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Persi Diaconis; Laurent Miclo
On quantitative convergence to quasi-stationarity
Annales de la faculté des sciences de Toulouse Sér. 6, 24 no. 4: Numéro Spécial : Conférence “Talking Across Fields” du 24 au 28 mars 2014 à l’Institut de Mathématiques de Toulouse (2015), p. 973-1016, doi: 10.5802/afst.1472
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Résumé - Abstract

The quantitative long time behavior of absorbing, finite, irreducible Markov processes is considered. Via Doob transforms, it is shown that only the knowledge of the ratio of the values of the underlying first Dirichlet eigenvector is necessary to come back to the well-investigated situation of the convergence to equilibrium of ergodic finite Markov processes. This leads to explicit estimates on the convergence to quasi-stationarity, in particular via functional inequalities. When the process is reversible, the optimal exponential rate consisting of the spectral gap between the two first Dirichlet eigenvalues is recovered. Several simple examples are provided to illustrate the bounds obtained.


[1] Ané (C.), Blachère (S.), Chafaï (D.), Fougères (P.), Gentil (I.), Malrieu (F.), Roberto (C.), and Scheffer (G.).— Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris (2000). With a preface by Dominique Bakry and Michel Ledoux.  Zbl 0982.46026
[2] Barbour (A. D.) and Pollett (P. K.).— Total variation approximation for quasi-stationary distributions. J. Appl. Probab., 47(4), p. 934-946 (2010).  MR 2752899 |  Zbl 1213.60126
[3] Barbour (A. D.) and Pollett (P. K.).— Total variation approximation for quasi-equilibrium distributions, II. Stochastic Process. Appl., 122(11), p. 3740-3756 (2012).  MR 2965923 |  Zbl 1251.60060
[4] Bobkov (S. G.) and Tetali (P.).— Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab., 19(2), p. 289-336 (2006).  MR 2283379 |  Zbl 1113.60072
[5] Champagnat (N.) and Villemonais (D.).— Exponential convergence to quasi-stationary distribution and Q-process. ArXiv e-prints, April 2014.
[6] Cloez (B.) and Thai (M. N.).— Quantitative results for the Fleming-Viot particle system in discrete space. ArXiv e-prints, December 2013.
[7] Collet (P.), Martínez (S.), and San Martín (J.). Quasi-stationary distributions.— Markov chains, diffusions and dynamical systems. Probability and its Applications (New York). Springer, Heidelberg (2013).  MR 2986807 |  Zbl 1261.60002
[8] Defosseux (M.).— Fusion coefficients and random walks in alcoves. ArXiv e-prints, July 2013.
[9] Del Moral (P.).— Mean field simulation for Monte Carlo integration, volume 126 of Monographs on Statistics and Applied Probability. CRC Press, Boca Raton, FL (2013).  MR 3060209 |  Zbl 1282.65011
[10] Del Moral (P.) and Miclo (L.).— Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM Probab. Stat., 7, p. 171-208 (2003). Numdam |  MR 1956078 |  Zbl 1040.81009
[11] Diaconis (P.), Amy Pang (C. Y.), and Ram (A.).— Hopf algebras and Markov chains: two examples and a theory. J. Algebraic Combin., 39(3), p. 527-585 (2014).  MR 3183482 |  Zbl 1291.05220
[12] Diaconis (P.) and Saloff-Coste (L.).— Comparison theorems for reversible Markov chains. Ann. Appl. Probab., 3(3), p. 696-730 (1993).  MR 1233621 |  Zbl 0799.60058
[13] Diaconis (P.) and Saloff-Coste (L.).— Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab., 6(3), p. 695-750 (1996).  MR 1410112 |  Zbl 0867.60043
[14] Diaconis (P.) and Saloff-Coste (L.).— Nash inequalities for finite Markov chains. J. Theoret. Probab., 9(2), p. 459-510 (1996).  MR 1385408 |  Zbl 0870.60064
[15] Diaconis (P.) and Saloff-Coste (L.).— What do we know about the Metropolis algorithm? J. Comput. System Sci., 57(1), p. 20-36 (1998). 27th Annual ACM Symposium on the Theory of Computing (STOC’95) (Las Vegas, NV).  MR 1649805 |  Zbl 0920.68054
[16] Diaconis (P.) and Stroock (D.).— Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab., 1(1), p. 36-61 (1991).  MR 1097463 |  Zbl 0731.60061
[17] Fill (J. A.).— Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab., 1(1), p. 62-87 (1991).  MR 1097464 |  Zbl 0726.60069
[18] Gyrya (P.) and Saloff-Coste (L.).— Neumann and Dirichlet heat kernels in inner uniform domains. Astérisque, (336), p. viii+144 (2011).  MR 2807275 |  Zbl 1222.58001
[19] Holley (R.) and Stroock (D.).— Simulated annealing via Sobolev inequalities. Comm. Math. Phys., 115(4), p. 553-569 (1988).  MR 933455 |  Zbl 0643.60092
[20] Jerrum (M.) and Sinclair (A.).— Approximating the permanent. SIAM J. Comput., 18(6), p. 1149-1178 (1989).  MR 1025467 |  Zbl 0723.05107
[21] Jacka (S. D.) and Roberts (G. O.).— Weak convergence of conditioned processes on a countable state space. J. Appl. Probab., 32(4), p. 902-916 (1995.)  MR 1363332 |  Zbl 0839.60069
[22] Jiang (Y.).— Mixing Time of Metropolis Chain Based on Random Transposition Walk Converging to Multivariate Ewens Distribution. ArXiv e-prints, April 2012.  MR 3325282
[23] Lierl (J.) and Saloff-Coste (L.).— The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms. ArXiv e-prints, October 2012.  MR 3170207 |  Zbl 1295.35240
[24] Méléard (S.) and Villemonais (D.).— Quasi-stationary distributions and population processes. Probab. Surv., 9, p. 340-410 (2012).  MR 2994898 |  Zbl 1261.92056
[25] Miclo (L.).— Remarques sur l’hypercontractivité et l’évolution de l’entropie pour des chaînes de Markov finies. In Séminaire de Probabilités, XXXI, volume 1655 of Lecture Notes in Math., pages 136-167. Springer, Berlin (1997). Numdam |  MR 1478724 |  Zbl 0882.60065
[26] Miclo (L.).— On eigenfunctions of Markov processes on trees. Probab. Theory Related Fields, 142(3-4), p. 561-594 (2008).  MR 2438701 |  Zbl 1149.60059
[27] Miclo (L.).— On hyperboundedness and spectrum of Markov operators. Available at, January 2013.
[28] Saloff-Coste (L.).— Lectures on finite Markov chains. In Lectures on probability theory and statistics (Saint-Flour, 1996), volume 1665 of Lecture Notes in Math., p. 301-413. Springer, Berlin (1997).  MR 1490046 |  Zbl 0885.60061
[29] van Doorn (E. A.).— Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. in Appl. Probab., 23(4), p. 683-700 (1991).  MR 1133722 |  Zbl 0736.60076
[30] van Doorn (E. A.) and Pollett (P. K.).— Quasi-stationary distributions for discrete-state models. European J. Oper. Res., 230(1), p. 1-14 (2013).  MR 3063313
[31] van Doorn (E. A.) and Zeifman (A. I.).— On the speed of convergence to stationarity of the Erlang loss system. Queueing Syst., 63(1-4), p. 241-252 (2009).  MR 2576013 |  Zbl 1209.90122
[32] Zhou (H.).— Examples of multivariate Markov chains with orthogonal polynomial eigenfunctions. Ph.D. thesis, dept. of statistics, Stanford University (2008).  MR 2712374
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