Spectral gap and convex concentration inequalities for birth-death processes

Liu, Wei; Ma, Yutao

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 58-69.

Dans ce travail, nous considérons un processus de naissance et de mort de générateur $ℒ$ et de probabilité invariante réversible $\pi$. Étant données une fonction strictement croissante $\rho$, et la norme lipschitzienne ${∥·∥}_{\text{Lip}\left(\rho \right)}$ par rapport à $\rho$, nous trouvons une représentation explicite de ${\parallel (-ℒ)}^{-1}{\parallel }_{Lip\left(\rho \right)}$. En guise d’une application typique, nous retrouvons une formule variationnelle de M. F. Chen pour le trou spectral de $ℒ$ dans $L^2\left(\pi \right)$. 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 $\pi$. Given an increasing function $\rho$ and the associated Lipschitz norm ${∥·∥}_{\text{Lip}\left(\rho \right)}$, we find an explicit formula for ${\parallel (-ℒ)}^{-1}{\parallel }_{Lip\left(\rho \right)}$. As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of $ℒ$ in $L^2\left(\pi \right)$. Moreover, by Lyons-Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth-death processes.

Classification : 60E15, 60G27
Mots clés : Birth-death process, spectral gap, Lipschitz function, Poisson equation, convex concentration inequality

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