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.
Mots clés : 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
@article{PS_2010__14__117_0, author = {Miclo, Laurent}, title = {On absorption times and {Dirichlet} eigenvalues}, journal = {ESAIM: Probability and Statistics}, pages = {117--150}, publisher = {EDP-Sciences}, volume = {14}, year = {2010}, doi = {10.1051/ps:2008037}, mrnumber = {2654550}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2008037/} }
Miclo, Laurent. On absorption times and Dirichlet eigenvalues. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 117-150. doi : 10.1051/ps:2008037. http://www.numdam.org/articles/10.1051/ps:2008037/
[1] Strong uniform times and finite random walks. Adv. Appl. Math. 8 (1987) 69-97. | Zbl
and ,[2] 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).
and ,[3] Characterizations of generalized hyperexponential distribution functions. Commun. Statist. Stoch. Models 3 (1987) 115-148. | Zbl
, and ,[4] Phase-type distributions and representations: some results and open problems for system theory. Int. J. Control 76 (2003) 566-580. | Zbl
and ,[5] Strong stationary times via a new form of duality. Ann. Probab. 18 (1990) 1483-1522. | Zbl
and ,[6] On times to quasi-stationarity for birth and death processes. J. Theoret. Probab. 22 (2009) 558-586. | Zbl
and ,[7] Separation cut-offs for birth and death chains. Ann. Appl. Probab. 16 (2006) 2098-2122. | Zbl
and ,[8] Total variation cutoff in birth-and-death chains. Probab. Theory Relat. Fields 146 (2010) 61-85. | Zbl
, and ,[9] The nonnegative inverse eigenvalue problem. Linear Algebra Appl. 379 (2004) 475-490. | Zbl
, and ,[10] Strong stationary duality for continuous-time Markov chains, Part I: Theory. J. Theoret. Probab. 5 (1992) 45-70. | Zbl
,[11] 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. | Zbl
,[12] On hitting times and fastest strong stationary times for skip-free processes. J. Theoret. Probab. 22 (2009) 587-600. | Zbl
,[13] Spectral polynomial algorithms for computing bi-diagonal representations for phase type distributions and matrix-exponential distributions. Stoch. Models 22 (2006) 289-317. | Zbl
and ,[14] PH-invariant polytopes and Coxian representations of phase type distributions. Stoch. Models 22 (2006) 383-409. | Zbl
and ,[15] Coincidence properties of birth and death processes. Pacific J. Math. 9 (1959) 1109-1140. | Zbl
and ,[16] Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin (1995). Reprint of the 1980 edition. | Zbl
,[17] Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes. J. Appl. Probab. 8 (1971) 391-398. | Zbl
,[18] Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980) 309-319. | Zbl
,[19] The spectral decomposition of a diffusion hitting time. Ann. Probab. 10 (1982) 207-219. | Zbl
,[20] 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. | Zbl
,[21] On functions which preserve the class of Stieltjes matrices. Linear Algebra Appl. 23 (1979) 141-156. | Zbl
and ,[22] 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). | Zbl
,[23] Characterization of phase-type distributions. Commun. Statist. Stoch. Models 6 (1990) 1-57. | Zbl
,[24] Phase-type distributions and invariant polytopes. Adv. Appl. Probab. 23 (1991) 515-535. | Zbl
,[25] Phase-type distributions: open problems and a few properties. Commun. Statist. Stoch. Models 15 (1999) 731-757. | Zbl
,Cité par Sources :