Spectral gap and convex concentration inequalities for birth-death processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 58-69.

In this paper, we consider a birth-death process with generator and reversible invariant probability π. Given an increasing function ρ and the associated Lipschitz norm · Lip(ρ) , we find an explicit formula for (-) -1 Lip(ρ) . As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of 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.

Dans ce travail, nous considérons un processus de naissance et de mort de générateur et de probabilité invariante réversible π. Étant données une fonction strictement croissante ρ, et la norme lipschitzienne · Lip(ρ) par rapport à ρ, nous trouvons une représentation explicite de (-) -1 Lip(ρ) . En guise d’une application typique, nous retrouvons une formule variationnelle de M. F. Chen pour le trou spectral de 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.

DOI: 10.1214/07-AIHP149
Classification: 60E15,  60G27
Keywords: Birth-death process, spectral gap, Lipschitz function, Poisson equation, convex concentration inequality
@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},
     pages = {58--69},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     doi = {10.1214/07-AIHP149},
     zbl = {1172.60023},
     mrnumber = {2500228},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/07-AIHP149/}
}
TY  - JOUR
AU  - Liu, Wei
AU  - Ma, Yutao
TI  - Spectral gap and convex concentration inequalities for birth-death processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
DA  - 2009///
SP  - 58
EP  - 69
VL  - 45
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/07-AIHP149/
UR  - https://zbmath.org/?q=an%3A1172.60023
UR  - https://www.ams.org/mathscinet-getitem?mr=2500228
UR  - https://doi.org/10.1214/07-AIHP149
DO  - 10.1214/07-AIHP149
LA  - en
ID  - AIHPB_2009__45_1_58_0
ER  - 
%0 Journal Article
%A Liu, Wei
%A Ma, Yutao
%T Spectral gap and convex concentration inequalities for birth-death processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 58-69
%V 45
%N 1
%I Gauthier-Villars
%U https://doi.org/10.1214/07-AIHP149
%R 10.1214/07-AIHP149
%G en
%F 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, Volume 45 (2009) no. 1, pp. 58-69. doi : 10.1214/07-AIHP149. http://www.numdam.org/articles/10.1214/07-AIHP149/

[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 | Zbl

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

[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

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

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

[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 | Zbl

[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

[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 | Zbl

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

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

[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 | Zbl

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

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

Cited by Sources: