Spectral gap and convex concentration inequalities for birth-death processes

Liu, Wei; Ma, Yutao

Liu, Wei ; Ma, Yutao

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009), p. 58-69 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Dans ce travail, nous considérons un processus de naissance et de mort de générateur $ℒ$ et de probabilité invariante réversible $π$ . Étant données une fonction strictement croissante $ρ$ , et la norme lipschitzienne ${∥\cdot∥}_{Lip (ρ)}$ par rapport à $ρ$ , nous trouvons une représentation explicite de ${∥ (- ℒ)}^{- 1} ∥_{Lip (ρ)}$ . En guise d’une application typique, nous retrouvons une formule variationnelle de M. F. Chen pour le trou spectral de $ℒ$ dans $L^{2} (π)$ . De plus, par la décomposition des martingales progressive-rétrogrades de Lyons-Zheng, nous obtenons des inégalités de concentration convexe pour des fonctionnelles additives de processus de naissance et de mort.

In this paper, we consider a birth-death process with generator $ℒ$ and reversible invariant probability $π$ . Given an increasing function $ρ$ and the associated Lipschitz norm ${∥\cdot∥}_{Lip (ρ)}$ , we find an explicit formula for ${∥ (- ℒ)}^{- 1} ∥_{Lip (ρ)}$ . As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of $ℒ$ in $L^{2} (π)$ . Moreover, by Lyons-Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth-death processes.

Publié le : 2009-01-01

MR 2500228 | Zbl 1172.60023

DOI : https://doi.org/10.1214/07-AIHP149
Classification: 60E15, 60G27

@article{AIHPB_2009__45_1_58_0,
     author = {Liu, Wei and Ma, Yutao},
     title = {Spectral gap and convex concentration inequalities for birth-death processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {45},
     year = {2009},
     pages = {58-69},
     doi = {10.1214/07-AIHP149},
     mrnumber = {2500228},
     zbl = {1172.60023},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2009__45_1_58_0}
}

Liu, Wei; Ma, Yutao. Spectral gap and convex concentration inequalities for birth-death processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) pp. 58-69. doi : 10.1214/07-AIHP149. http://gdmltest.u-ga.fr/item/AIHPB_2009__45_1_58_0/

Bibliographie

[1] S. G. Bobkov and F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1-28. | MR 1682772 | Zbl 0924.46027

[2] M. F. Chen. Estimation of spectral gap for Markov chains. Acta Math. Sin. New Ser. 12 (1996) 337-360. | MR 1457859 | Zbl 0867.60038

[3] M. F. Chen. Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci. Sin. (A) 42 (1999) 805-815. | MR 1738551 | Zbl 0936.35120

[4] M. F. Chen. Explicit bounds of the first eigenvalue. Sci. China (A) 43 (2000) 1051-1059. | MR 1802148 | Zbl 1054.60082

[5] M. F. Chen. Variational formulas and approximation theorems for the first eigenvalue. Sci. China (A) 44 (2001) 409-418. | MR 1831443 | Zbl 1012.34078

[6] M. F. Chen. From Markov Chains to Non-equilibrium Particle Systems, 2nd edition. Springer, 2004. | MR 2091955 | Zbl 1078.60003

[7] M. F. Chen. Eigenvalues, Inequalities and Ergodic Theory. Springer, 2005. | MR 2105651 | Zbl 1079.60005

[8] M. F. Chen and F. Y. Wang. Application of coupling method to the first eigenvalue on manifold. Sci. Sin. (A) 23 (1993) 1130-1140 (Chinese Edition); 37 (1994) 1-14 (English Edition). | MR 1308707 | Zbl 0799.53044

[9] M. F. Chen and F. Y. Wang. Estimation of spectral gap for elliptic operators. Trans. Amer. Math. Soc. 349 (1997) 1239-1267. | MR 1401516 | Zbl 0872.35072

[10] H. Djellout and L. M. Wu. Spectral gap of one dimensional diffusions in Lipschitz norm and application to log-Sobolev inequalities for Gibbs measures. Preprint, 2007.

[11] A. Guillin, C. Léonard, L. M. Wu and N. Yao. Transportation-information inequalities for Markov processes. Preprint, 2007. | Zbl 1169.60304

[12] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Stat. Assoc. 58 (1963) 13-30. | MR 144363 | Zbl 0127.10602

[13] A. Joulin. A new Poisson-type deviation inequality for Markov jump process with positive Wasserstein curvature. Preprint, 2007. | MR 2348750

[14] T. Klein, Y. T. Ma and N. Privault. Convex concentration inequalities and forward/backward stochastic calculus. Electron. J. Probab. 11 (2006) 486-512. | MR 2242653 | Zbl 1112.60034

[15] T. J. Lyons and W. A. Zheng. A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Astérique 157-158 (1988) 249-271. | Zbl 0654.60059

[16] Y. T. Ma. Grandes déviations et concentration convexe en temps continu et discret. PhD thesis, Université de La Rochelle (France) et Université de Wuhan (Chine), 2006. Available at http://perso.univ-lr.fr/yma/thesis.pdf.

[17] L. Miclo. An exemple of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | MR 1710983 | Zbl 0942.60081

[18] Z. K. Wang and X. Q. Yang. Birth-Death Processes and Markov Chains. Academic Press of China, Beijing, 2005 (in Chinese). | Zbl 0773.60065

[19] L. M. Wu. Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23 (1995) 420-445. | MR 1330777 | Zbl 0828.60017

[20] L. M. Wu. Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 35 (1999) 121-141. | Numdam | MR 1678517 | Zbl 0936.60037

[21] L. M. Wu. Essential spectral radius for Markov semigroups (I): discrete time case. Probab. Theory Related Fields 128 (2004) 255-321. | MR 2031227 | Zbl 1056.60068

[22] K. Yosida. Functional Analysis, 6th edition. Spring, 1999.