Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Spectral gap and convex concentration inequalities for birth–death processes

Wei Liu and Yutao Ma

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In this paper, we consider a birth–death process with generator $\mathcal{L}$ and reversible invariant probability π. Given an increasing function ρ and the associated Lipschitz norm ‖⋅‖Lip(ρ), we find an explicit formula for $\|(-\mathcal{L})^{-1}\|_{\operatorname {Lip}(\rho)}$. As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of $\mathcal{L}$ in L2(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 $\mathcal{L}$ 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 $\|(-\mathcal{L})^{-1}\|_{\operatorname{Lip}(\rho)}$. En guise d’une application typique, nous retrouvons une formule variationnelle de M. F. Chen pour le trou spectral de $\mathcal{L}$ dans L2(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.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 1 (2009), 58-69.

First available in Project Euclid: 12 February 2009

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60G27

Birth–death process Spectral gap Lipschitz function Poisson equation Convex concentration inequality


Liu, Wei; Ma, Yutao. Spectral gap and convex concentration inequalities for birth–death processes. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 58--69. doi:10.1214/07-AIHP149.

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