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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Abstract

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.

Résumé

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

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

Dates
First available in Project Euclid: 12 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1234469971

Digital Object Identifier
doi:10.1214/07-AIHP149

Mathematical Reviews number (MathSciNet)
MR2500228

Zentralblatt MATH identifier
1172.60023

Subjects
Primary: 60E15: Inequalities; stochastic orderings 60G27

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

Citation

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. https://projecteuclid.org/euclid.aihp/1234469971


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