absorption timesDirichlet eigenvaluesmixturesgeometric lawsexponential distributionsstrong random timeslocal equilibriaintertwiningbirth and death chains and processes
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 117 - 150
Copyright
© EDP Sciences, SMAI, 2010

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References

Aldous, D. and Diaconis, P., Strong uniform times and finite random walks. Adv. Appl. Math. 8 (1987) 69–97. CrossRef
D. Aldous and J. Fill, Reversible Markov chains and random walks on graphs. Monograph in preparation, available on the web site: http://www.stat.berkeley.edu/∼aldous/RWG/book.html (1994-2002).
Botta, R.F., Harris, C.M. and Marchal, W.G., Characterizations of generalized hyperexponential distribution functions. Commun. Statist. Stoch. Models 3 (1987) 115–148. CrossRef
Commault, C. and Mocanu, S., Phase-type distributions and representations: some results and open problems for system theory. Int. J. Control 76 (2003) 566–580. CrossRef
Diaconis, P. and Fill, J.A., Strong stationary times via a new form of duality. Ann. Probab. 18 (1990) 1483–1522. CrossRef
Diaconis, P. and Miclo, L., On times to quasi-stationarity for birth and death processes. J. Theoret. Probab. 22 (2009) 558–586. CrossRef
Diaconis, P. and Saloff-Coste, L., Separation cut-offs for birth and death chains. Ann. Appl. Probab. 16 (2006) 2098–2122. CrossRef
Ding, J., Lubetzky, E. and Peres, Y., Total variation cutoff in birth-and-death chains. Probab. Theory Relat. Fields 146 (2010) 61–85. CrossRef
Egleston, P.D., Lenker, T.D. and Narayan, S.K., The nonnegative inverse eigenvalue problem. Linear Algebra Appl. 379 (2004) 475–490. CrossRef
Fill, J.A., Strong stationary duality for continuous-time Markov chains, Part I: Theory. J. Theoret. Probab. 5 (1992) 45–70. CrossRef
Fill, J.A., The passage time distribution for a birth-and-death chain: Strong stationary duality gives a first stochastic proof. J. Theoret. Probab. 22 (2009) 543–557. CrossRef
Fill, J.A., On hitting times and fastest strong stationary times for skip-free processes. J. Theoret. Probab. 22 (2009) 587–600. CrossRef
Qi-Ming, He and Hanqin Zhang, Spectral polynomial algorithms for computing bi-diagonal representations for phase type distributions and matrix-exponential distributions. Stoch. Models 22 (2006) 289–317.
Qi-Ming, He and Hanqin Zhang, PH-invariant polytopes and Coxian representations of phase type distributions. Stoch. Models 22 (2006) 383–409.
Karlin, S. and McGregor, J., Coincidence properties of birth and death processes. Pacific J. Math. 9 (1959) 1109–1140. CrossRef
T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin (1995). Reprint of the 1980 edition.
Keilson, J., Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes. J. Appl. Probab. 8 (1971) 391–398. CrossRef
J.T. Kent, Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980) 309–319.
Kent, J.T., The spectral decomposition of a diffusion hitting time. Ann. Probab. 10 (1982) 207–219. CrossRef
J.T. Kent, The appearance of a multivariate exponential distribution in sojourn times for birth-death and diffusion processes. In Probability, statistics and analysis. London Math. Soc. Lect. Note Ser. 79. Cambridge Univ. Press, Cambridge (1983) 161–179.
Micchelli, C.A. and Willoughby, R.A., On functions which preserve the class of Stieltjes matrices. Linear Algebra Appl. 23 (1979) 141–156. CrossRef
M.F. Neuts, Matrix-geometric solutions in stochastic models, Johns Hopkins Series in the Mathematical Sciences: An algorithmic approach, Vol. 2. Johns Hopkins University Press, Baltimore, MD (1981).
O'Cinneide, C.A., Characterization of phase-type distributions. Commun. Statist. Stoch. Models 6 (1990) 1–57. CrossRef
O'Cinneide, C.A., Phase-type distributions and invariant polytopes. Adv. Appl. Probab. 23 (1991) 515–535. CrossRef
O'Cinneide, C.A., Phase-type distributions: open problems and a few properties. Commun. Statist. Stoch. Models 15 (1999) 731–757. CrossRef