Spectral gap and convex concentration inequalities for birth-death processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, p. 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 : https://doi.org/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},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     pages = {58-69},
     doi = {10.1214/07-AIHP149},
     zbl = {1172.60023},
     mrnumber = {2500228},
     language = {en},
     url = {http://www.numdam.org/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, Volume 45 (2009) no. 1, pp. 58-69. doi : 10.1214/07-AIHP149. http://www.numdam.org/item/AIHPB_2009__45_1_58_0/

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