Spectral gap and convex concentration inequalities for birth–death processes

Wei Liu; Yutao Ma

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 1, page 58-69
  • ISSN: 0246-0203

Abstract

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In this paper, we consider a birth–death process with generator ℒ and reversible invariant probabilityπ. Given an increasing function ρ and the associated Lipschitz norm ‖⋅‖Lip(ρ), we find an explicit formula for ∥ ( - ℒ ) - 1 ∥ Lip ( ρ ) . As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of ℒ inL2(π). Moreover, by Lyons–Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth–death processes.

How to cite

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Liu, Wei, and Ma, Yutao. "Spectral gap and convex concentration inequalities for birth–death processes." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 58-69. <http://eudml.org/doc/78021>.

@article{Liu2009,
abstract = {In this paper, we consider a birth–death process with generator $\mathcal \{L\}$ and reversible invariant probabilityπ. Given an increasing function ρ and the associated Lipschitz norm ‖⋅‖Lip(ρ), we find an explicit formula for $\Vert (-\mathcal \{L\})^\{-1\}\Vert _\{\operatorname\{Lip\}(\rho )\}$. As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of $\mathcal \{L\}$ inL2(π). Moreover, by Lyons–Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth–death processes.},
author = {Liu, Wei, Ma, Yutao},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Birth–death process; spectral gap; Lipschitz function; Poisson equation; convex concentration inequality; birth-death process},
language = {eng},
number = {1},
pages = {58-69},
publisher = {Gauthier-Villars},
title = {Spectral gap and convex concentration inequalities for birth–death processes},
url = {http://eudml.org/doc/78021},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 1
SP - 58
EP - 69
AB - In this paper, we consider a birth–death process with generator $\mathcal {L}$ and reversible invariant probabilityπ. Given an increasing function ρ and the associated Lipschitz norm ‖⋅‖Lip(ρ), we find an explicit formula for $\Vert (-\mathcal {L})^{-1}\Vert _{\operatorname{Lip}(\rho )}$. As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of $\mathcal {L}$ inL2(π). Moreover, by Lyons–Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth–death processes.
LA - eng
KW - Birth–death process; spectral gap; Lipschitz function; Poisson equation; convex concentration inequality; birth-death process
UR - http://eudml.org/doc/78021
ER -

References

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